# Source extract: FA-LRT-01.docx Local Relational Time, Version 56, Truth-First Layer 2 Synchronized Successor-12 This Layer 2 successor retains the complete metric-time theorem lineage while making explicit that physical events, carriers, interactions, records, instruments, and calibrations are admitted inputs rather than consequences of an ultimate-provenance theorem. ETP-CPR Batch 2 Constitutional Synchronization Truth-first starting point The Basement Necessity Program rigorously investigated whether physical actuality could be derived from or uniquely grounded in a more primitive basement. Within the declared languages, model classes, candidate families, evidence sources, and audit scope, no candidate basement was established as necessary, sufficient, unique, or least, and no physical-instantiation bridge achieved derived-with-scope status. This scoped negative result does not prove that no basement exists, that actuality is necessarily primitive, or that ultimate provenance is permanently unknowable. Physical actuality is admitted, not derived, as the minimal factual starting condition of the active theory. Ultimate provenance remains unresolved and reopenable. This manuscript asks what follows from its explicitly declared physical starting package under its stated hypotheses, operations, equivalences, access conditions, calibrations, resources, witnesses, and scope. Governing question Given an explicitly declared and physically admitted starting package, what additional structures follow, under what assumptions, with what necessity, sufficiency, uniqueness or nonuniqueness, robustness, scope, and empirical consequences? P.E.N. and nonretroactivity control The Paired-Expansion Nondefinability Theorem remains scoped to its declared language, formal reduct, admissible expansions, and model class. Admission is not derivation. No result concerning records, observers, objectivity, recoverability, order, clocks, duration, locality, topology, geometry, or related effective structure may retroactively establish the actuality, ontology, or ultimate provenance of the admitted inputs. Version 56 Record-orbit no-selection and recoverability-closure reconstruction Scott Muniz Dr. Ware Technology Services, Charleston, South Carolina, USA The principal theorem architecture and mathematical content identified by the completed preservation audit are retained. Complete retention of every historical formula, theorem, diagnostic, and implementation detail remains subject to the recorded genealogy and binary-package verification gates. Duration is not dynamics. Dynamics is not duration. Clock adequacy is a correspondence theorem. Editorial Preservation Rules Arrow conditions arise from synchronization plus memory, recoverability, autonomous ordering and entropy-production conditions. Conditional Temporal Direction Clock Construction, Clock Adequacy, Fisher-Clock Correspondence, Comparison Transducers, Synchronization and Holonomy. Bridge Structures Interactions -> Channels -> Open Dynamics -> Memory Structure -> Non-Markovian Evolution -> Autonomous Physical Architectures. Physical Evolution Recoverable Records -> Record Order -> Coordinate Charts -> Fisher Geometry -> Local Duration -> Endpoint Completion. Operational Reconstruction Pillar I is reorganized into Operational Reconstruction, Physical Evolution, Bridge Structures, and Conditional Temporal Direction. Master Architectural Principle Purpose: Theorem-preserving narrative reconstruction of Pillar I based on the N1-N7 audit program. No theorem, proof, benchmark, ledger entry, or claim-status classification is altered. BNP GOVERNING CONSTITUTION FOR LOCAL RELATIONAL TIME Constitutional role. This section fixes the physical interpretation, identity, nullity, carrier, operation, and claim-boundary rules governing every theorem and model in this manuscript. It does not add a new physical ontology or modify any mathematical result. Where a later theorem uses a narrower task datum, that datum is a declared projection of this constitution and not a replacement basement. BNP-C1. Physical bridge and one governing anchor The manuscript is physically interpreted only relative to the declared bridge J_phys connecting its formal objects to admitted preparations, observables, interactions, records, instruments, boundaries, calibrations, and empirical or constitutive interpretation. The bridge is not derived by the mathematical formalism and is not silently promoted to a theorem conclusion. The governing anchor is D_anchor=(P_model,J_phys,sim_phys,C_complete,S_scope), where P_model is the declared physical model class, sim_phys is the single complete-model physical-equivalence relation, C_complete is the complete physical carrier, and S_scope is the theorem scope. This anchor is fixed before theorem evaluation. A later change of model class, bridge, carrier, physical equivalence, boundary, sector, correlation, control, or branch structure is labeled a new model or an error correction. A change of selected output, detector, recovery task, clock domain, metric, tolerance, or witness family is labeled a new task or witness extension. BNP-C2. One basement-level physical identity Two complete LRT models M1 and M2 satisfy M1 sim_phys M2 only when an invertible typed transport preserves every physically admitted component of the complete model, all declared physical relations, and all outcome laws for witness families proved complete for the governing distinction, modulo transformations independently certified as proper gauge or presentation redundancy. Reduced, gauge-orbit, boundary, channel-representation, selected-output recovery, observer-experiment, clock-chart, comparison-kernel, and record-degradability equivalences are typed restrictions or induced task relations. For a covariant preregistered projection pi_T, complete physical equivalence implies task equivalence: M1 sim_phys M2 implies pi_T(M1) sim_T pi_T(M2). The converse is not licensed unless pi_T and its witness family are proved complete for the full governing physical distinction. BNP-C3. Typed nullity and witness discipline NULL means indistinguishable under a witness family proved complete for the stated domain and complete carrier. RESTRICTED-NULL means indistinguishable only under the declared task, access class, selected output, boundary completion, sector resolution, metric, or tolerance. UNDETERMINED means that carrier completeness, witness completeness, physical realizability, quotient descent, or scope is unresolved. Failure to find a separating witness establishes only the strongest status licensed by the proved completeness of the search. A positive witness is physically sound only when the witnessed difference descends through every admitted quotient and changes an admitted physical outcome or relation. Formal coordinate differences, unprobed spectra, auxiliary dimensions, Kraus presentations, tensor labels, gauge-dependent quantities, and preferred orbit representatives are not physical separation witnesses without the required descent and interpretation. BNP-C4. Complete carrier and operation classification Every theorem making an emergence, record, recovery, clock, resource, boundary, or arrow claim must identify the complete carrier relevant to that claim. The carrier includes, when admitted, systems, auxiliaries, discarded and retained environments, references, memories, controls, outcome registers, branch labels, boundaries, charges, sectors, batteries, work stores, apparatus, comparison protocols, and initial or generated correlations. Every operation used in a physical reconstruction is classified separately for mathematical admissibility, physical realizability, quotient descent, and complete null preservation. Partial trace, coarse graining, conditioning, postselection, ancillary extension, catalytic use, averaging, recovery, composition, and continuum completion are not presumed physically harmless merely because they are mathematically defined. Cross-program anti-aggregation principle. ETP distinguishes family-level certification from distribution-relative, quotient-relative, or presentation-relative summaries. No averaging, weighting, quotient, or selector may support a universal, worst-case, or provenance-complete emergence claim when that reduction suppresses an admitted distinction material to the claim. The aggregation rule, measure or weighting, null-set convention, boundary conditions, support, equivalence relation, and complete carrier must be declared before evaluation. A weighted or typical-case summary answers only its declared distribution-relative question; it does not replace a worst-case diagnostic or a complete provenance audit. BNP-C5. Provenance, non-nullity, and minimality Every certified non-null output must be traced to the physical bridge and to the specific inherited source distinctions, interactions, preparations, instruments, memories, boundaries, controls, references, resources, calibrations, or comparison protocols that support it. LRT reconstructs how admitted physical distinctions are organized, transported, recorded, recovered, compared, and metrized. It does not presently derive physical actuality or physical distinction from a certified-null basement. Terms such as minimal, maximal, largest, smallest, complete, closed, and unique are always indexed by the declared candidate class, order, complete datum, task, and theorem scope. Mathematical or task-relative minimality does not imply basement minimality, leastness, causal priority, temporal priority, ontological simplicity, or uniqueness of physical substance. Ground-frontier qualification. Where a companion basement-provenance analysis yields several distinct minimal certified ancestors under a declared strict dependence relation, those minima form an incomparable frontier rather than a uniquely selected ground. LRT therefore may not introduce an undeclared ranking, probability measure, boundary condition, simplicity criterion, or representative selector to collapse that frontier. Any such selector is additional physical or methodological data and must be included in the provenance ledger. BNP-C6. Scope, falsification, and ontology boundary Each theorem and model claim is read together with its declared model class, dimensional and algebraic setting, domains and topology, boundary and sector regime, causal or process assumptions, exact or approximate character, selected output, witness and access class, resources, falsifier, and extension boundary. Claims outside those declarations remain open extensions rather than consequences of the theorem. No result in this manuscript identifies the basement with Hilbert space, information, energy, relation, event, process, observer, record, time, geometry, field, quantum state, or mathematical structure. The manuscript may conditionally reconstruct physical sectors, observer access, records, recovery, statistical length, clock adequacy, comparison, and finite-step local-arrow structure after the corresponding bridge, carrier, and operation hypotheses are supplied. It may not infer ontology from formal existence, invariance, recoverability, minimality, or operational equivalence alone. Distinguishability-ontology boundary. Fisher information, accessible Fisher information, mutual information, channel capacities, recoverability metrics, and related statistical quantities quantify declared discrimination, estimation, recovery, or comparison tasks. Unless independently proved otherwise, they are not basement-level physical substances, conserved media, transported fluids, accumulated materials, or ontologically fundamental resources. Changes in these quantities are interpreted as changes in the operational accessibility of distinctions under the declared bridge, carrier, state family, measurement class, selected output, and observer protocol. BNP-C7. Controlling claim ceiling The constitution preserves all theorem statements within their proved scopes while controlling their physical interpretation. If a theorem family lacks a complete carrier, complete witness class, physical operation model, approximation convention, provenance closure, or nonfinite extension theorem, the manuscript retains the result as conditional, restricted, model-relative, or undetermined. No narrative summary, status table, or later application may upgrade that classification without a new proof or a documented correction of the governing datum. Phase E4 cross-reference and proof-import rule. A citation to a theorem imports only the cited conclusion under the theorem's complete operative hypotheses, definitions, carrier, selected output, equivalence relation, dimensional setting, and claim boundary. A later result must state or explicitly inherit every premise needed for the cited clause. Cross-reference does not supply an unproved converse, a stronger equivalence, physical realizability, witness completeness, approximate validity, compositional closure, or nonfinite extension. When several prior results are combined, their model classes, carriers, pictures, supports, outputs, parameter domains, measurement classes, approximation conventions, and equivalence relations must be jointly compatible. Current-authority and historical-reference rule. Current Version 55 theorem statements, correction notices, and claim-status ledgers control over historical roadmap, release, and completion language. Historical references identify genealogy only. A result reclassified as conditional, model-relative, open, restricted, or a research claim may be cited only at that status unless a new proof and formal ledger update restore stronger authority. Phase E6 cross-ledger authority rule. When the same claim appears in more than one current status location, the most recent scoped theorem or correction notice and the synchronized current claim-status ledgers control. Abstracts, roadmaps, Required Work, stage matrices, consequence tables, completion tables, and historical release records must be read consistently with that controlling status. A historical progression instruction is genealogy rather than current remaining work when a later scoped theorem has completed the referenced stage. No cross-ledger summary may convert a conditional, model-executed, restricted, open, undetermined, or historical-only result into a proved result. Stage-transition rule. A completed earlier stage supplies inputs or constraints for a later stage but does not discharge the later stage's independent hypotheses. Interaction does not itself establish a fragment boundary; output algebra does not establish implementable effects; generated access does not establish stability, readability, or recovery; recovery does not establish Fisher preservation or clock adequacy; local clock adequacy does not establish comparison consistency; and synchronization does not establish a thermodynamic arrow. Phase E5 quantifier, modality, and theorem-strength rule. Every universal, existential, uniqueness, maximality, minimality, necessity, sufficiency, genericity, stability, closure, or completion claim is indexed by the theorem's declared candidate class, order, equivalence relation, complete carrier, model scope, selected output, task, protocol, witness class, parameter domain, and exact or approximate convention. 'Unique' means unique within a stated candidate class and modulo a stated equivalence. 'Largest,' 'smallest,' 'maximal,' and 'minimal' refer only to a stated order. 'Necessary,' 'sufficient,' and 'exactly when' apply only under one compatible set of quantified hypotheses. 'Canonical' means invariant or uniquely determined only up to the equivalence proved by the cited theorem. 'Generally' means 'not universally' unless a topology or measure and a corresponding genericity theorem are expressly supplied. Exact channel stability, persistence, perturbative robustness, approximate recovery, and iterative closure are distinct predicates. Mathematical closedness and programmatic completion are not interchangeable. No summary or historical statement may enlarge these quantifiers or modalities. Phase E7 typed-notation rule. A symbol denotes one mathematical category within each governing scope. The governing physical anchor is D_anchor, while A_phys is reserved for the represented physical observable algebra. The finite-dimensional joint symmetry-observable algebra is J_joint, while J and J_M remain master-constraint operators. The event-domain output algebra is A_out(D); the admitted physical effect system is Eff_phys(D); and the admitted instrument family is Instr(D). Classical comparison kernels, quantum degrading channels, and deterministic comparison maps use direction-bearing subscripts. Historical formulas may retain their original notation when identified as genealogy, but current theorem statements, proofs, tables, and ledgers must use the typed registry or explicitly declare a local alias. Counterexample-strength rule. One counterexample disproves a universal assertion but does not establish topological genericity, measure-one prevalence, instability under arbitrary perturbation, or classification completeness. In such cases the current manuscript uses 'need not,' 'can fail,' or 'there exist admissible cases,' rather than an unqualified 'generic' claim. Phase D3 audit disposition. NONFINITE EXTENSION UNDETERMINED. The manuscript contains finite-dimensional theorems, a specific noncompact translation rigging-map benchmark, and several explicit nonfinite research frontiers. Except for results whose own hypotheses are already dimension independent, no finite-dimensional proof is promoted to separable infinite-dimensional Hilbert spaces, normal von Neumann algebras, Type II or Type III factors, unbounded generators, continuous direct integrals, noncompact group actions, continuum output nets, or infinite process tensors. Nonfinite provenance datum. A future nonfinite claim must declare the Hilbert or module class; represented C-star or von Neumann algebra; topology and predual; normality of states, channels, instruments, and conditional expectations; domains, cores, closures, and self-adjoint-extension data for unbounded operators; direct-integral or sector-measure structure; boundary and asymptotic data; reference, memory, and environment carriers; convergence mode; and the witness class used for physical equivalence and nullity. Finite-to-nonfinite transfer gate. Algebraic formulas that remain syntactically meaningful do not transfer automatically. Each extension must separately prove existence, normality or continuity, domain invariance, closure, quotient descent, complete-carrier control, and preservation of the claimed operational statistics. Compactness, finite generation, matrix Kraus representations, finite Wedderburn decomposition, attainment of suprema, finite-dimensional commutants, and norm equivalence may not be reused outside their proved domains. Specific status of existing results. C1 remains valid only under its stated finite-generator, dense common form-domain hypotheses and does not by itself solve infinite constraint families. The R translation example is one rigging-map execution rather than a general noncompact theorem. H1-H2, H10-H12, H22.2, C9-C17, the finite process-tensor executions, and the collision models retain their finite-dimensional scopes. G1, G7, G12, G13, and other measure-theoretic one-parameter statements may be dimension independent only when their stated path, measurability, integrability, and accessibility hypotheses are independently satisfied in the chosen nonfinite model. Type II and Type III boundary. A general nonfinite observer theory should be formulated with normal von Neumann subalgebras and normal completely positive maps rather than by assuming matrix blocks or tensor factors. Centers, commutants, conditional expectations, modular structure, and possible absence of minimal projections must be treated explicitly. Finite-dimensional Wedderburn factorizations and dimension-counting benchmarks do not establish the corresponding Type II or Type III claims. Unbounded and continuum boundary. For unbounded generators, clocks, charges, constraints, or Hamiltonians, equality of formal expressions is insufficient. The manuscript requires declared domains, invariant cores, closability, self-adjointness or extension choice, and compatibility of products and commutators. Continuum limits require a specified convergence notion and proof that the limiting carrier, channel, effect system, recovery task, Fisher path, and physical-equivalence relation exist and preserve the claimed distinctions. Current claim ceiling and falsifier. LRT may state that nonfinite extensions are open and that particular finite-dimensional results suggest candidate normal-algebraic formulations. It may not claim a general infinite-dimensional observer algebra, recoverable algebra, process-tensor compression, sector decomposition, boundary theory, or continuum clock theorem without a dedicated proof. A claimed extension is falsified or restricted by domain failure, nonnormality, nonattainment, divergent or representation-dependent limits, missing predual control, inequivalent self-adjoint extensions, incomplete carriers, or a separating normal protocol omitted from the witness class. BNP VERSION 03 AND DRAFT-07 LAYER-2 SYNCHRONIZATION Authority. This Version 56 successor incorporates the truth-first architecture of the Core Synthesis Draft-07 and the Basement Provenance and Instantiation Protocol Version 03. These authorities govern interpretation and dependency ordering. The existing LRT theorems, countermodels, numerical executions, historical records, and claim ceilings remain preserved unless a statement is explicitly narrowed below. Truth-First Temporal Reconstruction Constitution • LRT seeks the truth about temporal reconstruction rather than confirmation of ETP or relational potentiality. • Relational potentiality is a candidate basement, not an established physical substrate. • LRT does not select the basement and does not derive physical instantiation. • Logical necessity, formal sufficiency, physical necessity, physical sufficiency, and empirical adequacy remain distinct statuses. • No downstream temporal theorem may retroactively instantiate its carrier, state family, interaction, event, record, observer access, reference, calibration, resource, or comparison protocol. • Null, nonunique, orbit-valued, plural, underdetermined, and theory-refuting outcomes remain valid. • A formal clock coordinate, quantum state path, Fisher metric, record order, entropy balance, or convergence result cannot be promoted to physical actuality without an explicit bridge. • Every temporal claim inherits the complete carrier, witness family, equivalence relation, selected output, model class, parameter domain, and exact or approximate convention of its premises. Three-Target Firewall Target I, basement instantiation, asks whether any basement B is physically instantiated. Target II, definite-event generation, asks whether an instantiated basement yields definite instantiated events without an imported actuality selector. Target III, effective temporal reconstruction, asks whether instantiated events and records support order, duration, clocks, comparison, synchronization, and a local arrow. LRT addresses Target III. Progress on Target III does not establish Target II or Target I. Layer 0 and Layer 1 Input Certificate Layer-2 Claim Ceiling • LRT may conditionally reconstruct record order, accessible statistical length, calibrated local duration, clock adequacy, comparison transports, synchronization obstructions, and finite-step local-arrow structure. • LRT does not establish physical actuality, a unique basement, self-instantiation of relational potentiality, or a unique actuality selector. • LRT does not derive its event domain, interaction carrier, measurement semantics, resources, boundary, clock reference, or physical bridge from certified-null ancestry. • Recoverability does not imply readability, accessibility, persistence, Fisher preservation, clock adequacy, synchronization, or thermodynamic direction. • Fisher information measures discrimination in a declared experiment. It is not a temporal, informational, or physical substance. • Intrinsic S1 record-channel derivation and intrinsic S2 operational-boundary derivation remain open and are downstream of Layer 0 and Layer 1 certification. • No LRT result is laboratory evidence for ETP unless an ETP-specific physical implementation and measured dataset are independently supplied. Revised Research Priority The controlling upstream priority is the truth-first Basement Necessity and Instantiation program. LRT remains preserved as the authoritative Layer 2 conditional reconstruction program. Further LRT work should proceed only when it addresses a declared Layer 2 obligation or tests how alternative Layer 0 and Layer 1 inputs change temporal conclusions. No finite model chain should be presented as progress on basement instantiation. PHASE E7 CONTROLLED NOTATION REGISTRY Historical notation rule. Earlier version summaries and quoted genealogy may preserve their original symbols. Current theorem statements, proofs, status tables, and ledgers are controlled by this registry. Temporal parameter discipline. lambda is a gauge-orbit coordinate where declared; theta is a local statistical-model parameter or collision angle only in its declared section; T is elapsed time or relational phase only in its declared clock model; t_O is a calibrated clock reading. No equality among these quantities is implied by symbol reuse. K_{j<-i}: classical comparison kernel from record i to record j. D_{j<-i}: quantum degrading channel from record i to record j. Gamma_{j<-i}: deterministic comparison map from clock i to clock j. Phi and R: channels only when their domains and codomains are declared. Phi dagger and R dagger denote Heisenberg-picture adjoints. A recovery map is denoted Q or another locally declared symbol and is never inferred from a record-channel symbol alone. Governing map-signature template. For every operative channel theorem, let Phi_*: S(A_in) -> S(A_out) denote the declared Schrödinger-picture CPTP map on the stated input and output state carriers, and let Phi^dagger: A_out -> A_in denote its unital completely positive Heisenberg adjoint, defined by Tr[Phi_*(rho)X]=Tr[rho Phi^dagger(X)]. Any support projector, restricted carrier, parameter domain, parameter dependence or independence, and exact or approximate convention is part of the signature. A later theorem may import this template only by naming the controlling definition and specifying its actual carriers; otherwise the theorem must restate the signature locally. A_out(D): event-domain output algebra. S_out(D): physically supplied local output generators. Instr(D): admitted finite-outcome instrument family. Eff_phys(D): admitted physical effect system. A_eff(D): C*-algebra generated by Eff_phys(D). J_phys: physical-instantiation bridge. J and J_M: master-constraint operators. J_joint=C*(A_phys,U(G)): finite-dimensional joint symmetry-observable algebra. J_joint prime is its commutant. D_anchor=(P_model,J_phys,sim_phys,C_complete,S_scope): governing physical anchor. A_phys is reserved for the represented physical observable algebra. Version 55 Narrative Reconstruction ABSTRACT We formulate a restrained theory of local relational time supported by a globally constrained quantum description. The manuscript preserves the finite-generator master-constraint closure, compact and noncompact physical-sector benchmarks, seed-relative observer reconstruction, sectorwise factorization, accessible-Fisher geometry, dynamical-time/Fisher-clock correspondence, endpoint-regularized duration, energetic-resource separation, operational clock gluing, positive comparison holonomy, and explicit repeated-interaction open dynamics established in Version 13. Version 17 adds an exact record-channel stable-access selector. For a faithful finite-dimensional physical algebra, an active family of unital completely positive Heisenberg maps, and an operationally justified readable C*-algebra, the exact stable multiplicative algebra is intersected with the readable algebra. The result is the unique largest physical C*-algebra whose observables are both exactly stable and readable under the declared protocol. It is covariant under consistent physical *-isomorphisms. In the controlled collision benchmark, a thermally biased fragment selects the commutative pointer algebra, while null interaction, an unbiased fragment, or incompatible hidden channels can yield no nontrivial common exact access. This does not derive observers from the master constraint or bare potentiality. It closes exact stable-readable selection only relative to declared physical record dynamics. Consequently, the general Version-13 duration and gluing theorems remain valid, while individual model claims about Fisher rate, clock adequacy, endpoint behavior, or comparison-edge existence must be evaluated on the selected measurement class. The observer-access program is organized as a six-stage observer-emergence architecture: S0 physical interaction structure, S1 emergent candidate record channels, S2 operational boundary selection, S3 exact stable-readable core, S4 recoverable observer access, and S5 temporal-information extraction. S3 is complete within its stated finite-dimensional scope; S1 and S2 are the fundamental derivation targets; the finite-dimensional mathematical S4 channel-correction construction is complete for a fixed selected channel and supported sector; observer-relative S4 access additionally requires intersection with the declared readable or operational algebra, physical admissibility of the recovery, and proof of any S3-to-S4 inclusion; and S5 remains partially executed across complementary clock and collision benchmarks. A new dual-output collision audit executes the exact Kraus-product criterion on both outputs of one two-fragment interaction isometry. In the tested order of aligned strong, aligned weak, rotated strong, and rotated weak regimes, the reduced final-system recoverable-algebra dimensions are (2,2,1,1), whereas the complementary environmental dimensions are (2,1,2,1). The result proves within the model that exact recoverability is output-channel relative: intermediate system rotation can destroy nontrivial exact access through the final system while a perfect record already retained in the first fragment survives. The audit does not promote record recoverability to clock adequacy; approximate recovery, restricted readout, and elapsed-time Fisher analysis remain separate gates. Version 17 derives the total complementary record channel directly from a finite-dimensional interaction isometry, uniquely up to output isometry, without presupposing an observer. It proves that interaction data alone need not select a unique proper fragment boundary when output symmetries exchange candidate fragments. Given a covariant physical output net, qualified fragment-record channels can nevertheless be selected without observer labels. For each resulting channel, the commutant of the compressed Kraus-product algebra is proved to be the unique largest observable algebra corrected by one CPTP recovery channel. Version 17 proves that a record-derived comparison transducer is a parameter-independent Markov post-processing of the full record experiment, or at the quantum level a CPTP degrading channel between record channels. A conditional kernel from one joint state or prior average is insufficient. Joint compatibility does not imply transducibility. Deterministic maps arise exactly as Dirac kernels, positive-affine maps arise from common-coordinate constant-speed calibrated clocks, and loop comparison extends to kernel-valued stochastic holonomy. The retained-memory collision chain supplies an exact partial-trace transducer, while independently calibrated collision-clock streams remain to be executed. Version 43 opens Stage S1B, Record-Orbit Selection and Realization. It proves a datum-relative no-selection result: no covariant rule built solely from the S1A reconstruction datum can uniquely select a member of a nontrivial residual record orbit. The adversarial program excludes record-internal canonicalization, persistence ranking, basin or attractor selection, stability ordering, and comparison geometry as independent selectors whenever they are reconstructed only from orbit-preserving data. The investigation then separates observer-free recoverability from observer readability and identifies a candidate invariant recovery package consisting of recoverable distinction content and recovery geometry. All explicitly realized recoverability effects examined so far reduce to a change in what can be recovered, a change in how recovery occurs, or both. No constructive third recoverability axis or hidden modulus has been found. This is recorded as a provisional closure program, not as a completed completeness theorem: the universal necessity-of-manifestation statement and the corresponding completeness implication remain open. M9 completes the Stage 8S-8Z finite audit block by synchronizing recovery-target alignment, explicit record gates, event-net provenance, autonomous activation, controller readability, calibration regress, and physical-anchor admissibility. The block establishes quotient-level sufficiency for orbit-constant operational claims and preserves intrinsic S1 record-channel derivation, intrinsic S2 operational-boundary derivation, physical anchor selection, thermodynamic orientation, and nonfinite extension as open. S1/S2 closure update through Stage 7. Reduced endpoint dynamics, and even a complete reduced semigroup, need not select a unique discrete event architecture. Event reconstruction is therefore defined only up to refinement equivalence unless independently physical separators resolve the refinement. Individual support, action, record, exchange, and resource signatures are insufficient in general because higher-order compatibility and composition may differ while all marginal and pairwise signatures agree. The strongest current finite-dimensional candidate invariant adds the complete compatibility hypergraph and physical composition law; its witness-completeness result remains conditional on a fixed exhaustive candidate class, complete testers, complete carriers, and a fixed physical-equivalence relation. Intrinsic S1 record-channel derivation and S2 operational-boundary derivation remain open. S1/S2 closure update through Stage 8R. Stage 8 separates response-law existence, uniqueness, covariance, composition, normalization, and operational adequacy; distinguishes effect, instrument, endpoint-channel, and multitime-process equivalence; and makes complete-carrier and tester-class declarations mandatory for causal screening and memory claims. Finite-dimensional operational tester distances obey data processing under admitted superprocesses, while conditional-mutual-information recovery becomes an operational process theorem only after causal validity, physical recovery typing, complete-carrier coverage, and a proved tester-distance bound. A two-slot noncommutative product comb supplies an explicit causal feasible point, and a retained-memory two-collision benchmark demonstrates history dependence across a visible-system discard-and-reprepare break, partial phase-sensitive transmission, and strict obstruction to exact full-qubit recovery from the final system alone. No noisy nearest-causal optimization, minimal Markov-order theorem, minimal bond-dimension theorem, intrinsic S1 record-channel derivation, intrinsic S2 boundary derivation, endogenous-order theorem, or nonfinite process theorem is claimed. [86-90] S1/S2 closure update through Stage 9C. Stage 9 completes the planned finite provenance, operation, controller, resource, cycle, reservoir, witness, instrument, calibration, complete-carrier, sequential-process, falsifier, and feasibility audits. Stage 9C reconciles a 16-node dependency graph and ten controlling open obligations, consolidates 36 grouped theorem and countermodel records with scoped claim ceilings and falsifiers, and audits manuscript readiness. The research-program audit is closed with open scientific obligations: intrinsic S1 candidate-record derivation and intrinsic S2 operational-boundary derivation remain open; sequential process validity remains conditional; the device-relative protocol is model-executed; laboratory ETP evidence and nonfinite or device-independent closure are not established. This release is a final Stage 9 audit release, not proof of intrinsic S1/S2 closure or experimental confirmation of ETP. Version 18 adds the Autonomous Record-Formation Theorem Package. A finite-horizon, gap-matched battery dilation realizes controlled record collisions by a unitary commuting with the total bare energy on the declared supported sector. An internal clock or transport degree of freedom supplies conditional autonomous activation under an ideal-clock hypothesis. The thermally biased qubit-ladder benchmark preserves nontrivial fragment distinguishability and the exact stable-readable pointer algebra, while an unbiased fragment remains a valid null result. Retained fragments continue to recover earlier memories by partial trace. Exact first-law accounting separates heat from battery energy transfer, and the battery-inclusive identity ΔS_S+ΔS_W-βQ_S=I_tot′+D(ρ_F′||τ_F)≥0 retains battery entropy and final correlations rather than treating all stored energy as pure work. Finite battery capacity, finite-clock degradation, reusable-battery error, and an autonomous continuum limit remain open execution gates. Version 19 audits clock adequacy under nonunitary and non-Markovian evolution. The regime-independent Fisher-clock correspondence is proved for any absolutely continuous physical state path: global affine calibration to the dynamical coordinate holds exactly when the observer-accessible elapsed-time Fisher information is positive and constant almost everywhere. CP divisibility is shown to be neither sufficient nor necessary for that condition. Revival of parameter distinguishability about an encoded parameter is distinguished from Fisher information about elapsed time; reduced-state recurrence obstructs global instantaneous clock readings; and exact recovery does not imply clock adequacy. Markovian dephasing, Markovian amplitude damping, and oscillatory non-Markovian dephasing provide explicit counterexamples. A seven-class status system separates globally affine, approximately affine, piecewise, memory-extended, event-timer, record-only, and inadequate clocks. Version 20 executes the audit on an ETP-compatible composite collision model with a persistent qubit memory carrier. Excitation-preserving source-memory and memory-fragment collisions yield an exact amplitude recurrence and an exact Fisher partition for a relational phase T: F_T^A(n)=omega^2 q_A(n), where q_A is the conditional excitation weight in the accessible subsystem. The operational distinguishability of the encoded parameter has an exact additive Fisher partition across source, memory carrier, and retained fragments in this declared model. In a non-Markovian benchmark the reduced-source Fisher information revives and the reduced dynamics fails CP divisibility on invertible revival steps, while the distinguishability recoverable from retained fragments grows monotonically under the declared unrestricted retained-memory measurement class. Nevertheless adjacent retained-memory states have nonuniform Bures increments, so memory growth does not supply homogeneous duration for collision count. The execution distinguishes a locally calibrated relational-phase clock from an update-count clock and closes the finite-dimensional ETP-specific non-Markovian audit within its declared scope. Version 21 reformulates the unresolved symmetry-selection problem as a staged, falsifiable reconstruction program rather than claiming a completed symmetry theorem. The active finite-dimensional target is to reconstruct the maximal effective compact symmetry preserving a complete physical datum consisting of the represented physical observable algebra, admissible states, active channels, physically supplied nets, and operational relations. Unitary implementers acting trivially on all physical observables are quotiented as presentation redundancy. Subsequent stages classify gauge versus physical symmetry, recover connected generators and test master-constraint compatibility, derive sector and commutant structure, and escalate to Tannaka-type categorical reconstruction only if ordinary group reconstruction is insufficient. The program permits trivial, discrete, disconnected, multiply presented, or categorical symmetry outcomes and explicitly guards against circularly recovering a gauge group already assumed in constructing the physical Hilbert space. Version 22 completes SYM-1 within the closed finite-dimensional physical-datum scope. The stabilizer G_impl of the complete datum is proved to be a closed subgroup of U(H_phys), hence a compact embedded Lie group. The null implementers K_null form a closed normal subgroup, and G_eff=G_impl/K_null is a compact Lie group acting faithfully on A_phys and canonically isomorphic to the concrete effective image in Aut(A_phys). Covariance under physical unitary equivalence, maximality, a nested symmetry-breaking hierarchy, and physical-action nonuniqueness are proved. Full-matrix, nondegenerate-state, degenerate-state, and ETP nonfactor benchmarks are executed. The result reconstructs the maximal effective physical action, but not a unique unitary lift, covering group, central extension, or pre-reduction gauge group. Gauge-versus-physical classification therefore remains the next stage. Version 23 completes SYM-2 within the closed finite-dimensional, boundary-declared scope. Proper gauge transformations are proved to lie in K_null, while nullity alone is proved insufficient for gauge status. Every nonidentity element of G_eff is excluded from pure-gauge classification relative to the declared physical algebra. Gauge certification requires an unreduced equivalence-orbit realization and zero physical boundary or asymptotic charge. Physical, dynamical, operational, boundary, sector, and presentation roles form an overlapping classification signature rather than a mutually exclusive partition. The ETP nonfactor benchmark is classified accordingly. No nontrivial G_eff generator is thereby authorized as a master-constraint generator; SYM-3 must distinguish physical generators from separately gauge-certified unreduced generators. Version 24 formulates SYM-3 as a truth-first program of effective generator reconstruction, gauge-lift certification, and master-constraint recoverability. The stage distinguishes the canonical physical derivations determined by G_eff from nonunique operator representatives, retains disconnected symmetry data separately from the Lie algebra, and forbids nontrivial effective physical generators from being imposed as pure-gauge constraints. Gauge generators become eligible for C1 only after an independently supplied unreduced lift proves orbit equivalence, anomaly-compatible closure, and zero physical boundary or asymptotic charge. The stage targets a master-constraint compatibility class rather than presuming unique recovery: different gauge lifts, generator bases, positive coefficient matrices, and master operators may share the same physical kernel and reduced datum. Version 24 installs the theorem program and benchmarks; it does not report SYM-3 as proved. Version 25 completes the first finite-dimensional SYM-3 proof block. The identity component of G_eff reconstructs a finite-dimensional real Lie algebra g_eff whose differentiated action is an injective Lie-algebra representation by physical *-derivations of A_phys; the disconnected component group is retained separately. Every such derivation is inner on the finite-dimensional algebra. Abstract algebra implementers are unique modulo the center, while represented Hilbert-space generators are unique only modulo the commutant. A Hilbert-Schmidt minimum-norm representative exists only relative to a declared faithful representation and trace. Operator lifts may close only modulo a commutant-valued defect even though the physical derivations close exactly. Nonzero effective derivations are physical and cannot be declared pure gauge on the same reduced datum. Full-matrix, nondegenerate-state, multiplicity, ETP nonfactor, and diagonal SU(2) benchmarks are executed. For a fixed gauge-certified generator family, every positive-definite master metric has the same common kernel, and invertible basis changes preserve the constraint content. At that stage, gauge-lift equivalence and the full constraint-compatibility class became the next SYM-3 targets, subsequently completed in Version 26 within the declared finite-dimensional scope. Version 26 completes SYM-3B within a declared finite-dimensional admissibility scope. Gauge-lift comparison is stratified into reduced-physical equivalence, strict gauge-orbit equivalence, and strong master-operator equivalence. The corresponding compatibility objects are indexed by an admissibility specification and by the selected equivalence relation. Strong master equivalence implies gauge-orbit equivalence, which implies reduced equivalence, while both converses fail. A covariance theorem transports the quotient classes under complete physical equivalence. The occupancy trichotomy is empty, singleton, or multiple only relative to the declared reconstruction problem; reduced physical data alone do not determine occupancy. Two explicit bounded finite-dimensional U(1) lifts with different prephysical dimensions reduce to the same complete physical datum while remaining strictly gauge-orbit inequivalent. A fixed certified constraint family has a metric-independent physical kernel but generally admits multiple exact master operators. Therefore the finite-dimensional SYM-3 result is compatibility-class recovery, not unique reconstruction of an unreduced gauge theory or master constraint. Stable equivalence under auxiliary pure-gauge extension is identified as a separate optional quotient and is not assumed. Version 27 performs a manuscript-wide synchronization audit after the Version-26 completion of finite-dimensional SYM-3B. The staged program, C12 status, required-work list, observer-stage summaries, both claim-status ledgers, and reference chronology are aligned with C13. Within the bounded finite-dimensional, declared-domain scope, SYM-3 is complete through compatibility-class recovery. The remaining SYM-3 items are extensions rather than missing finite-dimensional premises: unbounded-generator domain equivalence, noncompact rigging-map equivalence, BRST/BFV stable equivalence, field-dependent structure functions, infinite constraint families, and explicit boundary or edge-charge realizations. The next proposed research question, distinguished-representative or master-metric selection, is retained as a new falsifiable investigation and is not retroactively claimed by the completed SYM-3B theorem package. Version 28 performs a theory-level narrative audit while preserving the complete theorem architecture. The manuscript now states the emergence path explicitly: a constrained substrate yields a gauge-reduced physical arena; physical algebras support observer-relative access; interactions generate candidate records; recoverability orders those records; accessible statistical geometry supplies local duration; clock adequacy selects faithful temporal coordinates; comparison transports determine whether local clocks glue globally; and autonomous record formation supports a conditional local arrow. Major sections now identify their precise role in this sequence. Historical limitations superseded by later finite-dimensional results are updated, while unresolved extensions are consolidated in status and scope tables so that the main text can emphasize established advances without weakening claim discipline. Version 29 adds three reviewer-facing clarifications without changing any theorem, proof, benchmark, or status classification. It states that constructive nonuniqueness of unreduced gauge realizations is physically expected when they induce the same reduced observable datum; it points the ARF2 ideal-clock hypothesis directly to the finite-clock localization, backreaction, timing, and degradation requirements in the Claim-Status Ledgers and S5 program; and it condenses the emergence chain into one orienting sentence at the beginning of the manuscript. Version 30 adds SYM-3C.1, Observer-Temporal Descent Invariance and Enriched Gauge-Lift Separation. For gauge-certified lifts that are equivalent at the complete reduced physical level, every listed observer-temporal construction that is defined from the transported reduced datum, whose operational choices are transported, and whose governing component theorem applies is transported rather than split. The result proves that observer restriction and relational-time construction cannot recover unreduced information erased by reduction. A compatibility class can be separated only by independently justified physical enrichment that produces a gauge-invariant operational distinction. The finite-dimensional benchmarks show that auxiliary gauge-lift dimension and positive master-metric rescaling remain invisible to on-shell observer-temporal data unless an admitted off-shell, boundary, reference-frame, or interaction probe distinguishes them. Version 31 adds SYM-3C.2, Boundary-Charge, Edge-Presentation, and Interface Separation. It proves that a boundary separates bulk-reduced-equivalent lifts only when independently justified boundary conditions, physical boundary observables, charges, interface channels, or gluing relations produce a gauge-invariant operational distinction. Bulk data alone need not select a unique boundary completion, center, charge representation, regional factorization, edge extension, or gluing rule. The finite-dimensional package classifies proper boundary gauge, physical boundary symmetry, unresolved boundary-null action, and presentation change; defines boundary-experiment equivalence; proves central charge-sector and interface-matching results; establishes edge-presentation neutrality; and specializes exact recoverability to boundary record channels. Explicit null, central-charge, equal-spectrum/different-dynamics, presentation-equivalent, and gluing-obstructed benchmarks are included. Boundary or edge structure advances relational time only through admitted records, recovery, Fisher geometry, and clock or comparison gates. Version 34 sanitation addendum. Version 34 completes the bounded finite-dimensional SYM-4 architecture after a manuscript-wide mathematical and narrative sanitation. A declared compact unitary implementation yields canonical isotypic sectors and multiplicity spaces, while the physical observable algebra and represented symmetry jointly generate J_joint=C*(A_phys,U(G)); the minimal central projectors of J_joint give the canonical central Wedderburn decomposition, without excluding noncentral reducing projections inside multiplicity spaces. Observer access then separates sector preservation, readability, recordability, and exact recoverability. For differentiable statistical models block diagonal in fixed orthogonal joint sectors, the SLD quantum Fisher information is the sum of the classical Fisher information of the sector weights and the sector-weighted intrasector quantum Fisher information. Parameter-independent channels contract SLD Fisher information. Exact recovery of the complete model family is sufficient for equality, but equality of SLD Fisher information for one family or tangent is not treated as a recoverability certificate. Positive Fisher information supplies local sensitivity only; relational clock adequacy additionally requires accessibility, identifiability, ordering or declared periodicity, calibration, stability, and compositionally consistent comparison transport. Version 35 establishes the bounded finite-dimensional foundation of SYM-5, Observer-Sector Selection and Accessibility Reconstruction. Sector and commutant data alone need not select a unique proper observer algebra. Given a physically supplied output net, admitted fragment-effect systems, and interaction-derived channels, accessible effects pull back to covariant seed systems whose generated algebras can be qualified by physical provenance, nontrivial accessibility, sector relation, persistence, recordability, one-channel recoverability, covariance, robustness, and compositional closure. The covariantly determined output object or output family may be empty, scalar, nonunique, a symmetry orbit, a physical-equivalence quotient, a Pareto family, or conditionally unique only after independently physical structure resolves residual equivalence. Version 35 also distinguishes representation sectors, joint central sectors, record sectors, and observer-central sectors; defines their incidence when a common central refinement exists; and integrates the exact stable, readable, and recoverable cores. Categorical or Tannaka escalation is moved to SYM-6 and remains conditional on independently supplied sector-category, intertwiner, composition, duality, and faithful-fiber-functor data. SYM-5 is not declared complete pending model-level benchmark execution, an ETP-grounded output net and effect systems, approximate robustness and resource analysis, and one nontrivial compositionally closed interaction execution. Version 36 completes the ETP-grounded physical output-net step within the bounded finite-dimensional, interaction-relative scope and establishes the theorem architecture for physical effect-system derivation. A total channel or bare tensor factorization does not identify elementary interaction events or a preferred proper output net. Given a physically declared interaction-event structure, event-domain poset, inclusion-compatible local output generators, functorial embeddings, and complete-datum covariance action, the generated algebras form a covariant isotonic output net. Physical event decompositions and output nets related by unresolved complete-datum automorphisms remain nonunique. Parameter-independent degradability defines a record-support preorder, while causal order, recoverability order, and duration remain distinct. The repeated-collision architecture supplies an explicit event-indexed accumulated-output net. On each output domain, physically admitted instruments generate a convex effect system closed under complement, coarse-graining, randomization, and declared physical transport; the output algebra alone does not determine that effect system. Pullback of physical effects through fragment channels supplies covariant access seeds for the SYM-5 observer-algebra construction. The output-net step is complete only in this declared scope. Effect-system model execution, approximate detector resolution, energy and superselection restrictions, and inclusion compatibility remain open. I. PROBLEM AND PRINCIPLE Role in emergence. This section states the central separation principle: order, duration, synchronization, and direction are distinct achievements. The rest of the manuscript constructs each layer and proves the conditions under which one layer can support the next. LRT is a Layer 2 conditional temporal-reconstruction program. It does not assume a background physical duration or one globally synchronized clock. It begins only after Layer 0 has supplied, justified, empirically identified, or explicitly postulated physical instantiation and Layer 1 has supplied a complete instantiated event-and-recoverability datum. From declared constraints, interactions, state paths, records, ordering relations, instruments, references, resources, and comparison structures, LRT tests when those inputs support observer-accessible statistical length, calibrated local duration, clock adequacy, synchronization, and a conditional local arrow. LRT does not derive the physical actuality of those inputs. shared ordinal structure != shared temporal metric != globally parallel time != thermodynamic direction. The framework treats record order, local duration, global synchronization, and temporal direction as separate mathematical layers. No layer may be substituted for another. Roadmap to relational time. Layer 0 addresses basement necessity and physical instantiation. Layer 1 supplies instantiated event and recoverability structure. LRT begins at Layer 2 and reconstructs temporal structure conditionally: physical reduction provides an arena; observer-access algebras identify what can be read; record dynamics and recoverability establish operational order; accessible Fisher geometry supplies local statistical length; clock-adequacy theorems determine when that length can represent calibrated local duration; comparison transports test global compatibility or holonomy; and autonomous record and entropy balances support only a conditional local arrow. Background duration is not inserted as a primitive, but every admitted carrier, event, path, interaction, calibration, reference, resource, and comparison remains part of the provenance. II. SYMMETRY-DERIVED CONSTRAINED SUBSTRATE Role in emergence. Relational time requires a physical arena before it requires a clock. Constraint reduction identifies the states and observables on which records, observers, and temporal comparisons can have invariant meaning. B0-B2. Symmetry, constraint, and physical sector Let a connected gauge group G act strongly continuously on H_kin with self-adjoint generators C_a. Gauge invariance U(g)psi=psi is equivalent on a common invariant domain to C_a psi=0 for every a. For a positive-definite M^{ab}, the closable form q_J[psi]=sum M^{ab} defines a positive master constraint with ker J = intersection_a ker C_a = H_kin^G. For compact G, Haar averaging defines the invariant projector. Noncompact groups require a rigging map or equivalent physical-inner-product construction. Anomaly freedom, domain control, and selection of G remain explicit gates. B3-B4. Orbit coordinate and conditional dynamics A selected reference sector with an absolutely continuous real spectrum for C_R admits a Fourier-dual orbit coordinate lambda with C_R=-i partial_lambda. In a deparametrizable sector C=C_R tensor I + I tensor H_S, C|Psi>=0, i partial_lambda|psi(lambda)>=H_S|psi(lambda)>. lambda is a gauge-orbit coordinate, not physical duration. III. MASTER-CONSTRAINT DOMAIN AND ANOMALY CLOSURE Role in emergence. This section secures the mathematical substrate and reconstructs its effective symmetry content. By separating genuine physical transformations from gauge redundancy, it prevents temporal change from being confused with motion along an unphysical gauge orbit. Constructive nonuniqueness is physically expected: distinct unreduced gauge representations can descend to the same gauge-reduced physical datum and are therefore indistinguishable by reduced observables alone unless additional physical structure separates them. C0. Scope of the closure The closure is representation-specific. Given a declared Hilbert space, strongly continuous unitary gauge representation, generators, and dense domain, the master constraint and physical state construction are executed. Selection of the physical symmetry from a Layer 0-certified or explicitly postulated instantiated datum remains open; bare formal potentiality does not supply physical selection. C1. Finite-Generator Master-Constraint Closure Theorem Theorem C1. Let C_1,...,C_N be self-adjoint on H. Assume D(q_J)=intersection_a D(C_a) is dense. Let M=(M^{ab}) be positive-definite Hermitian and define q_J(psi,phi)=sum_{a,b} M^{ab}. Then q_J is densely defined, nonnegative, and closed; it determines a unique positive self-adjoint J; and ker J=intersection_a ker C_a. Proof. If m_- and m_+ are the smallest and largest eigenvalues of M, then m_- sum_a||C_a psi||^2 <= q_J[psi] <= m_+ sum_a||C_a psi||^2. Hence the form norm ||psi||_q^2=||psi||^2+q_J[psi] is equivalent to the common graph norm. A graph-norm Cauchy sequence converges in H and in every C_a component; closedness of each C_a places the limit in all domains. Thus q_J is closed. The representation theorem gives a unique positive self-adjoint J. Positive definiteness gives q_J[psi]=0 exactly when every C_a psi=0. QED. Corollary C1.1. If the C_a generate a connected strongly continuous unitary representation U(G), then ker J=H^G. The result follows by differentiating invariant one-parameter subgroups and conversely exponentiating the vanishing generators across products of exponentials. C2. Anomaly criterion on a common invariant core [C_a,C_b]psi=i f_ab^c C_c psi for every psi in D_infinity. For generators of an actual strongly continuous representation of a finite-dimensional Lie group, the identity holds on the smooth-vector domain. This does not cover field-dependent structure functions or arbitrary formal constraints. C3. Compact benchmark: diagonal SU(2) H_kin=C^2 tensor C^2, C_a=(sigma_a tensor I+I tensor sigma_a)/2, J_SU2=sum_a C_a^2. With |s>=(|01>-|10>)/sqrt(2) and P_s=|s>}. Haar averaging gives P_phys=P_s. Numerical audit: all nine commutator residuals vanish; ||J_SU2-2(I-P_s)||=4.44e-16; ||J_SU2 P_s||=0. C4-C5. Noncompact translation benchmark and rigging map H_kin=L2(R), P=-i d/dx on H1(R), J_R=P^2 on H2(R). Zero lies in the continuous spectrum and the ordinary L2 kernel is trivial. On Phi=S(R), group averaging gives eta(phi)[psi]=2 pi conjugate(phi_hat(0)) psi_hat(0). Quotienting the null space {phi:phi_hat(0)=0} and completing yields H_phys^R isomorphic to C, with inner product 2 pi conjugate(phi_hat(0)) psi_hat(0). For phi=exp(-x^2) and psi=exp(-2x^2), direct group averaging gives 2.221441469079183, agreeing with the Fourier expression at numerical precision. C6-C7. Retention and frontier Within their declared compact and noncompact represented models, the SU(2) and R examples show how a physically interpreted constraint and reduction datum can select an invariant arena in which relational records and clocks may subsequently be analyzed. They do not derive the physical bridge, the constraint choice, or a universal physical-sector construction. The broader extension frontier is recorded in the scope matrix and claim-status ledgers. C8. STAGED PHYSICAL-DATUM SYMMETRY-RECONSTRUCTION PROGRAM The remaining ETP symmetry-selection problem is divided into stages. The program does not choose a preferred group and then fit the physical data to it. It asks what symmetry object, if any, is forced by declared physical structure. Null, discrete, disconnected, nonunique, and non-group outputs are admissible. C8.1. SYM-1: Finite-Dimensional Physical-Datum Symmetry Reconstruction and Nonuniqueness Version-21 target, completed in Version 22. Let D_phys=(H_phys,A_phys,Omega,{Phi_alpha},N,Rel_op), where A_phys is a faithfully represented finite-dimensional physical C*-algebra, Omega is an admissible state family, {Phi_alpha} is a declared channel family, N is any physically derived locality or output net, and Rel_op is the declared set of operational recovery, comparison, incidence, and composition relations. Version 22 reconstructs the maximal unitary implementer group preserving the complete closed datum and quotients implementers acting trivially on A_phys. The intended objects are G_alg={U in U(H_phys): U A_phys U^dagger=A_phys}, G_impl={U in G_alg: Ad_U preserves Omega,{Phi_alpha},N,Rel_op}, K_null={U in G_impl: U X U^dagger=X for every X in A_phys}, and G_eff=G_impl/K_null. Setwise versus pointwise preservation and permitted permutations of equivalent labels must be declared explicitly. Version-21 theorem target, now proved as C9.2-C9.3 under explicit closed-datum hypotheses: G_impl is a compact embedded Lie group, K_null is closed and normal, and G_eff is a compact Lie group acting faithfully on A_phys, covariantly under physical equivalence. Version-21 nonuniqueness target, now proved as C9.4: the datum determines the common effective physical action but cannot generally distinguish covering groups, central extensions, scalar-phase lifts, or other null implementations. C8.2. SYM-2: Gauge-versus-Physical Symmetry Classification Version-22 target, completed in Version 23. Proper gauge lies in K_null, but nullity alone is insufficient. Nontrivial G_eff actions are non-gauge relative to A_phys. Classification is a multilabel signature, and gauge certification requires unreduced orbit data plus zero boundary charge. C8.3. SYM-3: Generator Reconstruction and Master-Constraint Recovery Version-23 target, reformulated in Version 24 and completed in finite dimension through Versions 25-26. Version 25 reconstructs g_eff and its physical derivations, retains disconnected components separately, and restricts C1 to separately gauge-certified unreduced generators. Version 26 formalizes reduced, gauge-orbit, and strong master equivalence and classifies the corresponding compatibility quotients. The canonical finite-dimensional output is compatibility-class recovery rather than one uniquely reconstructed master operator. C8.4. SYM-3C.1: Observer-Temporal Descent Invariance and Enriched Gauge-Lift Separation Version-30 theorem package. This stage tests whether observer-access and relational-time structure can distinguish gauge lifts already equivalent at the complete reduced physical level. It proves a descent-invariance and no-selection result for every covariantly derived observer-temporal construction, then identifies independently justified gauge-invariant enrichment as the only admissible mechanism for separating such lifts. The stage is complete in the bounded finite-dimensional, declared-protocol scope and does not claim ontological identity of operationally indistinguishable lifts. C8.5. SYM-3C.2: Boundary-Charge, Edge-Presentation, and Interface Separation Version-31 theorem package. This stage specializes independently physical enrichment to boundaries and interfaces. It classifies boundary actions, proves nonuniqueness of boundary completion from bulk data, defines operational boundary equivalence and separation, analyzes central charge sectors and gluing, remains neutral between literal edge variables and relational or sectorwise encodings, and transfers the exact recoverability criterion to boundary record channels. It is complete in the bounded finite-dimensional algebraic scope; derivation of an ETP-specific boundary algebra or charge remains open. C8.6. SYM-4: Sector and Commutant Decomposition Version 34 completes SYM-4A through SYM-4D in the bounded finite-dimensional scope. SYM-4A reconstructs implemented-symmetry isotypic sectors and multiplicities. SYM-4B derives the canonical joint algebra generated by the physical observable algebra and represented symmetry. SYM-4C separates observer preservation, readability, recordability, and exact recoverability. SYM-4D supplies the sector-resolved Fisher decomposition, operational clock-adequacy gates, and comparison-transport criteria. No representation label, conserved charge, record, recoverable datum, or positive Fisher information is treated as duration without the declared accessibility, identifiability, ordering, and transport conditions. C8.7. SYM-5: Observer-Sector Selection and Accessibility Reconstruction Version 36 advances the observer-selection stage that follows the completed SYM-4 sector architecture. Sector structure alone cannot select a unique proper observer algebra. Within bounded finite dimension, an interaction-event datum now conditionally reconstructs a covariant isotonic physical output net, while admitted instrument families reconstruct physical effect systems whose channel pullbacks generate access seeds. Bare tensor cuts, total channels, and output algebras remain insufficient without event, locality, instrument, or resource structure. The canonical result may be empty, scalar, a symmetry orbit, several inequivalent admissible algebras, a Pareto family, or conditionally unique only after independently physical structure breaks residual equivalence. C8.8. SYM-6: Categorical or Tannaka Escalation If the reconstructed group action and joint algebra do not capture an independently supplied physical sector category, fusion law, intertwiner or bimodule composition, duality, and faithful fiber functor, escalate to a concrete categorical reconstruction. No categorical, weak-Hopf, or quantum-group symmetry is claimed before those hypotheses are physically supplied and verified. SYM-6 may not be used as a substitute for the SYM-5 task of deriving operational access from interaction and record structure. C8.9. Circularity and truth-first boundary Because the present H_phys was produced from declared constraint representations, reconstructing a symmetry from H_phys can be only a physical-datum consistency result unless the input datum is defined independently of the original group choice. A stronger gauge-selection claim requires either a symmetry-neutral preconstraint datum or a candidate-family selection theorem comparing multiple reductions against common ETP axioms. C8.10. SYM-1 benchmarks and completion criteria Full matrix benchmark: A_phys=M_d(C) with all states should test whether scalar phases collapse and the effective inner action is projective. Nondegenerate-state benchmark: a distinguished state with simple spectrum should exhibit state-induced reduction to its unitary stabilizer modulo null phases. Degenerate-state benchmark: degenerate eigenspaces should exhibit symmetry enhancement forced by physical degeneracy. ETP nonfactor benchmark: execute reconstruction on H_phys=(C^2 tensor C^2) direct-sum C^2 while successively preserving A_phys, the center, observer algebras, state data, and channel data. Version-21 completion criterion. Version 22 satisfies the datum, topology, closedness, effective quotient, covariance, nonuniqueness, falsification, and benchmark requirements. SYM-1 is complete within its declared closed finite-dimensional scope. C9. FINITE-DIMENSIONAL PHYSICAL-DATUM SYMMETRY RECONSTRUCTION AND NONUNIQUENESS C9.1. Closed physical datum Definition C9.1. Let D_phys=(H,A,Omega,Phi,N,Rel_op), where dim H=dAut(A), alpha_U(X)=UXU^dagger, is a continuous group homomorphism; (iv) K_null=ker(alpha) is a closed normal Lie subgroup; (v) G_eff=G_impl/K_null is a compact Lie group; (vi) the induced G_eff action on A is faithful; and (vii) G_eff is canonically isomorphic to the full effective image alpha(G_impl), the unique maximal datum-preserving unitary action on A. Proof. The identity stabilizes x_D. Products and inverses of stabilizers stabilize x_D, so G_impl is a subgroup. If U_n in G_impl converges in norm to U, continuity gives U_n x_D -> U x_D; the left side is constantly x_D, hence U x_D=x_D and G_impl is closed. U(d) is compact, so G_impl is compact. By the closed-subgroup theorem it is an embedded Lie group. Conjugation gives a continuous homomorphism alpha. Its kernel is normal; it is closed as the inverse image of the identity automorphism. The quotient by the closed normal subgroup is a compact Lie group. If a quotient class acts trivially on A, its representative lies in K_null, so the action is faithful. Finally the first isomorphism theorem gives G_impl/K_null isomorphic to alpha(G_impl). Every datum-preserving unitary belongs to the stabilizer by definition, so this image is maximal. QED. Claim boundary. If any preservation condition is not closed, the closed-subgroup conclusion need not follow. The theorem reconstructs symmetries of the complete represented physical datum. It does not by itself identify them as gauge redundancies or recover the group used before physical reduction. C9.3. Covariance under physical equivalence Theorem C9.3. Suppose a unitary W:H->H_prime transports every component of D_phys to a closed datum D_phys_prime, including states, channel labels, physical nets, and operational relations. Then G_impl(D_prime)=W G_impl(D) W^dagger, K_null(D_prime)=W K_null(D) W^dagger, and G_eff(D_prime) is canonically isomorphic to G_eff(D). Proof. U preserves D exactly when WUW^dagger preserves the transported datum. The same conjugation transports the kernels because WUW^dagger acts trivially on WAW^dagger exactly when U acts trivially on A. It therefore descends to the quotient. QED. C9.4. Physical-Action Nonuniqueness Theorem Theorem C9.4. Let pi_i:G_i->U(H), i=1,2, be continuous unitary representations whose images preserve the same complete datum. If alpha(pi_1(G_1))=alpha(pi_2(G_2)) as actions on the complete physical datum, then no reconstruction rule using only that datum and its induced physical action can distinguish G_1 from G_2. The canonical reconstructible object is their common effective image, not necessarily either original abstract group, covering group, central extension, or unitary lift. Proof. Every datum-accessible symmetry statement is determined by the induced transformations of observables, states, channels, nets, and operational relations. By hypothesis these transformations coincide. A datum-relative covariant selector therefore receives identical physical input and cannot prefer one presentation without extra structure. QED. Corollary C9.4a. Scalar-phase lifts and any subgroup represented entirely inside K_null are physically invisible to the declared observable action. C9.5. Nested Symmetry-Breaking Theorem Theorem C9.5. If D_2 contains every physical structure in D_1 on the same represented Hilbert space, then G_impl(D_2) is contained in G_impl(D_1). Proof. A unitary preserving the richer datum preserves each component of the poorer datum. QED. Thus G_A contains G_(A,Omega), which contains G_(A,Omega,Phi), which contains G_(A,Omega,Phi,N), which contains the complete G_impl. This filtration identifies which added physical structure breaks which symmetry. C9.6. Benchmark A: full matrix algebra Let H=C^d, A=M_d(C), let Omega be the full state space preserved setwise, and impose no additional symmetry-breaking structure. Then G_impl=U(d). The commutant of M_d(C) consists of scalar matrices, so K_null=U(1)I_d and G_eff=U(d)/U(1)=PU(d). The benchmark removes globally invisible phases and passes. C9.7. Benchmark B: nondegenerate distinguished state Let A=M_d(C) and pointwise preserve rho_star=sum_j p_j |j> is closed and defines a unique positive self-adjoint J_M with ker J_M=intersection_a ker C_tilde_a. The new SYM-3 question is not whether C1 works; it is whether its generator inputs and gauge status are actually recovered or independently certified. C11.8. Master-Constraint Nonuniqueness target Version-24 target, partially proved in Version 25 for a fixed certified generator family and invertible basis changes. A common physical kernel generally does not select a unique master constraint. Every positive-definite M gives the same common kernel for a fixed certified generator family. Invertible changes of generator basis alter the presentation and transform M while preserving the constraint surface. Distinct unreduced gauge lifts may also reduce to the same physical datum. Therefore existence, reconstruction, and uniqueness of J are separate claims. C11.9. Constraint-compatibility class Define Lift_J(D_phys;Sigma) to be the set of gauge-certified tuples (H_pre,G_tilde,U_tilde,g_tilde_0,{C_tilde_a},M,J_M,q,iota,B,Q_chg) that satisfy a declared admissibility specification Sigma and reduce to the complete physical datum. Version 26 distinguishes the quotients C_red=Lift_J/~_red, C_orb=Lift_J/~_orb, and C_J=Lift_J/~_J. The admissibility specification declares the permitted prephysical spaces, gauge groups and extensions, domains, boundary and charge conditions, reduction and physical-inner-product mechanism, and whether auxiliary pure-gauge stabilization is allowed. Empty, singleton, and multiple occupancy are meaningful only after Sigma and the equivalence level are fixed. C11.10. Falsification program Test scalar and central shifts that preserve the physical derivation but change operator representatives. Test multiplicity-space additions from A_phys prime that are physically invisible on the declared algebra. Test disconnected groups with the same identity component and Lie algebra. Test inequivalent covers and central extensions with the same effective infinitesimal action. Test distinct certified generator bases and positive coefficient matrices with the same common kernel. Test boundary transformations whose bulk action is null but whose charge is nonzero. Test gauge-fixed descriptions containing manifest symmetries that do not descend to physical symmetries. C11.11. Required benchmarks Full-matrix PU(d): reconstruct su(d) derivations and classify them as physical, not gauge, on M_d(C). Nondegenerate-state torus: reconstruct the commuting traceless diagonal derivations and test representative ambiguity by scalar additions. Null multiplicity model B(H_A) tensor I_M: demonstrate commutant ambiguity and the failure of nullity to certify gauge. ETP nonfactor model: reconstruct su(2) direct-sum su(2) physical derivations for the ordered effective action and retain the discrete Z_2 swap separately. Existing diagonal SU(2) kinematical benchmark: use its declared unreduced representation as a positive gauge-certified case and contrast it with the trivial effective continuous symmetry of the reduced singlet sector. C11.12. Completion gate and claim boundary Version-24 completion gate. Version 25 completed the effective Lie-algebra, derivation, commutant-ambiguity, no-go, benchmark, and fixed-family master-nonuniqueness components. Version 26 completes the finite-dimensional gauge-lift equivalence and compatibility-class block relative to a declared admissibility specification. SYM-3 is complete within that bounded finite-dimensional, declared-domain scope. Unbounded generators, noncompact rigging-map equivalence, BRST/BFV stable equivalence, field-dependent structure functions, and boundary-charge executions remain extensions. C12. EFFECTIVE LIE-ALGEBRA AND GENERATOR RECONSTRUCTION C12.1. Effective Lie-Algebra Reconstruction Theorem Theorem C12.1. Let G_eff be the compact Lie group reconstructed by C9, acting faithfully and smoothly by *-automorphisms alpha on the finite-dimensional physical C*-algebra A_phys. Then: (i) G_eff^0 determines g_eff=T_e G_eff; (ii) differentiation defines a Lie-algebra homomorphism delta:g_eff->Der_*(A_phys), delta_X(A)=d/dt|_(t=0) alpha_exp(tX)(A); (iii) delta is injective; and (iv) pi_0(G_eff)=G_eff/G_eff^0 must be retained separately. Proof. Differentiating the product and adjoint identities for alpha_exp(tX) gives the Leibniz and * rules, so delta_X is a *-derivation. Differentiation of the group homomorphism gives delta_[X,Y]=[delta_X,delta_Y]. If delta_X=0, the finite-dimensional linear evolution generated by delta_X is the constant automorphism flow. Faithfulness of alpha then gives exp(tX)=e for every t and hence X=0. Finally the Lie algebra sees only the identity component; disconnected components need separate data. QED. C12.2. Center-Fixing Corollary Corollary C12.2. Every delta_X annihilates Z(A_phys). Proof. A derivation maps the center into the center. The connected automorphism flow cannot nontrivially permute the finite set of minimal central projections, and its restriction to each scalar central block is the identity. Differentiation gives delta_X(Z)=0. QED. C12.3. Finite-Dimensional Inner-Derivation Theorem Theorem C12.3. For every X in g_eff there is a Hermitian h_X in A_phys such that delta_X(A)=-i[h_X,A]. Inside the abstract algebra h_X is unique modulo Z(A_phys). Proof. Write A_phys as a finite direct sum of full matrix algebras. C12.2 makes every block invariant, and every derivation of a full matrix algebra is inner. Two implementers of the same derivation differ by an element commuting with the whole algebra, namely a central element. QED. C12.4. Represented Generator Nonuniqueness Theorem Theorem C12.4. Let C_X and C_X prime be Hermitian operators on H_phys implementing the same physical derivation. Then C_X-C_X prime lies in A_phys prime. Conversely, addition of any Hermitian commutant element leaves the derivation unchanged. Thus the represented generator is the coset [C_X]=C_X+(A_phys prime)_sa. Proof. Equality of the commutator derivations gives [C_X-C_X prime,A]=0 for every A in A_phys, which is precisely membership in the commutant. The converse is immediate. QED. C12.5. Conditional Minimum-Norm Representative Theorem C12.5. Relative to a declared faithful finite-dimensional Hilbert-space representation and its Hilbert-Schmidt inner product, every generator coset contains a unique minimum-norm representative C_X perpendicular to (A_phys prime)_sa. Proof. The self-adjoint commutant is a closed real subspace of the finite-dimensional Hilbert space of Hermitian operators, so orthogonal projection gives a unique perpendicular representative. QED. Claim boundary. This representative is canonical only relative to the selected representation and trace. For A_phys=M_d(C), the condition is tracelessness. It is not selected by the abstract physical algebra alone and does not certify gauge status. C12.6. Commutant-Valued Lift-Defect Theorem Theorem C12.6. For Hermitian representatives C_X,C_Y,C_[X,Y], the defect Omega(X,Y)=-i[C_X,C_Y]-C_[X,Y] lies in A_phys prime. Proof. The physical derivations satisfy [delta_X,delta_Y]=delta_[X,Y] exactly. The Jacobi identity shows that the two displayed operator implementations induce the same derivation, so C12.4 places their difference in the commutant. QED. Consequently the physical derivation algebra is anomaly free at the represented observable level, while a chosen Hilbert-space lift may close only modulo an invisible commutant term. Whether that term is harmless, projective, boundary charged, or anomalous in an unreduced gauge lift requires separate certification. C12.7. Physical-Generator Constraint No-Go Theorem Theorem C12.7. If X is nonzero in g_eff, then delta_X is nonzero and no representative C_X is justified as a pure-gauge constraint on the same reduced physical datum. Proof. Injectivity in C12.1 gives delta_X nonzero, so some physical observable is moved. C10 requires proper gauge to act trivially on the complete reduced physical datum. Imposing C_X psi=0 would therefore add an invariant-sector restriction rather than follow from gauge redundancy. QED. This theorem does not prohibit an operator of similar algebraic form from being gauge in a different unreduced realization. Group and Lie-algebra names do not determine physical role; the acted-on space, observable algebra, reduction map, orbit relation, and boundary charges do. C12.8. Executed full-matrix and state-stabilizer benchmarks For A_phys=M_d(C), G_eff=PU(d), g_eff is isomorphic to su(d), and delta_T(A)=-i[T,A] for traceless Hermitian T. The only represented ambiguity is T+cI. Every nonzero derivation is physical, so no nonzero T is gauge certified. For a nondegenerate pointwise-fixed state, G_eff is U(1)^(d-1) and a basis is H_k=|k>}. Its gauge status comes from the declared unreduced orbit construction. On the one-dimensional reduced singlet sector, B(H_phys)=C and the effective continuous Lie algebra is zero. Thus a nontrivial unreduced gauge algebra and master constraint can reduce to a trivial effective Lie algebra. Reduced effective symmetry therefore does not reconstruct the original gauge algebra in general. C12.11. Fixed-Family Master-Constraint Nonuniqueness Theorem Theorem C12.11. Let C_tilde_1,...,C_tilde_N be one gauge-certified family satisfying C1. For every positive-definite Hermitian M, the associated J_M has ker J_M=intersection_a ker C_tilde_a. Hence the common physical kernel does not select a unique M, quadratic form, or master operator. Proof. If m_-(M)>0 and m_+(M) are the extremal eigenvalues, then m_- sum_a ||C_tilde_a psi||^2 <= q_M[psi] <= m_+ sum_a ||C_tilde_a psi||^2. Therefore q_M[psi]=0 exactly when every C_tilde_a psi=0. C1 identifies the zero set of the closed form with ker J_M. QED. C12.12. Generator-Basis Covariance Proposition C12.12. For an invertible basis change C_tilde prime=S C_tilde, the common kernel is unchanged. With M prime=(S^(-1))^dagger M S^(-1), the quadratic form itself is unchanged. Therefore generator basis and coefficient matrix are presentation dependent even when the master form is identical. C12.13. SYM-3 status after Version 27 synchronization Completed in Version 25: g_eff reconstruction and the injective physical derivation representation. Completed in Version 25: separation of connected Lie data from pi_0(G_eff). Completed in Version 25: inner implementation, center ambiguity, represented commutant ambiguity, commutant-valued lift defect, and the physical-generator constraint no-go theorem. Completed conditionally in Version 25: the Hilbert-Schmidt minimum-norm representative relative to a declared faithful representation and trace. Completed in Version 25: five finite-dimensional benchmarks, including the diagonal SU(2) separation test, plus fixed-family master-metric nonuniqueness and generator-basis covariance. Completed in Version 26: formal reduced, gauge-orbit, and strong master equivalence; compatibility-class covariance; occupancy trichotomy; explicit inequivalent lifts; and empty, singleton, and multiple certification gates. Finite-dimensional SYM-3 status: complete within the bounded, declared-domain scope. Open extensions: unbounded-generator domains, noncompact rigging-map equivalence, BRST/BFV stable equivalence, field-dependent structure functions, infinite constraint families, and continuum or ETP-specific boundary realizations. Bounded finite-dimensional boundary compatibility is completed by C15. New research frontier, not a retroactive completion requirement: test whether added physical structure selects a distinguished compatibility-class representative or master metric. Conclusion. The canonical finite-dimensional SYM-3 output is the reduced symmetry data (g_eff,delta,pi_0) together with the declared compatibility hierarchy (C_red,C_orb,C_J), not a unique unreduced gauge algebra or master constraint. C13 completes gauge-lift equivalence, covariance, occupancy, and the explicit inequivalent-lift benchmark within the bounded finite-dimensional, declared-domain scope. C13. GAUGE-LIFT EQUIVALENCE AND CONSTRAINT-COMPATIBILITY CLASSIFICATION Truth-first scope. Reduced physical data can erase gauge-orbit, representation, domain, boundary, and master-metric information. Version 26 therefore does not ask for a unique unreduced gauge theory unless uniqueness is proved relative to a declared admissibility problem and an explicit equivalence relation. C13.1. Admissibility specification Definition C13.1. Let Sigma=(T,G,D,B,Q_chg,R_red,S) declare the permitted prephysical Hilbert-space class T; gauge groups, Lie algebras, covers, and extensions G; common-domain and self-adjointness requirements D; boundary, edge, and asymptotic conditions B; charge assignments Q_chg; reduction and physical-inner-product mechanism R_red; and whether auxiliary pure-gauge stabilization is allowed S. Let Lift_J(D_phys;Sigma) be the set of all gauge-certified master-constraint lifts satisfying Sigma and descending to the complete physical datum D_phys. Occupancy is not defined without Sigma. C13.2. Three equivalence levels Definition C13.2. For certified lifts L_1 and L_2, write L_1 ~_red L_2 when a unitary physical equivalence V transports the complete reduced datum, including physical inner product, descended observables, states, channels, nets, operational relations, and admitted boundary data. Definition C13.3. Write L_1 ~_orb L_2 when L_1 ~_red L_2 and there exist a unitary W between the prephysical Hilbert spaces and a continuous gauge-group isomorphism phi such that W U_1(g) W^dagger=U_2(phi(g)), q_2 W=V q_1 on the declared reduction domain, and the complete orbit relation, domains, zero-charge condition, boundary data, and descended observable representation are transported consistently. Definition C13.4. Write L_1 ~_J L_2 when L_1 ~_orb L_2 and W J_1 W^dagger=J_2, equivalently when the closed master quadratic forms are transported exactly on their form domains. C13.3. Equivalence-Hierarchy Proposition Proposition C13.5. Strong master equivalence implies gauge-orbit equivalence, which implies reduced physical equivalence: ~_J implies ~_orb implies ~_red. Both converses fail in general. Proof. The forward implications follow from the definitions. Reduced equivalence does not imply orbit equivalence because different unreduced gauge representations can descend to the same complete physical datum. Orbit equivalence does not imply strong master equivalence because one certified generator family admits positive master metrics with the same common kernel and orbit structure but generally different positive spectra. QED. C13.4. Compatibility hierarchy Definition C13.6. Define C_red(D_phys;Sigma)=Lift_J(D_phys;Sigma)/~_red, C_orb(D_phys;Sigma)=Lift_J(D_phys;Sigma)/~_orb, and C_J(D_phys;Sigma)=Lift_J(D_phys;Sigma)/~_J. These classes answer distinct questions: equality of reduced physics, equality of unreduced gauge-orbit realization, and equality of the master operator. A singleton at one level does not imply a singleton at a stronger level. C13.5. Gauge-Lift Equivalence and Descent-Invariance Theorem Theorem C13.7. Let L_1 and L_2 be gauge-certified lifts satisfying C1 and the SYM-2 zero-boundary-action and zero-charge conditions. If L_1 ~_orb L_2, then their gauge orbit spaces are isomorphic, their invariant physical sectors are unitarily equivalent, their descended physical observable algebras are *-isomorphic, and their physical inner products and complete reduced data agree under V. If W transports the certified constraint family by an invertible matrix S, then W ker J_1=ker J_2. Proof. The intertwining identity maps each U_1 orbit bijectively to a U_2 orbit. It maps invariant vectors to invariant vectors because phi is onto. The relation q_2 W=V q_1 identifies the reduced states, while transport of the physical inner product and descended observable representation identifies the complete physical datum. If W C_1 W^dagger=S C_2 with S invertible, simultaneous vanishing of one family is equivalent to simultaneous vanishing of the other. C1 then identifies the common zero sets with ker J_1 and ker J_2. QED. Claim boundary. Kernel transport does not imply equality or unitary equivalence of J_1 and J_2 unless strong master equivalence is separately established. C13.6. Fixed-Family Master-Metric Classification Theorem Theorem C13.8. Let C_tilde=(C_tilde_1,...,C_tilde_N)^T be a fixed linearly independent gauge-certified family satisfying C1. Every positive-definite Hermitian M defines a positive self-adjoint J_M, and all such J_M have the same kernel. The simultaneous change C_tilde prime=S C_tilde and M prime=(S^(-1))^dagger M S^(-1) preserves the quadratic form exactly. Varying M while holding C_tilde fixed generally changes the form and nonzero spectrum. Therefore constraint-content recovery may be unique while exact master-operator recovery remains nonunique. Proof. Existence and the common-kernel statement follow from C1 and C12.11. Direct substitution proves simultaneous-basis covariance. For one nonzero certified generator C and m>0, J_m=m C^2. The kernel is independent of m, while every nonzero eigenvalue is scaled by m. Thus exact master operators remain distinct unless the declared equivalence explicitly quotients positive rescaling. QED. C13.7. Explicit inequivalent-lift benchmark Let K be any nonzero finite-dimensional physical Hilbert space with physical algebra B(K). Lift A has H_A=K tensor C^2, C_A=I_K tensor diag(0,1), J_A=C_A^2, U_A(t)=exp(i t C_A), and q_A(psi_0 tensor |0>+psi_1 tensor |1>)=psi_0. Lift B has H_B=K tensor C^3, C_B=I_K tensor diag(0,1,2), J_B=C_B^2, U_B(t)=exp(i t C_B), and q_B(sum_j psi_j tensor |j>)=psi_0. In both cases q_i U_i(t)=q_i, ker J_i is canonically K, and X in B(K) descends from X tensor I. Theorem C13.9. Lifts A and B reduce to the same complete physical datum but are not strictly gauge-orbit equivalent. Proof. Their physical kernels, descended observables, states, channels, nets, and operational relations can be identified through K. However dim H_A=2 dim K and dim H_B=3 dim K, so no unitary W:H_A->H_B exists. Therefore neither strict gauge-orbit equivalence nor strong master equivalence holds. QED. Interpretation. This benchmark proves genuine finite-dimensional multiplicity of unreduced lift presentations for one reduced datum whenever Sigma admits both constructions and does not quotient auxiliary stabilization. C13.8. Strict versus stable gauge equivalence Definition C13.10. Strict equivalence preserves the complete unreduced representation as in C13.3. Stable equivalence may additionally identify L with L tensor T for a separately certified contractible pure-gauge auxiliary sector T whose reduction is physically trivial, descended observable action is scalar, boundary and asymptotic charges vanish, and physical inner product is unchanged. Version 26 uses strict equivalence as the primary classification. Stable equivalence is an optional further quotient and is not inferred from physical triviality alone. C13.9. Occupancy trichotomy Definition C13.11. For fixed (D_phys,Sigma,~), let nu_~(D_phys;Sigma)=|Lift_J(D_phys;Sigma)/~|. Class E0 is empty when nu=0; Class E1 is singleton when nu=1; and Class E>1 is multiple when nu>1. Theorem C13.12. Exactly one occupancy class holds for every declared (D_phys,Sigma,~). Proof. The quotient is a set whose cardinality is zero, one, or greater than one. These alternatives are exhaustive and mutually exclusive. QED. C13.10. Reduced-Data Occupancy Underdetermination Theorem Theorem C13.13. The complete reduced datum D_phys alone does not generally determine whether the gauge-lift compatibility class is empty, singleton, or multiple. Proof. Multiple occupancy is realized by the inequivalent lifts in C13.7. Singleton occupancy results when Sigma admits one strict lift class and quotients only declared basis presentations at the constraint-content level. Empty occupancy results, for example, if Sigma simultaneously requires H_pre=H_phys, q=I, a nontrivial proper-gauge action, and orbit invariance q U(g)=q: then U(g)=I, contradicting nontriviality. Hence occupancy belongs to the declared reconstruction problem (D_phys,Sigma,~), not to D_phys alone. QED. C13.11. Covariance of the Compatibility Hierarchy Theorem C13.14. Let V transport the complete physical datum D_phys to D_phys prime, and let Sigma prime be the transported admissibility specification. Composition q prime=V q induces bijections of C_red, C_orb, and C_J with their primed counterparts. Occupancy status is therefore invariant under complete physical equivalence. Proof. Transport each descended physical component by V and leave, or covariantly transport, the unreduced data as required by Sigma prime. V^(-1) constructs the inverse map. Because all three equivalence relations were defined by covariant transport, the map descends to bijections of quotient classes. QED. C13.12. Empty, singleton, and multiple certification gates Empty-class witnesses include failure of orbit descent qU=q, nonzero admitted boundary or asymptotic charge, anomalous closure, absence of a dense common form domain, failure of self-adjoint or closable implementation, failure of a positive physical inner product or rigging map, failure of observable descent, or failure of the C1 master-form hypotheses. A formal kernel alone is insufficient. Singleton recovery requires existence plus a proof that every admitted lift is equivalent at the declared level. At the orbit level this requires uniqueness of the unreduced orbit realization under Sigma; at the strong master level it additionally requires uniqueness of the transported quadratic form. Finding one convenient lift does not prove uniqueness. Multiple occupancy is certified by two admitted lifts separated by an invariant preserved under the selected equivalence, including prephysical dimension, gauge-group isomorphism class, representation multiplicity, orbit-stabilizer type, boundary-charge representation, self-adjoint extension class, superselection structure, or master-operator spectrum. C13.13. SYM-3B completion and scope Conclusion. For a declared finite-dimensional admissibility specification, gauge-lift equivalence and constraint compatibility form a hierarchy rather than one unqualified class. Reduced data canonically reconstruct (g_eff,delta,pi_0) but do not determine an unreduced gauge orbit or exact master operator. The finite-dimensional SYM-3 program is complete at the level of covariant compatibility-class classification, with explicit null, unique-conditional, and multiple outcomes permitted. No unique ETP gauge group or master constraint is thereby claimed. Open extensions. Unbounded-generator domain equivalence, noncompact group averaging and rigging-map equivalence, BRST/BFV stable equivalence, field-dependent structure functions, infinite constraint families, and explicit boundary or edge-charge executions remain outside the completed finite-dimensional theorem. C14. SYM-3C.1 OBSERVER-TEMPORAL DESCENT INVARIANCE AND ENRICHED GAUGE-LIFT SEPARATION Truth-first scope. This stage asks whether observer-access, record, Fisher, clock, comparison, or holonomy structure can distinguish gauge lifts that already induce equivalent complete reduced physics. It does not assume that operational restriction reveals an unreduced presentation. Its valid outputs are descent invariance, an explicit enriched separation, or a null result in which no admitted physical experiment distinguishes the lifts. BNP theorem-family label. Status: proved only within the bounded finite-dimensional, complete-reduced-datum, declared-protocol domain stated in C14. The family establishes covariance, no-selection from covariantly derived data, and an operational separation criterion. It does not establish global physical equivalence, witness completeness for every enrichment, or ontological identity of unreduced presentations. BNP model and carrier scope. The governing carrier consists of the reduced physical Hilbert space and inner product, represented physical algebra, admitted states, active channels, physical nets, operational relations, boundary and charge data, observer-access algebras, readable and recoverable structures, measurement classes, record channels, clock domains, comparison protocols, and holonomy data actually included in D(L) and OT(D). Unadmitted off-shell, boundary, reference, resource, control, or discarded-output data are outside the proved carrier. BNP equivalence and operation status. The equivalence used in C14 is reduced physical equivalence transported by V, not complete basement-level E_phys unless the complete-model constitution is independently satisfied. Covariant generation, restriction, channel pullback, exact recovery, Fisher evaluation, clock classification, comparison factorization, and chart conjugation are certified only under the hypotheses stated in the cited component theorems. BNP witness status. A positive separating protocol is sound only when the enrichment is independently physical, gauge invariant, admitted by the enriched model, and produces different corresponding outcome laws. Equality of every tested outcome establishes RESTRICTED-NULL relative to the tested admitted protocol family unless that family is proved complete for the governing enriched distinction. Without such completeness, the global relation is UNDETERMINED. BNP falsifiers and extension boundary. The family is restricted or falsified by noncovariant constructions, undeclared lift labels, gauge-dependent statistics, failure to transport operational choices, incomplete carrier specification, or a clock that fails the existing adequacy gates. Infinite-dimensional normal algebras, unbounded domains, noncompact rigging maps, BRST/BFV stable equivalence, structure functions, and unexecuted boundary or edge-charge models remain outside this certification. C14.1. Observer-temporal enrichment datum Definition C14.1. For a gauge-certified lift L, let D(L) be its complete reduced datum, including the physical Hilbert space and inner product, represented physical observable algebra, admissible states, active channels, physical nets, operational relations, and admitted boundary and charge data. Let OT(D) denote each listed observer-temporal structure that is defined covariantly from D and for which the governing component theorem applies: observer-access algebras; exact stable-readable algebras; exactly recoverable algebras for a fixed selected channel, output carrier, and support; admitted measurement classes; interaction-derived record channels; accessible Fisher information; Fisher length; endpoint completion; calibrated clock charts; comparison kernels; quantum degrading channels; and loop-holonomy data. The parameter domain, parameter identification, regularity convention, selected output, support, measurement class, clock domain, comparison graph, edge orientation, composition law, and permitted relabelings are part of the transported datum whenever the corresponding construction uses them. The constructor OT is not assumed to create physical information. It organizes structures already contained in, or covariantly derived from, the complete reduced datum. Choices of readable closure, recovery protocol, measurement class, clock domain, and comparison protocol must be declared and transported as part of the construction. C14.2. Proper-Gauge Observer-Stability Corollary Corollary C14.2. Let U_tilde(g) be a certified proper-gauge transformation with q(U_tilde(g)Psi)=q(Psi) on the admitted reduction domain and with zero admitted boundary or asymptotic charge. Then every descended physical observable, observer-access algebra, and admitted reduced record statistic is constant on the corresponding gauge orbit. Consequently proper gauge cannot change accessible outcome probabilities, record order, accessible Fisher information, Fisher duration, calibrated clock readings, comparison-transducer statistics, or physical holonomy. Proof. Proper-gauge orbit invariance identifies q(U_tilde(g)Psi) with q(Psi). Descended observables and all admitted reduced states, channels, nets, and operational relations therefore have identical physical representatives. Observer-temporal quantities are functions of those transported physical structures and their admitted protocols, so they are unchanged. QED. Claim boundary. The converse is not asserted. Operational nullity does not certify gauge status; C10.4 still requires an unreduced orbit interpretation and the declared zero-charge conditions. C14.3. Observer-Temporal Descent-Invariance Theorem Map signature for C14.3. Let V:H_phys,1 -> H_phys,2 be the unitary implementing the declared reduced physical equivalence, and let Ad_V:A_phys,1 -> A_phys,2 be the induced *-isomorphism Ad_V(X)=VXV^dagger. State transport is rho -> V rho V^dagger. Every channel, output, effect, measurement, recovery, clock, comparison, and holonomy datum invoked below is transported through these declared carrier maps and the corresponding dual maps. Theorem C14.3. Let L_1 and L_2 be gauge-certified lifts satisfying the same declared admissibility problem, and suppose L_1 ~_red L_2 through a physical unitary equivalence V transporting the complete reduced datum. Assume each invoked component of OT is defined, constructed covariantly, and accompanied by transport of its full operative datum. This includes observer seeds or their physical generating construction; the complete H12 selector datum; the selected channel, input and output carriers, support projector, Kraus-product span, protected-algebra convention, and recovery class for H22.2; the parameter domain and identification, measurement class, path regularity, support and endpoint conventions, injectivity or periodicity domain, calibration and tolerance data for Fisher and clock results; and the comparison graph, edge orientation, domains, composition law, loops, and chart convention for holonomy. Then each such component of OT(D(L_2)) is the V-transport of the corresponding component of OT(D(L_1)), up to the admitted physical isomorphisms, observer relabelings, channel-output isometries, and clock-chart conjugations. In particular: (i) corresponding observer, exact stable-readable, and recoverable algebras are physically isomorphic; (ii) transported state-measurement outcome distributions agree; (iii) accessible Fisher information, Fisher length, endpoint admissibility, and clock-adequacy class agree on transported domains; (iv) existence of classical comparison kernels and quantum degrading channels is preserved; and (v) loop holonomy is preserved up to the previously established conjugacy relation. Proof. Reduced equivalence transports the physical algebra, inner product, states, channels, nets, operational relations, boundary data, and charges. H1 transports a generated observer algebra only when the seed system, or the physical construction producing it, is transported. H12 transports the exact stable-readable intersection only under transport of the complete selector datum D_O, including the physical algebra, active Heisenberg maps, output channel, effect system, readable closure, and permitted label identifications. H22.2 transports the compressed Kraus-product algebra and its commutant only after the selected output, input and output carriers, support projector, complete Kraus-product span, protected-algebra convention, and admissible recovery class are transported. Simultaneous transport of the declared statistical path and measurement class preserves admitted probabilities and accessible Fisher information. The relevant Fisher, endpoint, and clock results then preserve Fisher length, endpoint admissibility, and clock class under transport of parameter domains, regularity and support conventions, injectivity or periodicity domains, calibration data, and tolerances. Comparison kernels and degrading channels satisfy the transported factorization identities. Graph-level transport of the comparison network, including incidence, orientation, composition, loops, and chart conventions, preserves holonomy up to the established conjugacy relation. QED. C14.4. Derived-Enrichment No-Selection Theorem Theorem C14.4. Under the hypotheses of C14.3, no covariant selector whose input is only OT(D) can choose L_1 over L_2 as a distinguished gauge lift or master-constraint representative. Proof. By C14.3 the selector receives physically equivalent inputs. Covariance requires equivalent outputs. Preferring one unreduced presentation would therefore introduce structure not present in the declared input datum. QED. Interpretation. Observer restriction does not expose information already erased by reduction. Operational indistinguishability is a statement about the admitted physical theory; it does not prove ontological identity of the unreduced presentations. C14.5. Independently Enriched Separation Criterion Definition C14.5. An enrichment Enr is independently physical when it is not reconstructed solely from the common reduced datum and when its observables or protocols are gauge invariant under the certified proper-gauge action. Examples may include a prephysical interaction coupling, material reference system, admitted boundary or edge algebra, nonzero asymptotic charge, self-adjoint domain or extension datum, physically admitted off-shell probe, or resource and disturbance ledger for a reference protocol. Theorem C14.5. Relative to the declared enriched operational experiment, reduced-equivalent lifts L_1 and L_2 are operationally separated by an admitted enrichment Enr exactly when there exists a gauge-invariant physical protocol P in the enriched theory and an outcome event X such that p_(L_1,Enr)(X|P) differs from p_(L_2,Enr)(X|P). If no such admitted protocol exists, the lifts remain operationally equivalent relative to (D,Enr). Proof. A difference in an admitted physical outcome distribution is an operational witness separating the enriched experiments. Conversely, if every admitted enriched protocol has the same outcome law, no operational test in the declared theory distinguishes the lifts. QED. Claim boundary. Formal differences in prephysical dimension, coordinates, generator names, or unprobed spectra are not separation witnesses. The enrichment must be independently justified, gauge invariant, and connected to an admitted physical statistic. This biconditional characterizes separation only in the declared operational experiment; it establishes complete physical identity only if the admitted witness family is independently proved complete for the governing distinction. C14.6. Auxiliary-Lift Benchmark Apply C14.3 to the C13.7 lifts with prephysical spaces K tensor C^2 and K tensor C^3. They are not strictly gauge-orbit equivalent because their prephysical dimensions differ, but both reduce canonically to K with physical algebra B(K). For any shared reduced states, channels, observer algebras, measurement classes, record protocols, and clock tasks, all components of OT agree under the canonical physical identification. The benchmark therefore passes the no-selection test: unequal auxiliary dimension is not observable from the exact reduced theory. C14.7. Master-Metric On-Shell Invisibility Theorem Theorem C14.7. Let a fixed gauge-certified generator family define master operators J_M for positive-definite metrics M, all with the same physical kernel. If the complete observer-temporal experiment is confined to that kernel and all admitted physical states, observables, channels, and protocols are independent of the nonzero spectrum of J_M, then OT(D) is independent of M. Hence on-shell relational-time data cannot select a distinguished master metric. Proof. C12.11 and C13.8 give the common kernel. By hypothesis every admitted observer-temporal input is the same restriction to that kernel and contains no dependence on the complementary spectrum. C14.3 then gives identical observer-temporal outputs. QED. C14.8. Off-Shell and Boundary Enrichment Warning A master metric or gauge-lift representative may become distinguishable only when the declared physical theory admits additional structure that responds to it, such as leakage from the physical kernel, finite constraint penalties, off-shell spectral gaps, boundary charges, domain-sensitive scattering, or a material reference coupling. Such a result belongs to the enriched datum and must not be retroactively attributed to the reduced datum alone. If the proposed probe is not gauge invariant or is merely a coordinate convention, it does not separate the physical theories. C14.9. Relational-Reference and Clock Audit A prephysical gauge parameter, frame coordinate, or convenient deparameterization variable is not physical duration by designation. To enter OT(D) as a clock, it must be represented by admitted gauge-invariant relational observables or by an independently physical reference system, and it must pass the existing clock gates: operational identifiability, record-order compatibility, accessible measurement support, global or chartwise injectivity, and the G12 Fisher-speed criterion on the claimed domain. Periodicity, recurrence, finite localization, backreaction, and reference degradation remain possible failure modes. C14.10. Falsification gates and completion status The stage is falsified or restricted if any claimed observer-temporal construction is not covariant under the declared reduced equivalence; if an alleged selector uses undeclared lift labels or formal off-shell data; if a proposed enrichment is not gauge invariant; if the separating statistic depends on gauge choice; or if a clock coordinate fails the existing adequacy audit. Positive separation requires one explicit admitted protocol with distinct gauge-invariant outcome laws. Failure to find a separating witness is a null result only relative to the declared admitted protocol family. Complete physical equivalence follows only if that family is proved complete for the governing physical distinction; otherwise the complete physical relation remains undetermined. Conclusion. SYM-3C.1 is complete within the bounded finite-dimensional, complete-reduced-datum, declared-protocol scope. It proves observer-temporal descent invariance and the failure of purely derived observer-temporal data to select among reduced-equivalent gauge lifts or on-shell-equivalent master metrics. It also supplies the necessary-and-sufficient operational criterion for separation by independently justified enrichment. Infinite-dimensional normal algebras, unbounded domains, noncompact rigging maps, BRST/BFV stable equivalence, structure functions, and boundary or edge-charge execution is continued by C15 in bounded finite dimension. The next principal symmetry stage after SYM-3C.2 is SYM-4 sector and commutant decomposition. C15. SYM-3C.2 BOUNDARY-CHARGE, EDGE-PRESENTATION, AND INTERFACE SEPARATION Truth-first scope. A boundary does not automatically create a physical edge degree of freedom. This stage asks whether independently justified boundary conditions, observables, charges, channels, or gluing relations distinguish lifts that agree on the bulk-reduced datum. It remains neutral between extended edge-variable descriptions and sectorwise or relational encodings. Valid outputs include physical boundary separation, equivalent edge presentations, unresolved null action, nonunique completion, or failure of interface composability. C15.1. Boundary-enriched datum and physical image Definition C15.1. For a gauge-certified lift L, let D_boundary(L)=(D_bulk,A_boundary,Omega_boundary,G_boundary,K_boundary,Q_chg_boundary,Z_boundary,Phi_boundary,Meas_boundary,I_boundary,G_glue), where D_bulk is the complete bulk-reduced datum; A_boundary is the admitted boundary, corner, or interface observable algebra; Omega_boundary is the admitted state family; G_boundary preserves the enriched datum; K_boundary acts trivially on all admitted boundary-enriched observables and protocols; Q_chg_boundary is the declared charge family; Z_boundary=Z(A_boundary); Phi_boundary is the admitted boundary-channel family; Meas_boundary is the admitted measurement class; I_boundary records orientation, incidence, and composability; and G_glue specifies physical composition of adjacent regions. A chosen extended presentation A_boundary^ext must be distinguished from its physical image A_boundary^phys. A difference confined to auxiliary variables outside the physical image is not a physical separation witness. No naive regional Hilbert-space tensor factorization is assumed; centers and charge sectors are permitted. C15.2. Boundary-Enriched Descent and Ordered Classification Theorem Theorem C15.2. Apply the following ordered decision rule to an unreduced transformation g. First, if g acts nontrivially on A_boundary^phys, changes an admitted enriched protocol, or carries nonzero admitted boundary or asymptotic charge, classify it as a physical boundary symmetry. Otherwise, if g is independently certified as an unreduced orbit redundancy and acts trivially on the complete boundary-enriched datum, classify it as proper boundary gauge. Otherwise, if g changes only an auxiliary edge or frame presentation while preserving the complete physical boundary experiment and gluing predictions, classify it as a presentation transformation. All remaining enriched-null cases are unresolved boundary-null transformations. The four outputs are exhaustive and mutually exclusive under this ordered convention. Proof. Nontrivial boundary action or charge changes the enriched physical datum and is excluded from pure gauge by C10.6. If no such action exists and orbit redundancy is independently certified, C10.5 supplies proper-gauge status. If orbit certification is absent but only auxiliary presentation data change while the full experiment is preserved, the distinction is presentational. C10.4 prohibits promoting any remaining null action to gauge. QED. C15.3. Boundary-Completion Nonuniqueness and Occupancy Theorem Definition C15.3. Let Sigma_boundary=(B_geom,B_cond,Q,E,P,G) declare the admitted boundary geometries, boundary conditions, charge assignments, edge presentations, physical protocols, and gluing rules. Let Comp_boundary(D_bulk;Sigma_boundary) be the admitted boundary completions, quotiented by a declared boundary equivalence relation. Theorem C15.3. The complete bulk-reduced datum does not generally determine a unique boundary algebra, center, charge representation, edge extension, regional factorization, boundary condition, or gluing prescription. Relative to fixed Sigma_boundary and a fixed equivalence relation, the quotient Comp_boundary is empty, singleton, or multiple. Proof. Distinct admitted regional algebras may have different centers or presentations, and different extended or sectorwise constructions may encode the same physical interface experiment. Thus bulk data alone do not fix one completion. Once the admissibility specification and equivalence relation are fixed, the quotient is a set and its occupancy is zero, one, or greater than one. QED. C15.4. Boundary-Experiment Equivalence and Separation Theorem Definition C15.4. The complete boundary experiment E_boundary(L) is the family of all admitted probabilities p_L(x|rho,P), with rho in Omega_boundary and P an admitted gauge-invariant boundary protocol. Boundary-equivalence map signature. Let beta:A_boundary,1^phys -> A_boundary,2^phys be the declared unital *-isomorphism. Effects are transported by beta, and states are transported by the dual rule omega_2(beta(X))=omega_1(X). For each Heisenberg-picture boundary channel Phi_boundary,i^dagger on its physical boundary algebra, channel transport means beta o Phi_boundary,1^dagger = Phi_boundary,2^dagger o beta; equivalently, the Schrödinger preduals intertwine by the dual state transport. Theorem C15.4. Two bulk-reduced-equivalent lifts are operationally boundary equivalent when a physical *-isomorphism beta between their physical boundary algebras transports admitted states, charges, center sectors, channels, measurements, incidence relations, and gluing protocols and preserves every corresponding outcome distribution. They are operationally boundary separated exactly when an admitted gauge-invariant boundary protocol has different corresponding outcome distributions. Proof. Under complete transport, simultaneous transport of states and effects preserves Born probabilities. Conversely, a differing admitted outcome law is an operational witness that no complete experiment-preserving equivalence exists. QED. C15.5. Boundary-Charge Sector Theorem Theorem C15.5. Let A_boundary be finite dimensional with block form direct-sum_q(M_(n_q)(C) tensor I_(m_q)) and minimal central projectors P_q. Then: (i) the P_q define boundary superselection sectors exactly when every admitted boundary observable and channel preserves their blocks; (ii) Q_chg_boundary=sum_q qP_q is physically distinguishing only when admitted states have distinguishable sector weights or admitted protocols respond differently to q; (iii) matching charge spectra alone is insufficient for boundary equivalence; (iv) different charge spectra or measurable sector statistics suffice for separation; and (v) sectors never populated or addressed by any admitted protocol are not operationally distinguished. Proof. Central projectors label invariant blocks exactly under universal preservation by the admitted operations. Charge statistics are p(q)=Tr(rho P_q). Equality of formal spectra without transport of states, channels, and protocols does not equate experiments; a different admitted distribution or charge-conditioned response is measurable. QED. C15.6. Edge-Presentation Neutrality Theorem Theorem C15.6. Let two extended boundary constructions admit surjective physical reduction maps r_i:A_boundary,i^ext -> A_boundary^phys. If they have the same physical image, their kernels contain only auxiliary or gauge-null presentation variables, every admitted state and protocol factors through r_i, and their gluing predictions agree, then the constructions are physically edge-presentation equivalent even when their extended Hilbert spaces, frame variables, or raw edge algebras differ. Proof. Every admitted probability factors through the common physical image. Kernel differences cannot change an admitted statistic, and agreement of gluing predictions removes interface-level separation. QED. This theorem does not assert that all proposed edge extensions satisfy its hypotheses. C15.7. Interface Gluing Compatibility Theorem Theorem C15.7. Let adjacent regions R and Rbar have boundary charge projectors {P_q^R} and {P_qbar^Rbar}, and let mu encode the declared orientation-reversal or matching rule. Define P_glue=sum_q P_q^R tensor P_(mu(q))^Rbar. If physical composition requires this charge matching, then H_glue=P_glue(H_R tensor H_Rbar) is the largest subspace satisfying the declared central matching condition. Let H_comp be the declared physical composite input carrier after the regional interface and charge data have been fixed. In this theorem a gluing map is specifically an isometry G_glue:H_comp -> H_glue, so G_glue^dagger G_glue=I_Hcomp and Ran(G_glue) is contained in H_glue. It is admissible only if the induced *-monomorphism X -> G_glue X G_glue^dagger preserves the admitted interface observables on its range, respects orientation and incidence, and intertwines the certified proper-boundary-gauge actions. The isometric class preserves the declared physical inner product and state normalization. CPTP interface channels, partial isometries with smaller initial support, and algebraic quotient maps are different gluing classes and are not asserted by this theorem. Proof. The range of P_glue contains exactly the matched sector pairs. Support outside this range violates the declared matching condition. The remaining requirements are precisely descent, composability, and physical-normalization gates. QED. No universal charge-matching law is claimed beyond the declared interface rule. C15.8. Boundary-Record Recoverability Corollary Corollary C15.8. For an admitted finite-dimensional boundary-record channel R_boundary with Kraus operators K_i and supported-sector projector P, define N_boundary=C*({PK_i^dagger K_jP}_(i,j)). The unique largest boundary observable algebra exactly recoverable by one CPTP recovery channel is A_boundary^rec=N_boundary prime intersection B(PH). A proposed edge observable is operationally recoverable only if it lies in this algebra and is admitted by the boundary measurement protocol. Proof. Apply H22.2 to the boundary-record channel. QED. Recoverability establishes operational reconstruction, not a fundamental ontology for the edge variable. C15.9. Finite-dimensional benchmark suite Benchmark A, boundary-null. Two extended presentations may have different auxiliary edge spaces while both physical boundary algebras are C I. All admitted boundary probabilities are constant, so the lifts remain boundary equivalent. Benchmark B, positive central-charge separation. Let A_boundary,1=C direct-sum C with P_plus,P_minus and Q_1=P_plus-P_minus, while A_boundary,2=C I. If distinct sector distributions and the charge measurement are admitted, theory 1 has nontrivial charge statistics unavailable in theory 2; the lifts are boundary separated. Benchmark C, equal spectrum but different dynamics. Let both algebras be C direct-sum C with the same declared charge spectrum. If Phi_1^dagger(P_plus/minus)=P_plus/minus while Phi_2^dagger swaps P_plus and P_minus, and the signs or orientation labels are physically fixed, the complete boundary experiments differ. Equal spectra do not imply equivalence. Benchmark D, edge-presentation equivalence. Different extended frame spaces that reduce to the same center span{P_q}, with all admitted effects, states, and gluing maps factoring through the same sector statistics, satisfy C15.6. Benchmark E, gluing obstruction. For q in {0,1} and matching q=qbar, P_glue=P_0^R tensor P_0^Rbar+P_1^R tensor P_1^Rbar. The mismatched 0 tensor 1 and 1 tensor 0 sectors are noncomposable under the declared rule. Regional admissibility does not imply interface composability. C15.10. Relational-time consequence and audit gates Boundary enrichment affects relational time only through the chain: boundary structure -> gauge-invariant boundary record -> recoverable distinction -> record order -> accessible Fisher geometry -> clock or comparison structure. A charge is not duration, an edge mode is not automatically a clock, and a gluing constraint is not temporal synchronization. Every downstream temporal claim remains subject to H22, G12-G18, and H23. The stage fails or is restricted if the enrichment is chosen to favor a desired lift; if a separating variable is gauge dependent; if an auxiliary edge coordinate is treated as observable without a physical reduction map; if regional tensor factorization is assumed where only an algebra with center is justified; if charge spectra are compared without states and protocols; if a gluing rule is undeclared; or if recoverability is confused with clock adequacy. Conclusion. SYM-3C.2 is complete in the bounded finite-dimensional algebraic, declared-boundary, declared-protocol scope. It proves a boundary compatibility hierarchy with explicit null, equivalent-presentation, charge-separated, dynamics-separated, and gluing-obstructed outcomes. It does not derive an ETP-specific boundary or corner algebra, charge, boundary condition, or interface dynamics. Continuum non-Abelian boundaries, unbounded charges, normal type-II/type-III algebras, BRST/BFV boundary reduction, gravitational corners, asymptotic or soft sectors, and a boundary-record clock execution remain open. The next principal symmetry stage is SYM-4. C15.11. Definitive SYM-3 completion statement Completion statement. The bounded finite-dimensional SYM-3 program is complete through SYM-3C.2. Its completed scope includes effective Lie-algebra and physical-derivation reconstruction; separation of connected and disconnected symmetry data; generator-representative ambiguity modulo the commutant; the physical-generator constraint no-go result; gauge-lift certification and the reduced, orbit, and strong-master compatibility hierarchy; observer-temporal descent invariance and the no-selection theorem for reduced-equivalent lifts; boundary-completion nonuniqueness; edge-presentation neutrality; central charge-sector classification; interface-gluing compatibility; and boundary-record recoverability. No additional missing conceptual branch of SYM-3 is presently required within that declared scope. Scope boundary. This completion is not a claim that every conceivable constrained-system problem has been solved. Unbounded generators, noncompact rigging-map equivalence, BRST/BFV stable equivalence, field-dependent structure functions, infinite constraint families, normal type-II/type-III algebras, continuum non-Abelian boundaries, asymptotic or soft sectors, gravitational corners, and explicit ETP-specific boundary or interface models remain extensions or applications of the completed framework. They do not reopen the finite-dimensional SYM-3 theorem architecture unless an execution exposes a failed premise or a genuinely missing logical case. Transition. The symmetry program freezes SYM-3 at Version 31 and proceeds to SYM-4. Sections C16.1-C16.4 now execute the sector, commutant, observer-compatibility, Fisher, clock, and comparison program in the declared bounded finite-dimensional scope. C16. SYM-4 SECTOR AND COMMUTANT DECOMPOSITION Role in emergence. Constraint reduction and symmetry reconstruction determine a physical arena, but relational time requires a sharper question: which sector distinctions survive jointly in the observable, observer, record, and recovery structures from which temporal information is extracted? SYM-4 answers this without identifying representation labels with physical time. It separates sector existence, observable compatibility, operational readability, record formation, and recoverability before any Fisher or clock claim is made. C16.1. SYM-4A: Implemented-Symmetry Sector Decomposition C16.1.1. Implementation gate Let H=H_phys be finite dimensional, A=A_phys a faithfully represented unital finite-dimensional C*-algebra, G a compact group, and U:G→U(H) a continuous unitary representation satisfying U_g A U_g†=A and implementing the declared physical automorphism action. The abstract effective automorphism group alone does not select a unique Hilbert-space representation. U is additional represented data and may be replaced by a cover or central extension only when that implementation is independently declared and physically justified. C16.1.2. Canonical Isotypic Decomposition Theorem Theorem C16.1. There is a finite set of occurring irreducible classes λ and a unitary decomposition H ≅ ⊕_λ(V_λ ⊗ M_λ) such that U_g ≅ ⊕_λ(U_g^(λ) ⊗ I_Mλ). The isotypic subspace H_λ=V_λ⊗M_λ and multiplicity m_λ=dim M_λ=dim Hom_G(V_λ,H) are canonical up to unitary equivalence and label ordering. Proof. Compactness permits Haar averaging of an inner product, so the finite-dimensional representation is unitary and completely reducible. Grouping all irreducible summands equivalent to V_λ gives an isotypic component independent of the decomposition into individual copies. The intertwiner space Hom_G(V_λ,H) supplies the multiplicity space. QED. C16.1.3. Character projector construction For normalized Haar measure μ, the isotypic projector is P_λ=d_λ∫_G overline(χ_λ(g))U_g dμ(g). For finite G, P_λ=(d_λ/|G|)Σ_g overline(χ_λ(g))U_g. These projectors are orthogonal, commute with U(G), and sum to I on the represented support. C16.1.4. Symmetry commutant and multiplicity reconstruction Define S_U=C*(U(G)) and C_U=S_U′. Then S_U≅⊕_λ(B(V_λ)⊗I_Mλ), C_U≅⊕_λ(I_Vλ⊗B(M_λ)), and Z(S_U)=Z(C_U)=⊕_λ C P_λ. Hence dim(P_λ C_U P_λ)=m_λ² and m_λ=sqrt(dim(P_λ C_U P_λ)). C16.1.5. Copy-Selection Nonuniqueness Theorem Theorem C16.2. If m_λ>1, the λ-isotypic subspace is canonical, but its decomposition into individual irreducible copies is not canonical from U alone. Proof. Every I_Vλ⊗W with W∈U(M_λ) commutes with U(G) while rotating candidate copies. Additional physical data are therefore required to select a copy basis. QED. C16.1.6. Superselection gate and disconnected data A symmetry-isotypic decomposition is superselected relative to an operational algebra A_op exactly when [P_λ,A]=0 for every λ and A∈A_op, equivalently P_λ A P_μ=0 for λ≠μ. Lie-algebra data see only G⁰. The full represented group, including π₀(G), must be retained because disconnected components can permute connected-component sectors. C16.2. SYM-4B: Joint Symmetry-Observable Decomposition C16.2.1. Independent physical-algebra decomposition Independently of U, finite-dimensional structure gives H≅⊕_z(K_z⊗N_z), A≅⊕_z(B(K_z)⊗I_Nz), and A′≅⊕_z(I_Kz⊗B(N_z)), with minimal central projectors Z_z^A. The P_λ and Z_z^A encode different equivalence relations and need not commute. C16.2.2. Center-Orbit Theorem Theorem C16.3. Conjugation by U_g permutes the minimal central projections of A: U_gZ_z^A U_g†=Z_{g·z}^A. The identity component G⁰ fixes each Z_z^A. For every orbit O, Z_O^A=Σ_{z∈O}Z_z^A is central in A and commutes with U(G), hence reduces both structures. Proof. Automorphisms preserve the center and its minimal projections. A continuous map from connected G⁰ to a finite permutation group has trivial image. Orbit sums are invariant under the full group. QED. C16.2.3. Orbitwise Wedderburn constancy If g·z=w, spatial equivalence of the blocks and unitary equivalence of their represented supports force dim K_z=dim K_w and dim N_z=dim N_w. Equal dimensions are necessary, not sufficient, for belonging to the same orbit. C16.2.4. Stabilizer factorization For the stabilizer G_z, after choosing a Wedderburn identification of Q_zH, every U_g|_{Q_zH} factorizes as u_z(g)⊗v_z(g). The factors are unique only under the compensating phase transformation u_z→e^{iθ}u_z and v_z→e^{-iθ}v_z. The separate factors may be projective with inverse multipliers even when U is an ordinary representation. C16.2.5. Failure of universal Cartesian refinement The products R_{λz}=P_λZ_z^A form a canonical orthogonal common refinement exactly when [P_λ,Z_z^A]=0 for all λ,z. Covariance U_gAU_g†=A does not imply this commutation and does not imply that every physical observable preserves every symmetry sector. C16.2.6. Observable-link criterion The incidence space I_{λμ}=P_λ A P_μ records whether physical observables connect symmetry sectors. Sectors λ and μ are physically linked exactly when I_{λμ} is nonzero. This incidence family is more informative than separate multiplicity and Wedderburn dimension lists. C16.2.7. Fixed-point and joint algebras The symmetry-invariant physical observables form A^G=A∩U(G)′. This must be distinguished from A′∩U(G)′, which contains operators commuting with both the physical algebra and symmetry. Define the canonical joint algebra J_joint=C*(A,U(G)). Then J_joint′=A′∩U(G)′. C16.2.8. Canonical Joint Reducing Decomposition Theorem Theorem C16.4. There are finite-dimensional spaces L_r and R_r such that H≅⊕_r(L_r⊗R_r), J_joint≅⊕_r(B(L_r)⊗I_Rr), and J_joint′≅⊕_r(I_Lr⊗B(R_r)). The minimal central projectors Z_r of J_joint are the canonical minimal central projectors separating the indecomposable Wedderburn summands of the joint algebra. They give the coarsest canonical central decomposition determined by J_joint, although noncentral reducing projections may exist inside multiplicity spaces. Proof. J_joint is a finite-dimensional unital C*-algebra, so its faithful representation has Wedderburn form. Since A and U(G) generate J_joint, its central projections reduce both. QED. C16.2.9. Operational qualification J_joint is a mathematical compatibility envelope. It is an operational observable algebra only when the protocol admits the relevant implementers or generated effects. Joint sectors are operationally superselected relative to A_op exactly when every Z_r commutes with A_op. C16.3. SYM-4C: Observer, Record, and Recovery Compatibility C16.3.1. Four non-equivalent properties Sector preservation: operations do not transfer support between Z_r sectors. Sector readability: the observer algebra contains the relevant Z_r or admitted coarse sums. Sector recordability: a physical output channel carries distinguishable information about the sector. Sector recoverability: one physical recovery channel restores an algebra containing the sector information. C16.3.2. Observer Sector-Preservation Theorem An observer algebra A_O preserves every joint sector exactly when [Z_r,A]=0 for all r and A∈A_O, equivalently Z_rA_OZ_s=0 for r≠s, equivalently A_O⊆Z(J_joint)′. If A_O⊆J_joint, preservation is automatic. Readability is stronger: exact reading requires Z_r∈A_O^read. The largest exactly readable part of the joint center is Z_O^read=Z(J_joint)∩A_O^read. Channel-picture convention for C16.3.3-C16.3.8. Let Phi_*: S(B(H_in)) -> S(B(H_out)) be the declared Schrödinger-picture CPTP channel on the supported carrier, and let Phi^dagger: B(H_out) -> B(H_in) be its unital completely positive Heisenberg adjoint. Every support compression and sector projector used below is taken on the declared input or output carrier specified by the corresponding formula. C16.3.3. Channel sector blocks For Φ(ρ)=Σ_iK_iρK_i†, define K_i^{sr}=Z_sK_iZ_r. Φ preserves every sector exactly when K_i^{sr}=0 for s≠r for one, hence every, Kraus representation. Equivalently every K_i commutes with every Z_r. C16.3.4. Central Markov-Closure Theorem Theorem C16.5. A state-independent stochastic matrix T with p_s′=Σ_rT_{sr}p_r for all states exists exactly when Φ†(Z_s)∈Z(J_joint) for every s. In that case Φ†(Z_s)=Σ_rT_{sr}Z_r, T_{sr}≥0, and Σ_sT_{sr}=1. If Φ†(Z_s) is block diagonal but non-scalar inside an input sector, output sector probabilities depend on intrasector state and no global classical Markov closure exists. C16.3.5. Exact Sector-Record Theorem Let R be a record channel. The joint-sector distribution is exactly readable at the output exactly when there is an admitted output POVM {E_r} satisfying R†(E_r)=Z_r. Then Tr[E_rR(ρ)]=Tr[Z_rρ] for every input state. This recovers the classical label distribution, not necessarily quantum information inside sectors or intersector coherence. C16.3.6. Joint-compatible exact recovery For supported projector P and channel Kraus operators K_i, define N_{Φ,P}=C*({PK_i†K_jP}) and A_{Φ,P}^rec=N_{Φ,P}′∩B(PH). The unique largest exactly recoverable algebra contained in J_joint,P=P J_joint P is A_{Φ,P}^{J-rec}=A_{Φ,P}^rec∩J_joint,P. The recoverable central sector algebra is Z_{Φ,P}^rec=Z(J_joint,P)∩A_{Φ,P}^rec. A central projector Z_rP is recoverable exactly when it commutes with every compressed Kraus product PK_i†K_jP. C16.3.7. Observer-readable recovered sector information The observer-accessible recovered algebra is A_O^rec=A_{Φ,P}^rec∩A_O^read, and the readable recovered sector algebra is Z_O^rec=Z(J_joint,P)∩A_{Φ,P}^rec∩A_O^read. Recoverability without readability and readability without dynamical preservation are both possible. C16.3.8. Symmetrized recovery Theorem C16.5a (Covariant Recovery by Haar Averaging). Suppose G is compact, Φ is covariant under declared input and output unitary actions U_g and V_g, B is symmetry invariant, and Q_rec is one CPTP recovery satisfying (Q_rec∘Φ)†(X)=X for every X∈B. Define Q̄_rec(·)=∫_G U_g† Q_rec(V_g(·)V_g†)U_g dμ(g). Then Q̄_rec is CPTP, covariant, and corrects B. Proof. Convex integration preserves complete positivity and trace preservation. Haar invariance gives covariance. For X∈B, covariance of Φ and invariance of B transport the correction identity to every conjugate U_g X U_g†; averaging and changing variables return X. QED. The recovery need not be unique. C16.3.9. No-assembly warning The existence of a separate recovery Q_r for each sector does not imply one CPTP recovery correcting their union. A global protocol requires an accessible nondestructive sector instrument or a compatible assembly that is completely positive and trace preserving and treats cross-sector coherences consistently. C16.4. SYM-4D: Sector-Resolved Fisher Geometry, Clock Adequacy, and Comparison Transport Truth-first role. The sector architecture constrains where temporal information may reside, but it does not identify sector labels with time. SYM-4D asks when an admitted one-parameter state model carries accessible, identifiable, ordered, and transportable temporal information across the joint sectors reconstructed in SYM-4A/B/C. C16.4.1. Declared temporal statistical model and accessibility gate Let θ↦ρ_θ be a differentiable family of density operators on the supported physical space, with tangent ρ̇_θ, and let Z_r be the minimal central projectors of the joint algebra J_joint. A sector-resolved temporal claim is admissible only after declaring the parameter domain, the physical preparation family, the observer-readable algebra, the measurement class, and any channel through which the family is accessed. Quantum Fisher information is therefore a property of a declared statistical model, not of a sector decomposition alone. C16.4.2. Sector-Resolved Fisher Decomposition Theorem Theorem C16.6. Suppose ρ_θ=⊕_r p_r(θ) σ_{r,θ} on a regular parameter interval, where the mutually orthogonal sector projectors Z_r are independent of θ, p_r≥0, Σ_r p_r=1, and each occupied normalized state σ_{r,θ} is differentiable on its fixed sector support. Then, at regular points with p_r>0, the symmetric-logarithmic-derivative quantum Fisher information is F_Q(ρ_θ)=Σ_{r:p_r>0}(ṗ_r²/p_r)+Σ_r p_r F_Q(σ_{r,θ}). At vanishing weights the expression is interpreted by the finite one-sided support limit when it exists; changing supports and singular endpoints require the separate G7 endpoint audit. Proof. Fixed block orthogonality makes the SLD block diagonal. In occupied block r its scalar part is (ṗ_r/p_r)I and its traceless normalized-state part is the SLD of σ_{r,θ}. The cross term vanishes because Tr(σ_{r,θ}L_{r,θ})=0. Taking Tr(ρ_θL_θ²) yields the classical Fisher information of the sector weights plus the weighted intrasector quantum Fisher information. QED. Corollary C16.6a. If p_r is θ-independent, the classical sector-weight term vanishes. If σ_{r,θ} is θ-independent inside every sector, all temporal sensitivity is classical sector-weight sensitivity. If both are constant, the total Fisher information vanishes even though the sector structure may be nontrivial. C16.4.3. Accessible Fisher Information and Monotonicity For an admitted observer measurement class Meas_O, define F_O(θ)=sup_{M∈Meas_O} F_C(p^M_θ). Then 0≤F_O≤F_Q. For every declared parameter-independent CPTP access channel Φ, F_Q(Φ(ρ_θ))≤F_Q(ρ_θ). A recovery channel that restores the complete model family is sufficient for equality of SLD Fisher information under the forward channel. The converse is not asserted: equality of SLD Fisher information for one parameter family or tangent does not generally certify recoverability or statistical sufficiency. Exact recoverability of an unrelated algebra does not by itself imply Fisher preservation for the temporal tangent. C16.4.4. Operational Clock-Adequacy Criterion Definition C16.9. This definition specializes the G12-G18 clock audit to the SYM-4 joint-sector setting. A declared clock model is adequate on an interval I for an observer class only if: (i) accessibility, the admitted observer protocol obtains a nonconstant outcome family; (ii) local identifiability, F_O(θ)>0 almost everywhere on the regular part of I; (iii) global ordering, the accessible statistical states do not recur or alias within I except under a declared clock period; (iv) calibration, an estimator or relational coordinate and its uncertainty convention are specified; and (v) stability, the required distinguishability survives the admitted channel and coarse graining to the stated tolerance. The resulting clock is then classified under G18 as globally affine, approximately affine, piecewise, memory extended, event timer, record only, or inadequate. Positive Fisher information is necessary for local metrological sensitivity but is not sufficient for a globally adequate clock. Theorem C16.10 (No-Fisher-to-Time Implication). Neither F_Q>0 nor sector-resolved F_Q>0 implies a unique duration, a global temporal order, constant clock rate, or observer-independent elapsed time. Proof. Fisher information is local distinguishability along a chosen parameterization. Reparameterization rescales it, periodic models can have aliases, and different accessible measurements can retain different fractions of the quantum information. The missing conclusions require calibration, injectivity or declared periodicity, accessibility, and comparison structure. QED. C16.4.5. Constant-Speed and Reparameterization Gate Under a smooth monotone reparameterization τ=f(θ), Fisher information transforms as F_τ=F_θ(dθ/dτ)². On a regular one-dimensional segment with F_θ>0, use the manuscript convention L_O(θ)=(1/2)∫√F_O(u)du. Where L_O is a valid monotone coordinate, the reparameterized Fisher information is F_L=4 almost everywhere. This is a coordinate construction, not a derivation of physical duration. A physically privileged clock scale requires an independently justified calibration convention, generator, reference process, or interobserver comparison rule. C16.4.6. Sector Clock Adequacy and Assembly Theorem Theorem C16.11 (Sector-Clock No-Assembly Theorem). Sectorwise adequate clocks do not in general assemble into one global clock. A global assembly requires an accessible sector instrument or coherent joint protocol, compatible sector calibrations on physically admitted transition, comparison, or common-record domains, a common orientation, and transition or comparison maps satisfying consistency around every admitted cycle. Without these data, sector-dependent origins, rates, periods, or unreachable components leave only a family of local clocks. Proof. Each sector can support an identifiable one-parameter model while independent affine or periodic recalibrations preserve its local statistics. Pairwise transports remove this freedom only when they exist and satisfy path independence. QED. C16.4.7. Comparison Transport and Fisher Contraction Let D_{s←r} be a declared parameter-independent CPTP comparison channel from a model in sector r to one in sector s. Then Fisher monotonicity gives F_Q(D_{s←r}(ρ_{r,θ}))≤F_Q(ρ_{r,θ}). Exact recovery of the complete model family is sufficient for lossless Fisher comparison. Equality of SLD Fisher information for one temporal tangent is only a local information-preservation condition and does not by itself prove recoverability or equivalence of the full statistical experiments. For several sectors, compositional consistency requires D_{t←s}∘D_{s←r} to agree with D_{t←r} on the transported family. Nontrivial holonomy around a cycle signals path-dependent calibration rather than a single globally synchronized time coordinate. C16.4.8. Benchmarks and falsification gates The benchmark suite below is executed by explicit finite-dimensional constructions. Benchmark F uses ρ_θ=ρ independent of θ on a nontrivial direct-sum sector space, giving F_Q=0. Benchmark G uses |ψ_θ⟩=(|0⟩+e^{-iθ}|1⟩)/√2, for which the local pure-state QFI is 1 while ρ_{θ+2π}=ρ_θ, so global aliasing remains. Benchmark H uses two sector models with independent positive-affine calibrations and an inconsistent loop offset, producing comparison holonomy. Benchmark I applies complete dephasing inside each sector: sector probabilities are preserved while phase-dependent intrasector QFI is erased. Benchmark J uses a unitary sector swap followed by its inverse, preserving the complete statistical model although support moves between sectors. These examples falsify every shortcut from sector existence, preservation, readability, recordability, recoverability, or positive Fisher information directly to duration. C16.4.9. SYM-4D completion statement Completion statement. After the Version 34 sanitation corrections, SYM-4D is complete within the declared differentiable, fixed-sector, finite-dimensional, parameter-independent-channel scope. Its canonical outputs are the classical-plus-intrasector Fisher decomposition for orthogonal fixed sectors, the observer-accessible Fisher bound, the local clock-adequacy gates, the reparameterization boundary, and the comparison-transport consistency criterion. Approximate sufficiency, singular models, multiparameter incompatibility, non-Markovian process tensors, unbounded generators, and model-specific ETP clock executions remain extensions. C16.5. Integrated Benchmark and Falsification Suite C16.6. Relational-Time Consequence SYM-4A/B/C do not produce duration, and SYM-4D does not promote Fisher sensitivity to duration without the established clock gates. They determine the sector-resolved physical and operational arena from which temporal information may later be extracted. The valid chain is implemented symmetry → isotypic sectors → joint algebra J_joint → observer-readable and recoverable joint distinctions → accessible Fisher geometry → clock adequacy → comparison transport. A sector label is not duration, a conserved charge is not a clock, a record is not necessarily an adequate timer, and exact recovery does not imply constant elapsed-time Fisher speed. C16.7. Completion Statement and Transition Completion statement. SYM-4A, SYM-4B, SYM-4C, and SYM-4D are complete within the bounded finite-dimensional, compact implemented-symmetry, faithfully represented physical-algebra, declared-observer, declared-channel, exact-recovery, and differentiable statistical-model scope. Their outputs are the implemented isotypic decomposition; the joint algebra and commutant; the observer/readout/recovery hierarchy; the classical-plus-intrasector Fisher decomposition; the clock-adequacy gates; and the comparison-transport consistency criterion. Scope boundary. Infinite-dimensional normal algebras, noncompact groups, unbounded representations, antiunitary or graded symmetries, approximate and resource-constrained recovery, singular and multiparameter statistical models, process-tensor memory, and model-specific ETP executions remain extensions. No sector label or Fisher metric is promoted to physical duration without independently declared calibration and comparison structure. Transition. With the bounded finite-dimensional SYM-4 architecture complete, SYM-5 now asks how physically admitted interactions, output nets, effect systems, and operational relations select admissible observer-access algebras from the joint-sector arena. C17. SYM-5 OBSERVER-SECTOR SELECTION, ACCESSIBILITY RECONSTRUCTION, AND OPERATIONAL BOUNDARY EMERGENCE Role in emergence. SYM-4 reconstructs the arena of possible joint distinctions, but a sector is not an observer and a tensor factor is not an operational boundary. SYM-5 asks how physically admitted interactions and output structures turn some of those distinctions into accessible, stable, readable, and recoverable observer algebras. The stage permits null, nonunique, symmetry-orbit, and conditionally unique outcomes. It does not address consciousness or first-person experience. C17.1. SYM-5 reconstruction datum Definition C17.1. Let D_4=(H_phys,A_phys,G_eff,U,J_joint,J_joint′,Z(J_joint),Omega,Phi,N,Rel_op) denote the complete SYM-4 datum. A SYM-5 datum is D_5=(D_4,V,Net_out,Eff_out,C,ResTol), where V is a declared family of physical interaction isometries or dilations; Net_out={A_out(D_xi)}_{xi in Xi} is a physically supplied output net; Eff_out(D_xi) is the admitted output-effect system or measurement class on A_out(D_xi); C records causal, incidence, independence, compatibility, composition, memory, and control relations; and ResTol records resource bounds, robustness tolerances, and any approximation class. Tensor labels alone do not supply a physical output net. The net must arise from admitted interaction events, causal localization, emitted modes, ports, gauge-invariant local algebras, persistent memories, boundaries, or equivalent physical structure. C17.2. Sector-only Observer-Selection No-Go Theorem Theorem C17.2. Let Aut(D_4) be the automorphism group preserving every component of the complete SYM-4 datum. If two candidate proper observer algebras A_O,1 and A_O,2 satisfy A_O,2=alpha(A_O,1) for some alpha in Aut(D_4), then no covariant selector depending only on D_4 can select A_O,1 while excluding A_O,2. Proof. Because alpha preserves the complete input, alpha D_4=D_4. Covariance requires Sel(alpha D_4)=alpha Sel(D_4). Selecting exactly A_O,1 would therefore require the same unchanged input to select exactly A_O,2. If the candidates differ, unique asymmetric selection is impossible without additional physical structure. QED. Corollary C17.2a. The joint central projectors, represented-symmetry sectors, multiplicity spaces, and joint commutant do not generally derive one preferred observer algebra. Their canonical role is to constrain the space in which observer access can be reconstructed. C17.3. Interaction-to-access-seed reconstruction Channel-picture convention for C17.3. The reduced channel Phi_B,*: S(B(H_A)) -> S(B(H_B)) and complementary channel R_E,*: S(B(H_A)) -> S(B(H_E)) are Schrödinger-picture CPTP maps. Their adjoints Phi_B^dagger: B(H_B) -> B(H_A) and R_E^dagger: B(H_E) -> B(H_A) are the corresponding Heisenberg-picture maps. Every restricted fragment channel R_xi and pullback R_xi^dagger inherits its explicitly declared output carrier from the physical output net. Let V:H_A→H_B⊗H_E be a finite-dimensional physical interaction isometry. Its reduced and complementary channels are Phi_B(rho)=Tr_E[V rho V†] and R_E(rho)=Tr_B[V rho V†]. For a physical fragment algebra F_xi in the output net, let R_xi be the corresponding restricted output channel. Define the pulled-back accessible effect system S_xi=R_xi†(E_xi), the raw generated algebra A_O,xi^raw=W*(S_xi), and the joint-compatible candidate A_O,xi^{J_joint}=A_O,xi^raw∩J_joint when membership in J_joint is part of the declared protocol. The Kraus family entering an exact-recovery calculation must be constructed for the selected output: tracing E yields K_e= A_out(D_2) be a unital injective *-homomorphism. The family satisfies iota_D,D=id_Aout(D) and iota_D2,D3 o iota_D1,D2=iota_D1,D3 whenever D_1 subseteq D_2 subseteq D_3. Event-level symbols iota_ef denote the corresponding maps when individual events themselves are the admitted domains. Definition C17.3A.2 (Event domains and output net). Let K(I) be a physically admitted poset of event domains, closed under the declared causal, interaction-connected, memory-retention, or output-composition rule. Let A_out be the represented global output algebra and let S_out(D) be the physically supplied local output generators for D in K(I). Define A_out(D)=C*(S_out(D)). The output-net axioms are physical provenance; isotony under D_1 subset D_2; functorial consistency of embeddings; covariance under Aut_phys; declared causal compatibility; generated-union closure where permitted; and scalar or empty output admissibility. Theorem C17.3A.3 (Conditional Output-Net Reconstruction). Let K(I) be finite, let A_out be a represented finite-dimensional unital C*-algebra, and let S_out(D) be inclusion-compatible local generator systems with functorial embeddings and a covariant complete-datum action. Then A_out(D)=C*(S_out(D)) defines a finite-dimensional covariant isotonic output net. The net is independent of the chosen generating presentation and may contain scalar algebras. If physical automorphisms exchange proper domains, the canonical output is their full transported family rather than one preferred member. Proof. Finite-dimensional generated unital *-algebras are C*-algebras. Inclusion of the S_out generator systems implies inclusion of their generated algebras. Functoriality descends from the generator embeddings. Physical *-isomorphisms commute with generated-algebra closure. Different generating lists with the same generated algebra define the same output. Covariance and the SYM-5 no-go theorem prohibit asymmetric selection inside an unresolved automorphism orbit. QED. Theorem C17.3A.4 (Output-Event Decomposition No-Go). Suppose only an undifferentiated total physical input-output datum is supplied, with no intervention slots, interaction supports, ports, persistent memories, boundary interfaces, causal factorization, or local output generators. If two event decompositions produce the same complete experiment and are exchanged by a complete-datum automorphism, no covariant rule using only the undifferentiated datum can select one while excluding the other. Proof. The automorphism fixes the selector input while transporting one candidate decomposition into the other. Covariance therefore forbids a unique asymmetric output. QED. Record-support map signature. For output domains D_1 and D_2, let R_Dk:S(A_src) -> S(A_out(D_k)) be the declared Schrödinger-picture record channels from the common admitted source carrier. Let Delta_1<-2:S(A_out(D_2)) -> S(A_out(D_1)) be a parameter-independent CPTP map, with UCP adjoint Delta_1<-2^dagger:A_out(D_1) -> A_out(D_2). Definition C17.3A.5 (Record-support preorder). For output domains D_1,D_2 with record channels R_D1 and R_D2, write D_1 preceq_rec D_2 when a parameter-independent physical channel Delta_1<-2 satisfies R_D1=Delta_1<-2 o R_D2 on the complete admitted source model. This relation is reflexive and transitive. Mutual degradability defines the operational equivalence D_1 sim_rec D_2. The relation preceq_rec is not identified with causal precedence or temporal order. Theorem C17.3A.6 (Causal/record separation). Record degradability does not, without additional intervention or influence data, imply causal precedence, physical locality, or duration. Conversely, causal proximity need not imply recoverability. Therefore preceq_causal and preceq_rec must remain independently declared or reconstructed. Proof. Redundant causally separated fragments may encode mutually simulable classical records, while neighboring outputs can lose mutually unrecoverable quantum information. These possibilities satisfy one relation without the other. QED. Corollary C17.3A.7 (Collision-net execution). In the repeated-collision model, each fresh fragment F_n is distinguished by its declared interaction event e_n. For D_n={e_1,...,e_n}, A_out(D_n)=B(H_F1 tensor ... tensor H_Fn), with embedding X maps to X tensor I_F(n+1). The accumulated record satisfies R_1:n-1=Tr_Fn o R_1:n, so D_n-1 preceq_rec D_n. This executes an event-indexed isotonic accumulated-output net without an observer label. It does not derive the event sequence from bare potentiality or identify update count with duration. Definition C17.3A.8 (Net qualification hierarchy). Relative to the fixed event-domain system, declared local generators, and inclusion order, the inclusion-smallest generated output-net candidate, when it exists, contains all output algebras grounded by that datum. The record-qualified subfamily additionally satisfies declared distinguishability, persistence, common-record, recovery, redundancy, and robustness gates. The observer-qualified subfamily additionally supports a coherent operational closure and an admissible observer algebra. The physical net may not be defined circularly by requiring every output already to be a record or observer. Completion statement. The ETP-grounded physical output-net step is complete within the bounded finite-dimensional, interaction-relative scope: the datum, event and domain definitions, net axioms, conditional reconstruction, presentation independence, covariance, nonuniqueness boundary, record-support preorder, causal/record separation, hierarchy, and collision execution are established. Bare ETP potentiality does not uniquely determine elementary events or one universal output net. Infinite-dimensional, continuous-event, indefinite-causal, gauge-interface, approximately stable nets and compositionally closed nets under declared physical composition laws remain extensions. C17.3B. Physical Effect-System Reconstruction Role in emergence. The output net determines where mathematical output observables reside. Observer access additionally requires a physical account of which effects and instruments can actually be implemented. The full mathematical interval Eff(A_out(D))={E in A_out(D):0<=E<=I} is not automatically operationally available. Definition C17.3B.1 (Physical instrument datum). For each output domain D, let Instr(D) be a nonempty family of physically admitted finite-outcome instruments I^(a)={I_x^(a)}_x. Each I_x^(a) is completely positive and trace nonincreasing, and sum_x I_x^(a) is trace preserving. The primitive physical effects are Eff_0(D)={I_x^(a)dagger(I)}. Define Eff_phys(D)=cl_phys(Eff_0(D)), where the declared operational closure may include outcome coarse-graining, complement, classical randomization, admitted preprocessing, symmetry transport, and norm limits at a stated tolerance. Arbitrary products, spectral projectors, and all effects of A_out(D) are not included unless physically justified. Theorem C17.3B.2 (Effect-System Reconstruction). If Instr(D) is closed under classical randomization and outcome coarse-graining, then Eff_phys(D) contains 0 and I, lies in the positive unit interval of A_out(D), is convex, is closed under complement and admitted coarse-graining, and transforms covariantly with the complete physical measurement datum. It need not be closed under multiplication and therefore need not be a C*-algebra. The generated accessible algebra is A_eff(D)=C*(Eff_phys(D)). Proof. Instrument effects are positive, trace nonincrease gives E_x<=I, and normalization gives sum_x E_x=I. Grouping outcomes yields sums and complements; randomizing instruments yields convex combinations. Complete physical transport conjugates instruments and their effects. None of these operations supplies multiplicative closure. QED. Theorem C17.3B.3 (Effect-System No-Go). The output algebra A_out(D) alone does not determine Eff_phys(D). Two physical protocols may represent the same output algebra while admitting different couplings, probe states, controls, detector outcomes, energy budgets, superselection restrictions, or resolution tolerances. If those protocols are not included in the datum, no rule based only on A_out(D) can infer which mathematical effects are implementable. QED. Corollary C17.3B.4 (Effect-to-seed descent). For a fragment channel R_D and physical effect system Eff_phys(D), define S_D=R_Ddagger(Eff_phys(D)) and A_O,D^raw=C*(S_D). Complete physical equivalence transports Eff_phys(D), S_D, and A_O,D^raw covariantly. Thus the chain output domain -> physical instrument family -> effect system -> pulled-back access seeds -> observer algebra is explicit. Compatibility gate. If D_1 subset D_2, the algebra embedding A_out(D_1)->A_out(D_2) does not by itself guarantee that every physical effect or instrument on D_1 is implementable through D_2. Effect-system isotony, restriction, or simulation requires a declared instrument transport and is a separate model theorem. C17.3B completion statement. The abstract finite-dimensional physical effect-system reconstruction, effect-system no-go theorem, effect-to-seed descent, and instrument-transport compatibility gate are complete within their declared bounded scope. C17.3C completion statement. The controlled-collision and accumulated-memory executions, including full and restricted effects, superselection, detector noise, inclusion compatibility, and null classifications, are completed only in the declared model scope. Adjacency of the sections does not transfer the C17.3C model executions into the proof of C17.3B. C1 audit status. SYM-5D is reclassified as an active robustness theorem-reconstruction program. Its numbered items organize model-specific research targets and are not certified as a unified robustness theorem family. C2 audit status. SYM-5E is reclassified as an active resource-theory reconstruction program. Resource qualification, costs, Pareto analysis, and efficiency selection remain conditional on a complete resource model and are not certified as a unified theorem family. C4 audit status. SYM-5F is reclassified as an active benchmark-taxonomy and dependency-audit program. The listed families remain useful examples and falsification targets, but they do not validate the reclassified C17.3D robustness, C17.3E resource, or C17.3G composition programs. No general earliest-failed-gate theorem is certified until every gate has a complete definition and the benchmark suite is proved exhaustive for the declared candidate class. C3 audit status. SYM-5G preserves a compositional-closure research architecture and ten traceable proof obligations. No general C17.3G closure theorem is presently certified, and the full SYM-5 program is not closed end to end. Research Claim C17.3G.10 (Unified Compositional Program; not certified as a theorem). Construction closure, hierarchy compatibility, objective closure, iterated closure, and selection nonclosure are separate research targets. They may be combined only after the component theorems are established under one compatible composite datum. Research Claim C17.3G.9 (Iterated Observer Networks; not certified as a theorem). Finite iteration remains inside an observer family only when the one-step composition result is valid under iteration and its hypotheses remain stable as memories, resources, correlations, interfaces, and discarded carriers accumulate. Research Claim C17.3G.8 (Generated Closed Family; not certified as a theorem). A minimal nontrivial compositionally closed family can be defined only after fixing the seed family, physical composition operation, closure operator, equivalence relation, candidate universe, and ordering. No basement minimality or unique physical observer family is asserted. Research Claim C17.3G.7 (Hierarchy Compatibility; not certified as a theorem). The ordering of admissibility, robustness, resource qualification, and optional efficiency selection is a bookkeeping hierarchy. Preservation must be proved gate by gate for a declared physical composition. Research Claim C17.3G.6 (Failure-Index Behavior; not certified as a theorem). No general failure-index monotonicity is asserted. Composition can introduce a new carrier, correlation, incompatibility, resource conflict, control, or boundary condition that causes an earlier gate to fail than in either isolated constituent. Research Claim C17.3G.5 (Selection Nonclosure Program; not certified as a general theorem). Construction and optimization-based selection are distinct. Nonclosure of Pareto, efficiency, or robustness-maximal subclasses requires a declared candidate class, order, physical composition operation, and explicit counterexample family. Research Claim C17.3G.4 (Objective-Class Composition; not certified as a theorem). Common-record accessibility and redundancy must be proved for the composed channels, fragment structure, instruments, disturbance criterion, and independence assumptions. Objective constituents do not automatically produce an objective composite. Research Claim C17.3G.3 (Resource Closure; not certified as a theorem). Resource-qualified observers remain resource qualified only when a complete resource theory proves closure of the permitted composite implementation and budget. A compositional budget must be defined and proved sufficient. Research Claim C17.3G.2 (Robustness under Composition; not certified as a theorem). Composite robustness is model specific and requires one common metric, perturbation family, tolerance rule, and complete carrier. No general lower bound by the weakest constituent layer is asserted. Research Claim C17.3G.1 (Admissibility under Composition; not certified as a theorem). For a declared physical composition operation, the composite must independently satisfy every admissibility gate on the complete composite carrier. Correlations, shared carriers, interfaces, controls, boundaries, resources, and generated interactions can create a new failure. Role in emergence. SYM-5G is a compositional audit program asking when observer families remain inside the observer-selection architecture after a declared physical composition. Construction, comparison, iteration, and optimization-based selection are tested separately; closure is not presumed. C17.3G. SYM-5G Compositional Closure C3 audit disposition: RECLASSIFIED. The former C17.3G.1 through C17.3G.10 theorem statements lacked one fully typed composition operation, a complete composite carrier, control of correlations and shared environments, interface and feedback hypotheses, resource and robustness composition rules, and proofs at the breadth asserted. The identifiers are retained below as research claims and proof obligations. Complete composite datum. A future theorem must declare the physical composition operation and the complete composite carrier, including initial and generated correlations, shared or independent environments, memories, controls, interfaces, feedback, branches, selected and discarded outputs, boundaries, charges, resources, approximation conventions, and the governing physical-equivalence relation. Operation gate. A future closure proof must separately establish mathematical admissibility, physical realizability, quotient descent, complete null preservation where claimed, preservation of each observer gate, compatibility of witness families, and an explicit falsifier. Constituent certification does not by itself certify the composite. External alignment note. General process composition is formalism dependent; known no-go results for unrestricted process-matrix composition reinforce the need to prove closure only for a declared physical composition class. This does not prohibit tensor, sequential, or network closure in a specifically proved bounded model. Current claim ceiling. C17.3G may state that composition is an independent audit gate, that separately admissible observers need not remain jointly admissible, that construction and selection are distinct, and that model-specific closure can be investigated. It may not presently claim general admissibility, robustness, resource, objective, hierarchy, minimal-family, iterative, or unified closure. Research Claim C17.3F.2 (Unified Observer-Family Benchmark Program; not certified as a theorem). For a fixed, fully defined observer-selection hierarchy, k(O) may be used as a bookkeeping label for the earliest tested gate failed by O. The present benchmark suite is not proved exhaustive, and the reclassified robustness, resource, and composition gates do not yet support a unified benchmark theorem. Benchmark item F8. Null or scalar observer outputs remain valid outcomes for explicitly defined models. This item illustrates the null-classification discipline but does not validate completeness of the observer hierarchy. Benchmark item F7. Objective multiobserver families are a proposed composition test. Preservation of common-record accessibility must be proved for each declared composite model and is not inferred from constituent objectivity. Benchmark item F6. Efficiency-maximizing observers are a proposed optimization example. Any conclusion requires a complete resource model and a declared objective; no physical selection principle is validated here. Benchmark item F5. Pareto observer families illustrate that a declared multi-resource order may yield several nondominated candidates. Nonemptiness, compactness, uniqueness, and physical selection remain model dependent. Benchmark item F4. Resource-dominance is meaningful only after task equivalence and every resource coordinate are fixed. This item remains a proposed diagnostic under the C17.3E reconstruction gate. Benchmark item F3. A model may preserve a chosen observer algebra while a specified Fisher or discrimination statistic degrades. The relevant metric, perturbation family, and carrier must be declared before robustness conclusions are drawn. Benchmark item F2. Recoverability and operational discrimination are distinct gates in a fixed model. The example remains a separation target and does not establish completeness of the gate hierarchy. Benchmark item F1. Readability and exact recoverability are distinct in a fixed channel and protocol. The example remains a separation target and does not establish universal observer classification. Research Claim C17.3F.1 (Benchmark Classification Program; not certified as a theorem). A benchmark may be classified by the first fully defined gate it fails, but only relative to a fixed ordered gate system. The current suite is not proved exhaustive, and benchmark differences need not arise solely from failure location. Role in emergence. SYM-5F retains an observer-family benchmark and falsification program. It tests the completed exact admissibility components and supplies targets for the active robustness, resource, and composition programs. Its role is diagnostic rather than a completed validation of the entire SYM-5 hierarchy. C17.3F. SYM-5F Observer-Family Benchmark Execution Research Claim C17.3E.9 (Unified Resource-Constrained Observer Architecture; not certified as a theorem). Admissibility, robustness, resource qualification, and optional efficiency selection are distinct gates that may be composed only after their individual models and compatibility conditions are declared. Conditional uniqueness may be investigated when a well-defined objective has a unique maximizer, but no universal emergence or uniqueness theorem is asserted here. Proposed Principle C17.3E.8 (Resource-Efficiency Selection; not derived). A theory or application may choose to optimize a declared robustness-versus-cost objective within a fixed resource-qualified family. This is an optional selection criterion, not a consequence of observer admissibility and not evidence that nature selects the optimizer. Research Claim C17.3E.7 (Resource-Robustness Tradeoff Program; not certified as a theorem). Particular models may exhibit tradeoffs between a declared robustness metric and specified memory, energy, recovery, transport, communication, or redundancy costs. No universal monotone tradeoff, duality, or claim that robustness is itself one resource is asserted without a complete resource theory and proof. Research Claim C17.3E.6 (Resource-Compression Program; not certified as a theorem). If two observer realizations are equivalent for a preregistered task under the governing physical-equivalence relation and one is no more costly in every declared resource while strictly cheaper in at least one, the second is dominated relative to that resource order. Physical equivalence, complete task equivalence, and all resource coordinates must be proved rather than assumed. Research Claim C17.3E.5 (Pareto Observer Program; not certified as a general theorem). For a fixed multi-resource model and candidate family, nondominated observers may be organized by Pareto order. Whether the frontier is nonempty, compact, unique, or physically selected depends on the declared candidate space, resource coordinates, topology, and implementability assumptions. Research Claim C17.3E.4 (Informational-Cost Separation; conditional). Identical observer algebras do not by themselves determine implementation, sampling, estimation, memory, communication, or recovery cost. A quantitative separation becomes a theorem only after a task, cost functional, admissible implementation class, and comparison relation are specified. Research Claim C17.3E.3 (Exact-versus-Approximate Recovery Cost Program; not certified as a theorem). Exact and approximate recoverability are distinct operational regimes. Any cost comparison requires a declared code or algebra, selected output, recovery class, error metric, ancillary convention, implementation model, resource monotone, and one-shot or asymptotic regime. No general infinite-cost boundary for exact recovery, and no universal resource realizability or stability of epsilon-approximate recovery, is asserted here. Research Claim C17.3E.2 (Memory-Cost Program; not certified as a theorem). Retained memory can improve specified recovery, discrimination, or clock tasks in selected models, but any storage cost must be defined on the complete memory carrier and may depend on compression, correlations, reuse, approximation tolerance, and the operational task. No universal necessity or monotonic cost law is asserted here. Research Claim C17.3E.1 (Resource Qualification Program; not certified as a theorem). For a declared resource model, the physically implementable observer family may be restricted by explicit bounds on specified resource carriers and operations. This statement is a programmatic definition until the free objects, free operations, admissible implementations, ancillary rules, and resource constraints are fixed. Role in emergence. Admissibility and robustness do not by themselves establish physical realizability. C17.3E is retained as a resource-theory research program for determining which observer structures remain feasible under explicitly modeled memory, energy, recovery, transport, communication, and informational constraints. No resource-qualified observer theorem is certified until the relevant free objects, free operations, costs, tasks, ancillary rules, and composition laws are fixed. C1 status. C17.3D remains an active theorem-reconstruction program. The eight numbered identifiers are preserved for traceability but have research-claim status. Certification requires separate model-level proofs and a later synthesis theorem, if one exists. C17.3E. SYM-5E Resource-Constrained Access and Recovery C2 audit disposition. RECLASSIFIED. C17.3E.1 through C17.3E.9 are retained as a traceable resource-theory research program. They are not certified as general theorems in the present manuscript because no single resource theory, implementation class, or cost model supports their original breadth. Resource-theory gate. A future theorem must declare the resource-bearing objects, free objects, resourceful objects, free operations, physically implementable operations, admissible ancillary and catalytic extensions, initial and final correlations, composition law, one-shot or asymptotic regime, operational task, cost functions or monotones, approximation metric, and falsifier. Extensional resource non-generation is not by itself a physical implementation theorem. Exact-versus-approximate recovery boundary. Approximate quantum error correction requires a specified figure of merit and error convention. Existing general formulations use explicit criteria such as worst-case entanglement fidelity and complementary-channel optimization. Those results support a metrized recovery program, but they do not imply that exact recovery universally has infinite physical cost or that approximate recovery is always resource realizable. Current claim ceiling. C17.3E may presently state that resource constraints can further restrict an admissible observer family, that different implementations of the same task may have different declared costs, and that Pareto or efficiency analyses are available after a concrete resource model is supplied. It may not claim universal memory-cost necessity, exact-recovery cost divergence, robustness-resource duality, a physically mandated efficiency selector, or end-to-end observer uniqueness. C2 status. The nine numbered identifiers are preserved for genealogy and future reconstruction. Certification now requires model-level definitions and proofs, followed by a later synthesis theorem only if a unified resource architecture survives the required ancillary, catalytic, correlation, composition, and approximation attacks. Research Claim C17.3D.8 (Unified Robustness Architecture; not certified as a theorem). For O=(F,E,A,M), the labels R0 through R4 organize proposed questions concerning output-net, effect, observer-algebra, recoverability, and informational persistence. The order symbols do not currently denote a proved numerical or logical ordering, and no universal bifurcation-boundary theorem is asserted. Research Claim C17.3D.7 (Memory-Amplification Program; not certified as a theorem). Retained-memory accumulation may improve a declared discrimination or estimation task in specific independent, correlated, or non-Markovian models. Whether information accumulates, saturates, revives, or degrades, and whether the observer algebra changes, must be established from the complete memory channel and admitted readout. Research Claim C17.3D.6 (Resolution-Threshold Program; not certified as a theorem). A declared decision or detector threshold can change a threshold-classification discontinuously when an underlying continuous statistic crosses it. This concerns the chosen thresholded classification and does not by itself prove discontinuity of the physical experiment, output net, effect system, or observer algebra. Research Claim C17.3D.5 (Approximate-Recovery Stability Program; not certified as a theorem). Exact and approximate recovery must be compared using a declared channel distance, code, selected output, ancillary convention, recovery error, and tolerance. No generic instability of exact recovery or universal continuity of epsilon-approximate recovery is asserted here. Research Claim C17.3D.4 (Operational Fragility Target; not certified as a theorem). Algebraic accessibility and operational sensitivity are distinct quantities. Model-specific perturbations may alter Fisher information, discrimination performance, or threshold-relative resolution while leaving a chosen algebra unchanged; the magnitude and continuity behavior require an explicit statistical model and metric. Research Claim C17.3D.3 (Observer-Algebra Stability Target; not certified as a theorem). Continued separation by an admitted effect may support model-specific preservation of a generated observer algebra, but algebraic equality requires explicit generator, closure, threshold, and perturbation hypotheses. Structural robustness is not inferred from separation alone. Research Claim C17.3D.2 (Output-Net Persistence Target; not certified as a theorem). A future result may test whether a specified output net is invariant or continuously comparable under perturbations that preserve explicitly defined event support, domains, interfaces, and reconstruction rules. Local constancy is not asserted without a topology and proof. Research Claim C17.3D.1 (Layered Robustness Program; not certified as a theorem). Output-net, effect-system, observer-algebra, recoverability, and informational stability are proposed as distinct audit layers. Any implication between layers must be proved for a declared perturbation model; no general implication or converse-failure theorem is asserted here. Program role. SYM-5C identifies physically accessible distinctions. SYM-5D defines a robustness research program asking which output-net, effect, observer-algebra, recovery, and informational structures persist under explicitly metrized perturbations. No general C17.3D robustness theorem is certified at the present stage. C17.3D. SYM-5D Approximate Robustness C1 audit disposition: RECLASSIFIED. The former C17.3D.1 through C17.3D.8 theorem statements lacked a common perturbation space, topology or norm, quantified neighborhood, support-stability hypotheses, continuity constants, recovery metric, threshold convention, and proofs at the breadth asserted. They are retained below as numbered research claims and proof obligations, not theorems. External alignment note. Established continuity results in quantum information are metric and hypothesis dependent. Examples use trace or operator-norm distances for state entropies, diamond or completely bounded norms for channel approximation, and input-dimension or energy constraints for channel information quantities. This supports reclassification rather than adoption of an unqualified generic stability claim. Reconstruction gate. A future theorem must declare the complete perturbed carrier; admissible perturbation family; norm, metric, or topology on states, channels, effects, instruments, algebras, memories, and outputs; support and rank conditions; threshold and tolerance conventions; constants and dimension or energy dependence; ancillary stabilization; selected output; recovery criterion; and explicit counterexample or falsifier. Layer-to-layer implications require separate proofs and may not be inferred from the ordering of labels R0 through R4. Current claim ceiling. C17.3D may state that robustness is usefully separated into structural, operational, recovery, and informational questions and that the listed mechanisms are research targets or model-specific possibilities. It may not claim general local constancy, generic instability, continuity of approximate recovery, monotone robustness layers, unchanged observer algebras, or memory amplification without a theorem satisfying the reconstruction gate. Five null classes are identified: thermally unbiased outputs, dynamically trivial interactions, completely randomized detectors, trivial effect systems, and resolution-relative nullity. These correspond to physically distinct failure modes and must not be merged. Theorem C17.3C.4 (Collision-Net Effect Inclusion). Effect-system inclusion is guaranteed only when the physical protocol admits extension by ignoring appended fragments. Output-algebra isotony alone does not imply instrument isotony. Theorem C17.3C.3 (Detector-Noise Threshold). Detector noise continuously contracts accessible information and collapses observer access to the scalar case only at complete randomization. Algebraic accessibility and operational usefulness must therefore be distinguished. Theorem C17.3C.2 (Energy-Superselection Pointer Recovery). Energy-restricted effects can still recover the pointer algebra whenever the interaction generates population-level distinctions. Superselection restrictions reduce sensitivity without necessarily destroying observer emergence. Theorem C17.3C.1 (Controlled-Collision Effect Dichotomy). For the executed collision model, the generated observer access algebra is either the scalar algebra or the complete source pointer algebra. A single physically admitted separating effect is sufficient to reconstruct the full pointer algebra, while no physically admitted separating effect yields only scalar access. Role in emergence. Version 37 executes the physical effect-system program on the repeated-collision output net and distinguishes algebraic accessibility from detector quality, resolution, and resource sensitivity. C17.3C. SYM-5C Effect-System Execution Effect-system status. The bounded finite-dimensional theorem architecture is established. Full closure requires execution on the repeated-collision output net, restricted-detector and null benchmarks, energy and superselection constraints, approximate detector resolution, and compatibility under output-domain inclusion. C17.4. Observer-sector incidence classification Let {Z_r} be the minimal central projectors of J_joint and let {Z_z^O} be the minimal central projectors of a candidate observer algebra A_O. Define H_zr=Z_z^O Z_r H whenever Z_z^O and Z_r commute. The nonzero incidence family records whether the observer resolves, coarse-grains, refines within, or is blind to the joint sectors. If the projectors do not commute, replace the projector product by the pair of compressed bimodules Z_z^O J_joint Z_r and Z_r A_O Z_z^O; no fictitious Cartesian sector grid is asserted. Proposition C17.4. If A_O⊆J_joint, then every Z_r commutes with A_O, so each observer-central projector decomposes as Z_z^O=sum_r Z_z^O Z_r and the incidence blocks give a canonical common central refinement. If A_O is not contained in J_joint, commutation must be tested and may fail. Proof. Z_r is central in J_joint. The remaining statements follow from finite-dimensional central decomposition. QED. Interpretive boundary. Representation sectors, joint central sectors, record sectors, and observer-central sectors are different equivalence relations. Equality among them is a result to be proved in a model, not a definition. C17.5. Admissible observer-algebra gates Definition C17.5. AdmObs(D_5) is the family of candidate observer algebras whose supporting channels and protocols satisfy every declared gate below. A failed gate may yield exclusion or a weaker status; an empty family is an admissible physical result. Physical provenance: The interaction, output fragment, and effect system arise from admitted physical structure rather than arbitrary refactorization. Nontrivial accessibility: At least two admitted source preparations or record sectors produce distinguishable output statistics, unless the declared result is the scalar null algebra. Sector relation: Preservation, readability, mixing, coarse-graining, or blindness relative to the joint sectors is explicitly classified. Persistence: The relevant distinctions survive for the claimed interval or sequence under an exact or declared approximate stability criterion. Recordability: Admitted output effects pull back to the proposed record distinctions. Recoverability: One admitted recovery channel corrects the whole claimed algebra when exact recovery is asserted. Covariance: Physical equivalence transports the complete candidate and every gate. Robustness: The norm, perturbation class, and tolerance for approximate persistence or recovery are declared. Compositional closure: Successive interactions, memories, controls, and comparison maps remain compatible with the claimed observer domain. C17.6. Exact admissible observer core For candidate xi and supported projector P, let A_stab,xi be the exact stable algebra from the active Heisenberg maps, A_read,xi the physically justified readable C*-algebra generated from S_xi, and A_rec,xi=[C*({P K_i†K_j P})]′∩B(PH) the largest exactly recoverable algebra for the fragment channel. Define A_adm,xi=A_O,xi^{J_joint}∩A_stab,xi∩A_read,xi∩A_rec,xi. Here the K_i are the Kraus operators of the specifically declared fragment or output channel xi; substituting Kraus operators of its complement changes the recovery problem and is not an equivalent representation change. Theorem C17.6. A_adm,xi is a finite-dimensional unital C*-algebra and is the unique largest subalgebra of A_O,xi^{J_joint} satisfying the declared exact stability, readability, and one-channel recoverability gates simultaneously. It is independent of the Kraus representation and transforms covariantly under complete physical equivalence. Proof. Each constituent is a finite-dimensional C*-algebra; intersections preserve that property. Every algebra satisfying all gates lies in every constituent and therefore in the intersection. Kraus-family isometries preserve the generated compressed Kraus-product algebra, and physical equivalence conjugates all constituents. QED. No equality among A_stab,xi, A_read,xi, and A_rec,xi is asserted. A record can be readable but unrecoverable, recoverable but not fixed, persistent but operationally inaccessible, or sector blind. C17.7. Family structure and conditional uniqueness The family AdmObs(D_5) carries algebra inclusion, physical-equivalence orbits, operational compatibility, and an accessibility preorder. Write A_1≼_acc A_2 when every admitted statistical experiment of A_1 is a parameter-independent post-processing of an admitted experiment of A_2. Larger algebra, greater stability, greater recoverability, stronger redundancy, lower resource cost, and stronger robustness need not be optimized by the same candidate. Theorem C17.7 (Conditional Uniqueness). A unique admissible observer algebra is selected only if the declared gates leave one physical-equivalence class and the required ordering criterion has a unique greatest member in that class. Otherwise the canonical result is the complete admissible family, its physical-equivalence quotient, or its Pareto-maximal subset under the declared operational criteria. Proof. Selection from a set requires both an admissibility predicate and a declared comparison rule. If either leaves incomparable or symmetry-related candidates, unique choice is not determined by the datum. QED. C17.8. Objective multiobserver subclass Define AdmObs_obj(D_5)⊆AdmObs(D_5) by adding a common-record requirement across the active channel family, informative encoding in the required number of disjoint physical output algebras, independently admitted fragment access, nondisturbance to the declared tolerance, the required fragment-independence condition, covariance, and robustness. Redundancy alone is insufficient when different fragments encode incompatible records. Corollary C17.8. Objective observer access is a qualified subclass of admissible observer access, not the definition of all observers. A single nonredundant but informative recoverable access channel may support an observer algebra without satisfying objectivity. C17.9. Observer-to-time descent For every admissible observer algebra and measurement class, the existing Fisher hierarchy, clock audit, comparison-transducer tests, and holonomy criteria apply. Observer selection changes temporal geometry only through the induced accessible statistical experiment. Neither admissibility nor exact recovery implies positive Fisher information, injective ordering, constant Fisher speed, clock adequacy, or global synchronization. Theorem C17.9. Complete physical equivalence of two SYM-5 observer candidates, including transport of states, channels, measurement classes, recoveries, and comparison protocols, preserves their accessible Fisher information, Fisher length, G18 clock status, comparison-edge existence, and loop-holonomy conjugacy class. Proof. This is the SYM-5 specialization of C14.3 and the previously proved observer-equivalence and comparison-transport theorems. QED. C17.10. Benchmark and falsification suite C17.11. Status and completion gate Historical-status notice. This subsection records the Version 36 SYM-5 status. Later bounded-scope executions may supersede individual open items, but this subsection remains controlling for the broader intrinsic S1 and S2 derivation problems unless a later theorem explicitly closes the same problem under the same scope. Version 36 extends the finite-dimensional SYM-5 foundation through two additional layers: conditional reconstruction of a covariant isotonic ETP physical output net from declared interaction-event structure, and physical effect-system reconstruction from admitted instruments. The output-net step is complete in the bounded finite-dimensional, interaction-relative scope. Effect-system theorem architecture is established, while model execution remains open. The sector-only observer-selection, interaction-to-seed, observer-sector incidence, exact admissible-core, family and conditional-uniqueness, objective-subclass, and observer-to-time results remain intact. Current Version 55 status after Phases C1 through C4. SYM-5 is not complete end to end. Completed bounded finite-dimensional results include the sector-only no-selection theorem, interaction-to-seed covariance, conditional output-net reconstruction, effect-system reconstruction, exact admissible observer core, and conditional uniqueness architecture. C17.3D robustness, C17.3E resources, C17.3F benchmark completeness, and C17.3G composition remain active reconstruction programs. Intrinsic S1 candidate-record derivation and S2 operational-boundary derivation remain open. Null, scalar, orbit-valued, nonunique, and restricted outcomes remain valid. Phase C4 dependency rule. No abstract, roadmap, benchmark, conclusion, Required Work item, claim-status ledger, or release-history statement may cite C17.3D, C17.3E, C17.3F, or C17.3G as a proved theorem family. Historical completion language is preserved only when explicitly labeled historical. Future certification requires model-level definitions, complete carriers, operation classifications, witnesses, falsifiers, and proofs. Transition to SYM-6. Categorical or Tannaka escalation is postponed until an independently supplied sector category, intertwiners, composition, duals, and a faithful fiber functor contain physical structure not captured by the group action and joint algebra. SYM-6 is not required merely because observer selection is nonunique. Provenance and reproducibility. The cumulative findings, sanitation rules, stage checkpoints, source registry, independent recovery-map reproduction, event-countermodel verification script, execution outputs, and SHA-256 manifest are preserved in S1S2_CLOSURE_COMPLETE_BACKUP.zip. External discovery results must be bibliographically and claim-by-claim verified before final reference integration. Stage 8, Composition-Origin and Higher-Order Compatibility Audit, must determine whether ETP-specific autonomous dynamics derives I_composition from the other five components or whether it remains independent. The decisive tests are an ETP-specific compatibility countermodel; conditions under which locality, commuting support, independent controls, or resource separability make the compatibility complex flag-like; and a complete RC cursor-detector-memory audit for genuinely higher-order incompatibility. S12.8. Next research gate Stage 7 controlling claim ceiling. The original five-component event package is refuted as complete. The six-component package is the strongest current candidate and has a conditional finite witness-completeness theorem. Universal ETP completeness, exhaustive architecture classification, nonfinite extension, and basement derivation of the separator family remain open. S2-OP2 remains downstream of effect-system, operational-boundary, and compositional closure. Individual admissibility does not imply joint operational closure. S2-OP1 remains open. Endpoint channels and marginal event signatures do not uniquely derive operational boundaries. Any selector must descend through refinement and compatibility quotients and must include physically admitted effects, instruments, interfaces, memories, controls, resources, complete-carrier structure, and composition laws. S1-OP2 remains open at universal recovery-semantic scope. H22.2 remains complete for the fixed finite-dimensional exact algebra-correction problem, while equality of I_rec is not proved sufficient for every recovery-family equivalence. S1-OP1 is substantially advanced but not closed. The program now contains support and intervention no-go theorems, conditional support and event-family constructions, autonomous model instantiation, exact endpoint countermodels, refinement classification, and compatibility-package enlargement. The physical origin and universal completeness of the separator and composition structures remain open. S12.7. Current S1 and S2 disposition The Stage 7 composition finding is naturally part of recovery geometry when it changes recovery realization structure. It is separately required in the pre-recovery event invariant because event architecture must be classified before a recovery geometry is derived. Universal completeness of I_rec=(I_content,G_rec) remains open. Benchmark S1-RC-N1. For qubit dephasing with p=0.3, the maximal exactly correctable algebra is the diagonal algebra span{I,Z}. The identity recovery and the complete-dephasing recovery are distinct CPTP maps but both correct that protected algebra, while their actions differ on coherent states. Thus a unique maximal recoverable algebra does not imply a unique recovery map. The benchmark does not prove universal map underdetermination, universal incompleteness of I_rec, or a third recovery-semantic axis. S12.6. Recovery-map benchmark and recovery-semantic boundary Theorem S12.7.4, Conditional Finite Witness Completeness. For a fixed bounded finite-dimensional candidate class with fixed carriers and interfaces, complete preparations and effects for every retained carrier, complete testers for every admitted context and ordered composition, every required adaptive strategy, complete support, exchange, and resource witnesses, no omitted material carrier, and a fixed equivalence relation, equality of all complete witness probabilities determines I_evt^(6) up to declared transport. Event-equivalence follows only when equivalence in the candidate class is defined by that complete physical signature. The strongest current candidate is I_evt^(6)=(I_support,I_action,I_record,I_exchange,I_resource,I_composition), where I_composition contains the compatibility hypergraph, composition operation and domain, event incidence, typed order relations, and physically distinct contexts. Theorem S12.7.2, Composition-Component Necessity. Any invariant intended to classify event architectures under an equivalence that preserves joint implementability must determine the compatibility hypergraph and physical composition law up to transported isomorphism. Theorem S12.7.1, Five-Component Event-Package Incompleteness. The five-component package is not complete when compatibility and composition are not contained in one of its components. Pairwise compatibility also does not generally determine higher-order compatibility. Countermodel S12.7-C1. Let two architectures share events {a,b,c} with identical individual support, action, record, exchange, and resource signatures. In Architecture A, every subset, including {a,b,c}, is jointly implementable. In Architecture B, every singleton and every pair is implementable, but the triple is not. The first five invariant components and all pairwise compatibility data agree, while the compatibility hypergraphs are nonisomorphic. The initial candidate invariant was I_evt^(5)=(I_support,I_action,I_record,I_exchange,I_resource). It classifies individual event signatures but need not classify which events are jointly implementable or how they compose. S12.5. Minimal event-separator package and compatibility countermodel Proposition S12.6.5, Refinement-Stable Boundary Requirement. Any S2 boundary selector must assign physically equivalent boundaries to refinement-equivalent event presentations. Otherwise the result depends on presentation information removed by the governing quotient. Theorem S12.6.3, Refinement-Separator Theorem. One admitted physical separator with a different outcome law proves refinement inequivalence. Separators may be intermediate interventions, persistent records, retained environments, independently controllable couplings, gauge-invariant exchanges, boundary or port distinctions, complete resource differences, or multitime response differences. A formal insertion point is not a separator. Theorem S12.6.2, Separator-Free Refinement Equivalence. A refinement is equivalent to its coarse presentation only conditionally: the composed transformations agree on every admitted state; no intermediate control, output, record, resource, boundary, support, charge, or disturbance distinction is admitted; every compatible external experiment factors through the same coarse process; and the witness family is complete for the declared refinement question. Without witness completeness, the status is restricted-null rather than universal physical equivalence. Theorem S12.6.1, Continuous-Family Refinement Nonselection. Even a complete reduced semigroup Lambda_(t+s)=Lambda_t composed with Lambda_s does not, by itself, select a discrete finite partition of an interval. Every finite partition supplies a valid factorization, while no partition point becomes a physical event without an independently admitted separator. Channel divisibility is therefore not event ontology. Definition S12.6.1. A noncircular pre-event datum may include the physical algebra, state family, transformation family, independently physical controls, invariant supports, conserved exchanges, admitted witnesses, physical automorphisms, and the governing physical-equivalence relation. It is pre-event only when it does not already insert the event count, event labels, preferred partition, event-specific memories, or intervention slots that the reconstruction purports to derive. S12.4. Pre-event datum and refinement equivalence Theorem S12.5.1, Endpoint-Channel Event-Architecture Underdetermination. Strictly event-inequivalent finite architectures can induce the same endpoint CPTP channel. More generally, q_n=(1-(1-2p)^(1/n))/2 gives Phi_(q_n)^n=Phi_p for every positive integer n. Endpoint dephasing therefore does not identify event count. This does not establish complete physical equivalence: admitted intermediate interventions, persistent memories, retained environments, resource exchanges, or multitime statistics may separate the realizations. Let Phi_p(rho)=(1-p)rho+p Z rho Z be the qubit dephasing channel. It preserves diagonal entries and multiplies coherence by 1-2p. Set p=0.3 and q=(1-sqrt(0.4))/2. Then (1-2q)^2=0.4=1-2p, so Phi_q composed with Phi_q equals Phi_0.3 on the complete qubit operator space. The first presentation contains one non-null event, while the second contains two ordered non-null events. S12.3. Endpoint equivalence and exact event countermodel Theorem S12.4.1, Autonomous Event-Datum Instantiation. A complete finite-dimensional autonomous model with invariant coupling supports, internal controls, memories, detectors, retained and discarded outputs, incidence, resources, and covariance can instantiate an internal event datum without an external observer. A time-independent generator does not make its decomposition intrinsically unique. The RC battery-clock-cursor-detector-memory construction is therefore a model-relative event realization, not a proof that bare ETP structure uniquely selects that architecture. Theorem S12.3.4, Conditional Event-Family Theorem. A finite nonempty partially ordered family of physically realized, null-quotiented, support-localized interventions has support-minimal non-null elements forming a covariant event-atom family. An infinite descending chain of non-null interventions with no minimal element need not supply event atoms. Theorem S12.3.1, Undifferentiated-Dynamics Intervention No-Go. One undifferentiated transformation need not determine one intervention decomposition. If invertible mixing of tangent directions preserves the complete datum, only the tangent subspace or invariant decomposition is reconstructible, not a preferred basis of elementary interventions. S12.2. Intervention structure and autonomous event realization Theorem S12.2.3, Covariant Intervention-Support Extraction. Given independently physical interventions, composition and compatibility relations, and consequence algebras that descend through physical equivalence, the support-minimal non-null consequence algebras form a covariant interaction-support family. A support becomes a candidate record support only when its induced channel separates admitted source states under the declared physical effect family. This establishes record capacity, not persistence, recoverability, readability, boundary emergence, or clock adequacy. Theorem S12.2.1, Undifferentiated-Transformation Selection No-Go. If a complete-datum automorphism exchanges two proper output supports while preserving every input component, no covariant selector using only that datum can uniquely select one while excluding the other. This is a no-selection theorem, not a proof that physical support never becomes distinguished after independently physical enrichment. The common unresolved dependency for intrinsic S1 record derivation and S2 boundary derivation is the map from a preboundary physical interaction datum to a covariant intrinsic record-support family. S1 logically precedes S2 in the present staging unless both are reconstructed simultaneously from a deeper complete physical object. A valid selector may return an empty family, a scalar family, an automorphism orbit, or several inequivalent candidates. S12.1. Dependency root and intrinsic-support boundary Scope and authority. This controlled integration records the S1/S2 closure research through Stage 7. It supplements C17.3A, C17.3B, C17.3G, H20, H22, and H22B without replacing their proved results or enlarging their claim scopes. The cumulative scientific and provenance authority is S1S2_CLOSURE_PROGRAM_MASTER_LEDGER.md. Results remain bounded to the declared finite-dimensional candidate classes, carriers, equivalence relations, witness families, and exact or approximate conventions stated below. S1/S2 CLOSURE RESEARCH INTEGRATION THROUGH STAGE 7 S1/S2 STAGE 8 RESPONSE-LAW, PROCESS, AND MEMORY INTEGRATION THROUGH STAGE 8R Scope and narrative role. Stage 8 tests whether the compatibility and composition component identified at Stage 7 follows from the remaining event data or requires additional response-law, tester, process-carrier, and memory structure. The integrated result is a controlled sequence of finite-dimensional theorems, countermodels, and model executions. It strengthens the process-level architecture without claiming intrinsic S1 record-channel selection, intrinsic S2 boundary selection, universal recovery-semantic completeness, endogenous temporal order, or a nonfinite process theorem. Stage 8 authority and evidence rule. The scientific authority for Stages 8A through 8R is S1S2_CLOSURE_PROGRAM_MASTER_LEDGER.md together with the preserved stage audits, scripts, outputs, and manifests. The statements below are classified individually as definitions, finite-dimensional propositions, conditional theorems, countermodels, or model executions. No compressed summary imports a stronger proof than its controlling artifact. S12.8A-8G. Pre-event datum and response-law provenance Status. Cumulative conditional reconstruction and no-selection findings. Full stage hypotheses and provenance remain controlled by the cumulative master ledger; this paragraph is a referee-facing synthesis, not a replacement proof. Stages 8A through 8G separate typed physical carriers, pre-event restrictions, selectors, asymmetries, complete-invariant reconstruction, and response laws. The cumulative conclusion is that a pre-event datum can constrain admissible responses only relative to its declared carrier, restriction class, covariance group, witness family, and equivalence relation. No response law is derived from event labels, outcomes, or temporal ordering that the reconstruction is intended to explain. Residual symmetry, balanced orbits, or missing restriction carriers can leave the admissible response family nonunique or empty. S12.8H. Response-Law Axiom Independence and Separation Proposition S12.8H. The six response-law predicates listed below are logically independent unless one declared theorem supplies the missing implication. Proof status: finite countermodel separation within the declared candidate classes. Response-law existence, uniqueness, covariance, composition, normalization, and operational adequacy are logically distinct predicates. Equality of support, action, record, exchange, resource, and compatibility data does not by itself identify one response instrument unless the response law is included in the governing datum or is derived by an independent theorem. A separating tester establishes inequivalence only within the declared candidate and access classes. S12.8I. Balanced-Orbit Resolution and Enlarged Outputs Theorem S12.8I, Balanced-Orbit No-Selection. Under a nontrivial residual orbit with no globally fixed candidate, every covariant deterministic selector built only from orbit-preserving data fails to select one representative. Enlarged or stochastic outputs remain orbit-level constructions unless independently physical asymmetry is added. When candidate outputs lie in a balanced residual orbit, no covariant deterministic selector chooses one representative from orbit-preserving data alone. Stochastic or enlarged-output constructions may return the orbit, an invariant mixture, a relation, or an equivalence class, but they do not create a preferred representative. Any asymmetric resolution must be traced to independently physical structure. S12.8J. Instrument Provenance, Neutrality, and Composition Proposition S12.8J, Neutrality Noninheritance. Equality of outcome effects does not imply equality of post-response maps, and one-slot neutrality does not imply neutrality under composition, memory retention, adaptive control, or conditioning. Outcome effects, post-response maps, and complete instruments must remain distinct. A neutral outcome distribution does not imply neutral post-response dynamics, and local neutrality need not be stable under composition, retained memory, adaptive control, or conditioning. The complete operation signature and its physical provenance are therefore part of the event and boundary audit. S12.8K. Finite Instrument Equivalence and Complete Witnesses Theorem S12.8K, Finite Instrument Tomography. Within a fixed finite-dimensional matched outcome class, equality of all outcome-map Choi operators is necessary and sufficient for instrument equality; a reference-assisted tomographically complete one-slot witness family is complete for that declared class. For finite-dimensional matched outcome-indexed instruments, instrument equivalence means equality of every completely positive outcome map, while effect equivalence means equality only of the corresponding effects. Equality of the outcome-map Choi operators is necessary and sufficient for instrument equality. A reference-assisted tomographically complete one-slot witness family is complete for the declared finite-dimensional instrument class. Equal effects can nevertheless conceal distinct post-response dynamics that later adaptive tests separate. S12.8L. Multitime Completion and Tester Hierarchies Countermodel family S12.8L. Equality of single-time marginals, endpoint channels, or a fixed nonadaptive tester family does not imply equality under a richer multitime adaptive tester hierarchy. Single-time marginals, endpoint channels, and fixed nonadaptive tests need not determine a multitime process. Process-level equivalence is indexed by the admitted tester hierarchy, including slot structure, references, memory, adaptivity, branch access, and ordered composition. A finite-memory representation is a resource-relative model statement, not a proof of minimal Markov order, minimal bond dimension, instrument-independent memory, or indefinite causal order. S12.8M. Complete-Carrier Causal Breaks and Order Provenance Theorem boundary S12.8M. Causal screening is asserted only for the declared complete carrier and admitted tester family. This instrument-relative boundary is consistent with the quantum Markov-order literature and does not establish instrument-independent finite order. [86-88] A causal break screens the remote history only relative to a declared complete carrier and a complete admitted tester family. Re-preparing the visible system can leave history in a retained memory, environment, reference, control, or outcome register. CP divisibility of reduced dynamics is neither equivalent to process Markovity nor sufficient for complete screening. Finite Markov order remains instrument relative. A directed influence relation yields an order only after asymmetry, acyclicity, and quotient compatibility are independently established; an order-neutral datum cannot covariantly select a direction without added structure. S12.8N. Approximate Screening, Memory Strength, and Witness Minimality Proposition S12.8N. Tester-class enlargement cannot decrease the operational screening error, and compatible finite replacement steps obey the stated telescoping bound. Horizon-uniform screening requires summable or otherwise globally controlled errors. For tester classes Test_R contained in Test_Rprime, operational screening error is monotone under enlargement: epsilon_R <= epsilon_Rprime. Under the total-variation convention, epsilon_R bounds admitted event probabilities and bounded-payoff expectations. Compatible finite replacement steps obey epsilon_total <= min(1, sum_k epsilon_k). Terminal marginals can remain identical while retained-history tests distinguish processes; therefore finite-horizon compression does not imply exact finite quantum Markov order. Minimal witness claims are meaningful only relative to a fixed candidate class and declared resource order. S12.8O. Process Divergences, Conditional Mutual Information, and Horizon Uniformity Conditional theorem schema S12.8O. State-level conditional-mutual-information recovery is not imported as a process theorem. Operational process screening follows only if the reconstructed object satisfies the recursive causal constraints, the recovery is a physical causal process, the complete carrier is covered, and a proved bound dominates the declared tester distance. [89] Operational tester distance is contractive under admitted superprocesses whose pulled-back testers remain in the original class. A divergence between normalized Choi-like states is not automatically an operational process divergence. A conditional-mutual-information recovery statement becomes an operational screening theorem only after causal-process validity, physical recovery typing, complete-carrier coverage, and domination of tester distance are proved. Summable incremental errors yield a horizon-uniform telescoping ceiling; a constant positive local error gives only min(1,n epsilon). Reduced-state forgetting is weaker than complete process screening. S12.8P. Process Carrier, Causal Normalization, and Recovery Realizability Proposition S12.8P. Positivity, causal normalization, physical recoverability, and task-complete carrier sufficiency are separate gates. Failure at any gate prevents promotion to a deterministic process-recovery theorem. A finite process carrier is sufficient for a declared task only when equality on its complete admitted testers implies equality for every physical experiment in that task. Positive Choi-like operators must also satisfy the recursive causal trace constraints. A physical recovery must be completely positive with admitted references, have the correct deterministic or branch trace behavior, respect slot causality, and close on the declared carrier and resources. The exact diagonal H-M-F benchmark realizes a trace-preserving recovery, while a positive subnormalized retained branch demonstrates that positivity does not establish deterministic-process validity. The current ETP corpus does not determine one smallest complete process carrier without presupposing part of S2. S12.8Q. Noncommutative Two-Slot Benchmark and Sector Assembly Benchmark S12.8Q. The amplitude-damping/phase product comb is an exact finite causal feasible point. The dephasing |+> and |-> pair is an exact counterexample to assembling one full-qubit recovery from sectorwise diagonal correction. An amplitude-damping collision followed by a phase unitary supplies a noncommutative two-slot product comb. Its Choi operator is positive and satisfies the recursive comb constraints, providing an explicit feasible point in the finite SDP set. Positive scaling preserves positivity while violating deterministic normalization, so positivity alone is insufficient. Complete dephasing preserves each one-dimensional diagonal sector but admits no CPTP recovery of arbitrary qubit coherences because the orthogonal states |+> and |-> have the same dephased output. Sectorwise correction therefore does not imply one global recovery. The product-comb result is memoryless and does not derive its slot order. S12.8R. Memory-Bearing Collision Comb and Coherence-Recovery Boundary Numerical Benchmark S12.8R. The values below are model-executed at theta=0.63 under the declared two-collision partial-SWAP protocol. The benchmark is not a process-tomography result, a minimal-memory theorem, or a noisy nearest-causal optimization. A retained memory qubit is carried across two partial-SWAP collisions with theta=0.63. Between collisions the visible system is discarded and re-prepared in |0>. With memory retained, the final-system trace distances are 0.22661900463748627 for histories 0 versus 1 and 0.47604517079525777 for histories + versus -. Resetting the memory makes both tested distances zero. The future coherences for the + and - histories have opposite signs, proving partial phase-sensitive transmission. Because the orthogonal input trace distance contracts below one, no CPTP map acting only on the final system can exactly recover both coherent inputs. The unitary-dilation process is explicitly causal and therefore has zero distance to the causal feasible set; no noisy nearest-causal projection, SDP optimum, or dual certificate is claimed. Stage 8R methods and reproducibility signature. Let U_theta=cos(theta)I-i sin(theta)SWAP act on each declared qubit pair. The initial visible-system histories are |0>, |1>, |+>, and |->, the retained memory is not reset between the two collisions, and the visible system is discarded and re-prepared in |0> between collisions. The compared output is the final visible system after tracing the retained memory and any other unselected carrier. Trace distance uses D(rho,sigma)=(1/2)||rho-sigma||_1. The reset-memory comparator applies the declared memory reset before the second collision. The preserved Stage 8R script, machine-readable output, and artifact manifest control the reported digits and must accompany submission data. This subsection records the analytical protocol and convention; it does not claim independent rerunning within the manuscript binary. Stage 8 Integrated Claim Boundary The integrated Stage 8 sequence proves finite, declared-class separations among effects and instruments, endpoint and process equivalence, reduced and complete carriers, exact and approximate screening, Choi-like and operational divergences, state and process recovery, sectorwise and global correction, memoryless and memory-bearing combs, and partial coherence transmission versus exact full-algebra recovery. It does not close intrinsic S1 or S2, derive a unique response law or process carrier, establish a process-operational CMI theorem for physical ETP, prove minimal Markov order or bond dimension, derive endogenous order, or extend the results to continuum, infinite-dimensional, device-independent, or indefinite-order settings. Stage 8 Manuscript Disposition M8-R1 referee classification. The integrated section is now explicitly typed and synchronized at abstract, authority, methods, claim-status, required-work, bibliography, and release-control levels. Full proof details remain in the authoritative stage artifacts where this manuscript identifies a synthesis rather than reproducing the complete derivation. Stages 8H through 8R are integrated as theorem, proposition, countermodel, and benchmark statements under their declared scopes. The cumulative evidence supports a complete-carrier, tester-relative process architecture and a retained-memory quantum benchmark. It does not replace the still-open intrinsic record-channel and operational-boundary derivations. Subsequent work should begin from this synchronized binary and the cumulative S1/S2 master ledger rather than from isolated stage appends. M9 STAGE 8S-8Z MANUSCRIPT SYNCHRONIZATION Scope and authority. This controlled integration extends the Stage 8 manuscript record from Stage 8R through Stage 8Z. The cumulative scientific authority is the canonical S1S2_CLOSURE_PROGRAM_MASTER_LEDGER.md together with the preserved stage audits, scripts, outputs, manifests, and checksums. Each result remains limited to its declared finite model, carrier, tester, target, metric, symmetry action, and exact or approximate convention. No compressed synthesis upgrades a conditional theorem, countermodel, numerical benchmark, model execution, open problem, or nonfinite extension. S12.8S. Process-Operational Recovery Bridge For two orthogonal targets recovered by one CPTP decoder with worst-case trace-distance error epsilon, the selected-output trace distance must be at least 1-2 epsilon. The aligned binary symmetric channel saturates this bound under the declared target, output, decoder, prior, and metric. Exact recovery of a selected protected subsystem can coexist with maximal history dependence in an independent retained carrier, so selected-output recovery does not imply complete-carrier screening. A state-level conditional-mutual-information recovery statement becomes an operational process theorem only after causal representation, physical superprocess, complete-carrier, and tester-distance domination gates are established. S12.8T. Complete-Carrier Target Alignment and Algebra-Indexed Memory Memory and recovery are indexed by the target distinction family, complete carrier, preparations, testers, metric, and horizon. In the finite CNOT-dephasing benchmark, the selected system exactly preserves the diagonal algebra, while coherent plus and minus targets have identical system and environment marginals but orthogonal joint system-environment outputs. Selected-system full-qubit recovery therefore has worst-case error at least one half, while inverse-CNOT complete-carrier recovery is exact on the tested family. Diagonal-algebra recovery, full-algebra recovery, and complete-carrier memory are distinct claims. S12.8U. Explicit Collision-Record Gate Execution In the supplied four-fragment collision-event net, the ideal strong-collision model passes persistence under the declared retention law, basis-record distinguishability, Z-recordability, exact Z-recoverability, fourfold redundancy, common-record consistency, fragment-permutation covariance, and local distinguishability robustness. The null interaction fails the distinguishability gate. The execution yields a nonempty covariant candidate record-channel family in the declared model and a null family at zero interaction, but it does not derive the event net or a unique observer domain intrinsically. S12.8V. Output-Event-Net Provenance, Decomposition Stability, and Boundary Nonuniqueness A support-derived event net requires represented local algebras, a physically distinguished interaction-term decomposition, and readable activation provenance. Final states alone do not identify a unique schedule. Local basis changes and fragment permutations preserve the ideal record family, while an entangling refactorization can preserve global branch distinguishability and reduce perfect single-fragment redundancy. Conserved charges may restrict a decomposition without uniquely selecting it. In a transitive fragment orbit, permutation-invariant gates do not select one labeled boundary. S12.8W. Autonomous Activation and Event-Order Reconstruction A finite time-independent controller model reconstructs active interaction-support classes when orthogonal readable controller states and an injective activation signature are supplied. Controller transitions and the support map then reconstruct relational succession covariantly. Autonomy alone is insufficient when signatures are noninjective. Undirected adjacency or a uniform controller state does not fix orientation, and cyclic symmetry does not select a first event. Controller factorization, pointer basis, support map, preparation, and readability remain supplied model data. S12.8X. Controller Readability and Circularity For two pure controller pointer states with overlap c, optimal assignment error is (1-sqrt(1-c^2))/2. Independent product records reduce overlap to c^n under the declared independence model, while the unsharp-Z benchmark displays a measurement-information versus coherence-retention tradeoff. Controller labels cannot certify their own semantics because interpretation requires an independently specified correlation model and tester semantics. Independently grounded common records can break the immediate circle conditionally, but their provenance remains part of the complete physical datum. S12.8Y. Independent Calibration Network and Regress Termination A connected exact binary calibration graph reconstructs node conventions from relative edge relations up to one global complement. Cycle consistency can falsify incompatible reports but does not prove absolute correctness. One independently physical anchor fixes a representative. Independent common-target redundancy can reduce assignment error, whereas common-mode complement corruption defeats arbitrary copy number. The calibration regress terminates without an anchor only for gauge-invariant relative claims represented by the global-complement equivalence class. S12.8Z. Physical Anchor Admissibility and Gauge-Invariant Semantics A representative-selecting anchor must pass physical provenance, readability, residual-symmetry noninvariance, target independence, robustness, and complete-carrier gates. For the declared global-Z2 complement action, one physically certified nonzero odd scalar has trivial stabilizer and selects one representative. An ungrounded fixed label is a gauge convention; a flip-invariant parity constraint may restrict an orbit but cannot select its member; and a vanishing or uncertainty-dominated odd quantity does not robustly break symmetry. Orientation breaking and first-event selection are separate symmetry reductions. Theorem S12.8Z.5, Quotient Sufficiency If every admitted preparation, process, tester probability, recovery claim, and composition rule is constant on residual symmetry orbits, then the quotient datum is sufficient for those operational claims and no representative-selecting anchor is required. Absolute labels, unique orientation, first-event identity, unique record representatives, and unique observer boundaries cannot be recovered from quotient data when they vary along a residual orbit. Such claims must either be removed or supported by an admissible physical anchor. M9 Integrated Stage 8 Claim Boundary Stages 8S through 8Z complete the finite audit sequence for recovery-target alignment, complete-carrier memory, explicit record qualification, event-net provenance, autonomous activation, controller readability, calibration regress, anchor admissibility, and quotient semantics. The block does not derive an intrinsic ETP record channel, operational boundary, physical anchor, temporal orientation, first event, thermodynamic arrow, device-independent certification, or nonfinite extension. Stage 8 is programmatically complete within its declared theorem, countermodel, and model-execution scopes. Intrinsic S1 and S2 remain open and become the principal Stage 9 frontier. M9 Claim-Status Synchronization M9 Referee Disposition PASS WITH CLAIM-CEILING PRESERVATION. The Stage 8S-8Z block strengthens complete-carrier, target, provenance, calibration, symmetry, and quotient discipline. It does not justify strengthening the abstract into an intrinsic emergence-of-time claim. Publication-facing summaries must retain S1 and S2 as open and must distinguish gauge-invariant quotient reconstruction from representative-level physical selection. M9 STAGE 9B.1-9B.11 INTERIM MANUSCRIPT SYNCHRONIZATION Scope and authority. This controlled, checksum-distinct integration synchronizes the Stage 9B provenance program through Stage 9B.11. The scientific authority is S1S2_CLOSURE_PROGRAM_MASTER_LEDGER.md together with the preserved stage audits, scripts, machine-readable outputs, manifests, and checksums. The integration preserves the active Theory of Emergent Timeless Potentiality (ETP) name and retains Theory of Relational Recoverability only as a viable candidate name. No summary below replaces a stage proof or enlarges a theorem beyond its declared carrier, candidate class, tester family, metric, horizon, operation class, physical-equivalence relation, or exact/approximate convention. M9 Stage 9B abstract synchronization Stages 9B.1 through 9B.11 extend the intrinsic S1/S2 audit from strict preboundary data to process carriers, operation algebras, controller and reset provenance, catalytic and infinite-horizon closure, stationary open currents, and reservoir-gradient origins. The cumulative result is constructive but restrictive: exact finite models support conditional response, recurrence, recovery, current, and resource-accounting statements once the necessary carriers and operations are supplied, while the audited data do not intrinsically derive a unique response law, intervention algebra, process carrier, controller semantics, reusable reset resource, output boundary, nonequilibrium reservoir, or thermodynamic orientation. Intrinsic S1 candidate-record derivation and S2 operational-boundary derivation therefore remain open. 9B.1-9B.4: Preboundary data, qualification, testers, and slot-free process candidates The first Stage 9B block separates a strict preboundary datum from structures that already contain the event, slot, tester, or boundary organization under investigation. Candidate carriers and qualification laws must be independently physical and covariant. A candidate-independent tester family must be complete for the declared comparison before indistinguishability can support more than restricted nullity. Slot-free correlation or process candidates can encode operational behavior only after causal validity, composition, carrier completeness, and physical tester semantics are supplied. These results constrain admissible reconstructions but do not select one intrinsic candidate from residual symmetry or underdetermination. 9B.5-9B.7: Operation algebra, controller, reset, reference, and budget provenance The second block treats the operation algebra and history-composition law as independently typed physical data unless they are derived by a separate theorem. A controller state, pointer basis, activation support, interface, and readable transition signature can conditionally reconstruct a finite event order, but covariance prevents a unique semantic assignment when automorphisms exchange the candidate controller descriptions. Reset operations, reference tokens, work stores, batteries, clocks, discarded outputs, and final correlations belong to the complete carrier. A local subsystem reset does not establish cyclic return of the complete implementation, and a reusable-resource claim requires return under the declared physical equivalence rather than recovery of only the selected machine marginal. 9B.8-9B.9: Catalytic return, complete cycles, and infinite-horizon closure Exact catalysis, correlated catalysis, and approximate return are distinct resource claims. Exact return of a catalyst marginal permits residual correlations unless complete product return is separately required. For exact finite reversible dynamics, periodic nonidentity recurrence can persist indefinitely when the task itself is reversible and no persistent external output is claimed. This positive recurrence result does not establish productive closure. Any distinguishable work token, record, heat increment, changed load, emitted carrier, or accumulated sink state enlarges the complete accounting universe. A bounded finite resource coordinate cannot exhibit strictly positive drift forever, and constant positive per-cycle error supplies no nontrivial horizon-uniform guarantee under the finite telescoping bound. 9B.10-9B.11: Stationary currents, open autonomy, and nonequilibrium resource origin Stationarity permits nonzero circulation: for an oriented finite network, internal balance Bj=0 need not imply j=0. A stationary machine marginal can therefore support a current between supplied reservoirs or into a load. Such behavior establishes, at most, open autonomy under fixed couplings. It does not establish energetic or material closure. Current, heat, work, source, and sink labels are boundary relative and remain physically meaningful only when the carrier assignment, orientation, exchanged quantity, coupling law, and accounting maps are declared together. A finite reservoir gradient depletes under uncompensated productive transfer. Maintaining it requires compensating inflow, an external or internal replenishment carrier, an ideal-reservoir approximation, or an enlarged resource account. Treating a prepared nonequilibrium reservoir as free relocates the resource-origin problem upstream rather than resolving it. Consolidated Stage 9B theorem and claim boundary Within the declared finite and model-relative scopes, the integrated block establishes conditional reconstruction rules, no-selection and underdetermination results, complete-carrier expansion requirements, finite reset and positive-drift obstructions, exact reversible recurrence, stationary-current bookkeeping, finite sink-capacity limits, finite gradient depletion, maintenance balance, and explicit provenance classifications. It does not establish an intrinsic ETP operation algebra, process carrier, controller, reset law, reference token, resource monotone, source reservoir, sink, work/heat split, physical boundary, stationary-current orientation, nonequilibrium preparation law, thermodynamic arrow, device-independent closure, or nonfinite theorem. Thermodynamic-direction firewall The Stage 9B resource and current results do not derive a thermodynamic arrow. They show how reset cost, resource depletion, maintained gradients, stationary currents, output sinks, and finite-horizon closure can be audited after the physical carriers, boundaries, reference structures, and accounting laws are supplied. A direction becomes physically meaningful only under independently justified asymmetry, orientation, reservoir, record, and entropy-production data. No gradient, current sign, controller transition, reset convention, or fixed bath parameter is promoted to an intrinsic temporal orientation. Stage 9B evidence and reproducibility rule The exact finite benchmarks and countermodels are evidentiary components of the scoped results and remain controlled by their scripts, outputs, manifests, and SHA-256 records in the Recovery archive. Numerical or exhaustive finite verification supports only the model and claim class stated by its artifact. External literature supports distinctions among autonomy, open reservoirs, limit cycles, resource theories, and preparation costs, but it does not establish the ETP-specific intrinsic-origin claims. Every publication citation must be verified at point of use before final reference integration. Interim Stage 9B disposition and next frontier Historical Stage 9B.1-9B.11 checkpoint. At that earlier synchronization point, Stage 9B.12 was the next gate. That frontier is superseded by the completed Stage 9B.12-9B.26 and Stage 9C.1-9C.4 program recorded below. M9 STAGE 9B.12-9C.4 FINAL STAGE 9 SYNCHRONIZATION Authority and scope. This section synchronizes the Stage 9 record beyond the earlier Stage 9B.1-9B.11 checkpoint. The cumulative S1S2_CLOSURE_PROGRAM_MASTER_LEDGER.md and preserved stage artifacts control over compressed narrative. No conditional, finite, numerical, model-executed, restricted, open, or underdetermined result is upgraded. ETP remains the active name; TRR remains a candidate only. Stage 9B.12-9B.26 consolidated scientific result Stages 9B.12-9B.13 prove finite provenance and minimal-basis constraints while preserving that cyclic explanation does not self-ground and minimal sufficiency need not be unique. Stages 9B.14-9B.17 execute the finite reservoir-discrimination program. The controlling Stage 9B.17 correction is zero optimized distance through collision 78; positive numerical minima begin with the longer prefix through collision 90; the grid study is not an analytic global certificate; and the model classes intersect in closure as heterogeneity approaches zero. Stages 9B.18-9B.24 construct labeled and label-free witnesses, finite instruments and estimators, bounded anytime-valid e-processes, calibration and drift audits, exact rank nonidentifiability, and explicit witness-family counterexamples. Stages 9B.25-9B.26 establish selected-view equality with complete-carrier and future inequivalence, local-reset export, a device-relative simulator protocol, and a feasibility ceiling. These results are finite and carrier-, instrument-, tester-, calibration-, and model-relative. Stage 9C.1 dependency reconciliation The reconciled architecture contains 16 controlling dependency nodes and 51 directed dependencies. Every current path to intrinsic S1 or S2 depends on at least one underived physical bridge, physical-equivalence rule, complete carrier, operation/tester architecture, reference/access structure, or boundary-defining datum. Ten controlling obligations remain: three S1 obligations, four S2 obligations, one physical-experiment obligation, one nonfinite-extension obligation, and one manuscript-control obligation. Intrinsic S1 and S2 therefore remain OPEN. Figure M9-C1. Stage 9C.1 dependency graph. Node colors record ROOT/OPEN, OPEN, CONDITIONAL, MODEL-EXECUTED, and UNDERDETERMINED status. Stage 9C.2 theorem, countermodel, ceiling, and falsifier consolidation The cumulative ledger yields 109 raw lexical theorem, proposition, corollary, countermodel, or counterexample entries and 36 grouped publication-facing controlling records. Each controlling record keeps together the identifier, result class, exact scope, evidence basis, claim ceiling, falsifier or restriction, and approved manuscript verb. “Proves” is reserved for analytic results with local hypotheses; “establishes conditionally” names supplied structure; “demonstrates” describes exact or numerical model execution; “supports” marks bounded evidence; “is consistent with” preserves nondiscriminating compatibility; and “falsifies” applies only to a declared quantified target. Incomplete searches remain RESTRICTED-NULL or UNDERDETERMINED. Figure M9-C2. Grouped classes in the Stage 9C.2 controlling result register. Grouping does not replace the preserved raw inventory. Stage 9C.3 publication and closure audit The audit found strong scientific claim discipline but major publication-control defects in the interim manuscript: stale front-matter chronology, active-looking historical frontier language, incomplete Stage 9 claim-status tables, absent Stage 9C synthesis, compressed theorem locality, incomplete point-of-use reproducibility mapping, and incomplete Stage 9 bibliographic verification. The present successor remediates the first five defects at manuscript level and records the remaining release qualifications instead of silently certifying them. Figure M9-C3. Stage 9C.3 publication-readiness audit underlying the controlled remediation. Current Stage 9 claim-status ledger Point-of-use reproducibility crosswalk Final Stage 9 release disposition FINAL STAGE 9 AUDIT RELEASE WITH OPEN SCIENTIFIC OBLIGATIONS. The planned Stage 9 audit architecture is closed. Intrinsic S1 and intrinsic S2 remain open. Sequential process validity remains conditional. The device protocol remains model-executed. Laboratory ETP evidence, device-independent falsification, and nonfinite closure are not established. Historical freezes and frontiers remain genealogy only where superseded by this checksum-distinct successor. Post-Stage-9 frontier. Further work must either address a declared open scientific obligation with genuinely new physical structure or proof, transfer a boundary-dependent obligation to Local Relational Geometry with provenance intact, or perform journal-specific packaging. No new finite toy-model chain may be presented as intrinsic S1/S2 closure without closing the root dependencies recorded by Stage 9C.1. IV. OBSERVER-ALGEBRA RECONSTRUCTION, FACTORIZATION, AND RECORD-CHANNEL SELECTION Role in emergence. Once the physical arena is fixed, temporal information must be accessible somewhere. This section derives the algebraic form of observer-relative access and identifies stable and recoverable records that can carry temporal distinctions. H0. Observer-access starting point At Layer 2, the operative primitive is an accessible observable algebra, not a tensor factor. This is a temporal-reconstruction input, not a basement ontology. Work in a finite-dimensional physical algebra A_phys that is already physically admitted under the Layer 0 bridge and Layer 1 event datum. Proposition H0: neither H_phys nor (H_kin,J) uniquely selects a nontrivial proper observer algebra. A unitary symmetry W commuting with J maps any candidate A_O to another equally constraint-compatible algebra. Additional invariant access data are necessary. QED. H1. Seed-relative Observer Reconstruction A_O=W*(S_O)=S_O double-prime. Theorem H1. For a unital self-adjoint S_O in finite-dimensional A_phys, A_O is the unique smallest finite-dimensional von Neumann algebra containing S_O. Any B containing the same seeds contains A_O. Under a physical *-isomorphism phi, W*(phi(S_O))=phi(W*(S_O)). Proof: intersect all unital *-subalgebras containing the seeds, use finite-dimensional closure and the double-commutant theorem, and note that *-isomorphisms preserve generation. QED. H2. Finite-Dimensional Algebraic Factorization H_phys ~= direct-sum_z (H_O,z tensor H_Oc,z), A_O ~= direct-sum_z (B(H_O,z) tensor I_Oc,z). The commutant acts conversely and the center is direct-sum_z C I_z. Tensor factors arise only sectorwise, up to local unitaries, multiplicity relabeling, and permutation of isomorphic sectors. The observer state is the restriction omega|A_O; partial trace is a derived representation. H3-H5. Physical equivalence, inclusion, and invariance Physical equivalence requires a physical *-isomorphism, observer relabeling, transported states, admissible measurements, and comparison protocols. Finite-dimensional equivalence is unitarily implementable after matching block multiplicities. Joint observers are algebraic joins; common content is the intersection. If A_1 is contained in A_2 and M_1 is contained in M_2, then F_1^acc <= F_2^acc. Under physical equivalence, outcome distributions, accessible Fisher information, Fisher length, endpoint admissibility, comparison-edge existence, and holonomy conjugacy classes are invariant. H6. Executed nonfactor benchmark For H_phys=(C^2 tensor C^2) direct-sum C^2, the declared first- and second-qubit seed families give A_1=[B(C^2) tensor I_2] direct-sum C I_2 and A_2=[I_2 tensor B(C^2)] direct-sum C I_2. Their shared center has dimension 2; each observer algebra has dimension 5; their join has dimension 17; and B(H_phys) has dimension 36. Thus joint access need not recover the full physical algebra. H7. Finite-dimensional observer reconstruction and progression Version 13 establishes finite-dimensional observer reconstruction and representation stability relative to declared seeds. Versions 17 and 22 then advance the emergence chain by deriving total record channels from interactions and proving the maximal finite-dimensional recoverable algebra. The remaining derivation and dimensional extensions are consolidated in the observer-stage status tables. H8. Scope of dynamics-relative selection Version 17 embeds the dynamics-relative exact-core construction as Stage S3 of the observer-emergence architecture. It identifies the largest exact algebra that is simultaneously stable under the declared physical channel family and readable through the record protocol, thereby converting physical interaction data into a precise observer-access layer. H9. Selector datum and operational support D_O=(A_phys, Omega, {Phi_a,t^dagger}, E_O^dagger, Eff_O, Cl_O). Active-map signature for H9-H12. Let P be the declared support projector, with A_phys,P=P A_phys P; when P=I this is A_phys. For each channel label a and each t in the declared time or control domain I_a, Phi_a,t^dagger:A_phys,P -> A_phys,P is a UCP Heisenberg-picture map, with Schrödinger predual Phi_a,t,*:S(A_phys,P) -> S(A_phys,P). Let E_O,*:S(A_phys,P) -> S(B_O) be the declared record channel and E_O^dagger:B_O -> A_phys,P its UCP adjoint. The joint index domain, support, continuity assumptions, and exact or approximate convention are part of D_O. A_phys is faithfully represented and gauge reduced; Omega separates the supported algebra; Phi_a,t^dagger are active unital completely positive Heisenberg maps; E_O^dagger:B_O->A_phys is the adjoint record channel; and Eff_O is the directly admitted output-effect system. The datum already contains an output carrier and protocol, so the construction is canonical relative to D_O, not a derivation from bare potentiality. H10. Exact stable multiplicative algebra MD(Phi^dagger)={X: Phi^dagger(X*X)=Phi^dagger(X*)Phi^dagger(X), Phi^dagger(XX*)=Phi^dagger(X)Phi^dagger(X*)}. A_stab=intersection_(a,t)[MD(Phi_a,t^dagger) intersection Fix(Phi_a,t^dagger)]. Lemma H10. A_stab is a unital C*-subalgebra. Proof. Each multiplicative domain is a unital C*-subalgebra. Fixed elements in the multiplicative domain are closed under products and adjoints. Intersections preserve the C*-subalgebra property. QED. H11. Direct readability and operational closure V_O^read=span_C({I} union {E_O^dagger(E):E in Eff_O}). The raw pullback system is not silently treated as an algebra. Let A_read,O be the C*-algebra justified by the declared operational closure, including only physically admitted sequential composition, adjoints, repeated experiments, and classical postprocessing. Full W*(V_O^read) closure is used only when those operations justify it. H12. Record-Channel Stable-Access Selection Theorem A_O^exact=A_stab intersection A_read,O. Theorem H12. For a faithful finite-dimensional selector datum with justified readable C*-algebra, A_O^exact is a unital finite-dimensional C*-algebra and the unique largest subalgebra whose elements are both exactly stable in the multiplicative sense under every active map and readable under the declared protocol. A generating list is not unique; the selected algebra is invariant. If A_O^exact=C I, the model supplies no nontrivial exact common stable-readable observer record. The Version-13 reconstruction and factorization theorems then apply to A_O^exact. Proof. Both component sets are unital C*-subalgebras, so their intersection is. Any algebra satisfying both gates lies in both and hence in the intersection. Generator independence follows because the output is the algebra, not a presentation. QED. H13-H15. Discriminability, covariance, and residual symmetry A selected self-adjoint X is discriminating when two admissible states give different expectations. If Omega separates a nontrivial A_O^exact, some selected observable distinguishes states. If every admitted output distribution is preparation independent, operational reduction leaves no nontrivial readable distinction. Under physical *-isomorphisms transporting states, dynamics, output effects, channels, and operational closure consistently, A_O^exact transforms covariantly. If an automorphism preserves the complete selector datum while exchanging inequivalent candidates, no covariant rule based only on that datum can choose one uniquely. H16. Recoverable and approximate access R_t<-s^dagger o Phi_t,s^dagger(X)=X, epsilon_stab(X;I)=max_a sup_(t in I)||Phi_a,t^dagger(X)-X||. Exact fixed access is sufficient but not necessary for recoverable or finite-time records. The epsilon-stable set need not be closed under products. Version 17 retains exact-core closure and additionally proves the finite-dimensional exact recoverable-algebra theorem; approximate recovery remains open. Phase D2 audit disposition. RESTRICTED AND FORMALIZED. Version 55 contains a complete exact finite-dimensional recovery theorem, but it does not yet contain a general approximate-recovery theorem. Approximate recovery remains a task-relative research program and may not be inferred from exact algebra correction, nonzero discrimination, small channel perturbation, or successful recovery on a selected state family. Approximate-recovery datum. Any future approximate-recovery claim must fix the source model, code projector or protected algebra, selected output channel, complete retained and discarded carrier, admitted recovery class, ancillary and reference extensions, error metric, worst-case or average-case quantifier, one-shot or asymptotic regime, tolerance, resource restrictions, and composition rule. Changing any of these after evaluation defines a new task or model under BNP-C1 rather than repairing the original claim. Permitted metrics and non-equivalence. Candidate figures of merit include worst-case entanglement infidelity, purified or trace distance on a declared state family, and diamond-norm distance between the recovered logical channel and the identity or target channel. These choices answer different operational questions and are not interchangeable without a proved comparison bound on the declared domain. Average-state fidelity does not by itself certify worst-case channel recovery, and one-state recovery does not certify algebra-wide recovery. Selected-output and complete-carrier rule. Approximate recovery is output-channel relative. The reduced-system and complementary-environment outputs must be audited separately, exactly as in H22.6A. A small error on a selected output does not establish complete-model equivalence, and tracing out a carrier that contains residual distinguishability cannot be used to improve the recovery classification. Current theorem ceiling. LRT may state that approximate recovery is an open quantitative extension of H22.2 and that model-specific bounds may be sought for the weak nonorthogonal collision regimes. It may not claim a unique approximate recoverable algebra, universal continuity, universal physical realizability, universal stability, or an exact-to-approximate cost law until the declared metric, carrier, operation class, witness family, and proof support those conclusions. Falsification and completion gate. A proposed approximate-recovery theorem is restricted or falsified by a change of result under an admitted ancillary extension not covered by the metric, failure of the bound for a declared input or reference system, output-channel dependence omitted from the statement, average-case success with worst-case failure when worst-case recovery is claimed, hidden postselection or branch conditioning, or violation of the stated tolerance under the declared composition rule. Completion requires an explicit theorem and proof or a reproducible optimization certificate for the fixed recovery datum. H17. Collision-model execution and null tests U_theta=P_0 tensor I+P_1 tensor R_theta, R_theta=exp(-i theta sigma_y). The reduced system channel fixes P_0 and P_1 and multiplies coherence by cos(theta). For |cos(theta)|<1, A_stab=W*(P_0,P_1) ~= C direct-sum C. For fragment energy effect Q_0, E_F^dagger(Q_0)=q_0P_0+q_1P_1. If q_0 != q_1 and the effect is admitted, the pullback and identity generate W*(P_0,P_1), so A_S^exact=W*(P_0,P_1). The selector recovers the pointer algebra encoded by the controlled interaction when the outgoing fragment distinguishes it. It does not show that bare potentiality selected the controlled unitary, pointer decomposition, fragment boundary, or measurement. H18-H19. Fisher consequence and completion status The selected algebra constrains, but does not by itself exhaust, the implementable POVM class: Meas_O is contained in POVM(A_O^op). If the selected class is smaller than a previously assumed class, accessible Fisher information can only decrease or stay equal by H4. The general clock, endpoint, and gluing theorems remain valid; model-specific clock adequacy, endpoint asymptotics, and edge existence must be recomputed. Version 17 retains exact stable-readable selection relative to declared D_O, derives an observer-free total complementary record channel from an interaction isometry, proves the maximal finite-dimensional recoverable algebra, and establishes a no-go boundary for unique fragment selection. Approximate recovery, infinite-dimensional extension, and intrinsic comparison transports remain open. H20. Staged Observer-Emergence Architecture The observer-access program is divided into logically separate stages. The staging is constitutive, not cosmetic: it prevents an exact-core theorem from being misrepresented as a derivation of observers from bare potentiality, and it exposes the distinct problem of deriving candidate record channels and operational boundaries from the interaction structure itself. S0 interaction structure -> S1 candidate record channels -> S2 operational domains -> S3 exact core -> S4 recoverable algebra -> S5 temporal information H20.1. Stage S0: Physical Interaction Structure Input: a gauge-reduced physical algebra, admissible state family, and physical dynamics, with no observer label, preselected record channel, access seeds, or measurement protocol. The task is to characterize invariant interaction structure, information-flow relations, persistent correlations, candidate fragment families, and symmetry or locality restrictions. Stage S0 is open. D_S0=(A_phys,Omega,{Phi_a,t}) H20.2. Stage S1: Emergent Candidate Record Channels Input: the Stage-S0 interaction structure. Output: an equivalence class or family of candidate information-bearing channels derived from the dynamics rather than labeled in advance as observer outputs. A candidate must pass explicitly declared persistence, distinguishability, recoverability, redundancy, common-record, covariance, and robustness gates. The strengthened target is not to prove that every interaction creates a record channel, but to produce a covariant selection rule that may return an empty family. R(D_S0)={E_xi : xi in Xi and G_pers,G_dist,G_rec,G_red,G_cov,G_rob are satisfied} The gates are logically independent. Persistence alone does not imply readability; distinguishability alone does not imply durability; redundancy across incompatible records does not define a common channel; and an interaction may legitimately yield R(D_S0)=empty set. Stage S1 is open and is the first stage aimed directly at deriving candidate record channels from the interaction structure. RC-11 synchronized S1 status. The frozen ARF3 autonomous single-fragment model has now been audited across the integrated qualification gates. It passes distinguishability only for r != 1/2, fails redundancy because the frozen datum contains one fragment, and does not yet type or execute persistence, candidate-specific recovery, common-record compatibility, covariance, or a full physical robustness gate. It therefore does not certify a nonempty fully record-qualified S1 family. Stage S1 remains open, and a multi-fragment extension is required before the failed redundancy gate and the other unresolved gates can be retested in one common physical realization. RC-12 synchronized S1 status. A three-fragment extension of the declared dephasing-collision architecture discharges the RC-11 single-fragment redundancy failure at the exact strong-collision, zero-drift point. For theta_1=theta_2=theta_3=pi/2, each supplied fragment E_k carries orthogonal pointer-conditioned states, has exact recoverable algebra W*(P_0,P_1), and admits the same computational-basis decoder; the maximum audited decoder residual is 8.66 x 10^-17. The three local commutant nullspaces are span{I,Z}, so common-record compatibility and threefold redundancy pass. Persistence passes only on a separately declared identity-retention branch, and covariance is proved only for the S_3 permutation action on the prepared environmental sector. Exact zero-error robustness fails under every tested nonzero collision-angle deviation, while the symmetric no-drift approximate branch has trace distance cos(delta) and equal-prior Helstrom error (1-cos(delta))/2. Candidate provenance remains conditional on the supplied event-indexed factorization, full complete-datum covariance remains open, and Stage S1 therefore remains open as an intrinsic reconstruction problem. RC-14 analytic drift-scaling synchronization. For the declared strong controlled-dephasing stream with arbitrary finite N and intercollision drifts alpha_1,...,alpha_(N-1), the initial-pointer contrast at fragment E_k is c_k=product_(j=1-2epsilon. For every nonempty retained subset F, with m(F)=min F, the subset trace distance and optimal recovery error equal those of E_m(F) alone. Adding later fragments gives no collective advantage for recovery of the initial pointer; every subset containing E_1 is exact, and after erasing E_1 the full remainder E_2...E_N performs no better than E_2. Therefore this frozen model does not realize a nontrivial threshold code, secret-sharing access structure, or irreducibly distributed ledger for the initial pointer task. These conclusions do not extend to source coherence, noncommuting inputs, drift estimation, correlated fragments, detector restrictions, resource costs, or temporal Fisher adequacy. Stage S1 remains open intrinsically. H20.3. Stage S2: Operational Boundary Selection Input: the candidate record-channel family R(D_S0). The task is to determine which mutually composable channels, controls, memories, recovery maps, and readout instruments form a coherent operational domain. The output is an observer-domain candidate O_mu together with its supported operational closure Cl_mu. No claim concerning consciousness or subjective experience is made. A physical record carrier is not automatically an observer domain. Bnd(R)={O_mu=(B_mu,M_mu,Cl_mu) : closure, consistency, finite recovery, and covariance gates hold} Stage S2 is open. Its purpose is to derive an operational boundary rather than insert a tensor cut or observer label by hand. H20.4. Stage S3: Exact Stable-Readable Core Input: a Stage-S2 operational domain and its physical channel data. Output: the exact stable-readable core A_O^exact defined and proved in H9-H18. This stage is complete in faithful finite dimension relative to the declared selector datum, including the closure rule. It may return the trivial algebra C I. A_O^exact=A_stab intersection A_read,O H20.5. Stage S4: Recoverable Observer-Access Algebra Input: an operational domain, a fixed selected channel and output, a supported sector, a declared readable or operational algebra, and admitted recovery protocols. H22.2 supplies the unique largest algebra mathematically correctable by one CPTP recovery for the fixed channel and supported sector. The observer-relative S4 algebra is the intersection of that channel-correctable algebra with the declared readable or operational algebra. Physical admissibility of the recovery and inclusion of the S3 exact core require the corresponding channel, support, readability, and protocol hypotheses and must be proved for the declared S4 datum. A_O^exact subseteq A_O^rec subseteq A_O^op The finite-dimensional mathematical channel-correction construction is complete through H22.2 for a fixed selected channel and supported sector. Observer-relative S4 completion is conditional on the declared readable or operational intersection, physical admissibility of one algebra-wide CPTP recovery, and proof of any S3 exact-core inclusion. Approximate, resource-constrained, and normal infinite-dimensional recovery remain open extensions. H20.6. Stage S5: Temporal-Information Extraction Input: a declared exact, recoverable, or full operational measurement class. Output: task-relative accessible Fisher information, duration, clock adequacy, comparison-edge availability, and possible holonomy. This stage is partially complete through the Fisher-clock, endpoint, and gluing theorems, but remains open for interaction-derived collision record channels. Meas_O^exact subseteq Meas_O^rec subseteq Meas_O^op => F_O^exact <= F_O^rec <= F_O^op H20.7. Stage Interfaces and Falsification Discipline Each stage has an explicit null output. S1 may return no candidate record channel; S2 may return no coherent operational domain; S3 may return C I; S4 may fail to admit one recovery for a proposed algebra; and S5 may yield zero or nonconstant temporal Fisher speed. Later stages may not be used to retroactively alter earlier definitions merely to force a nontrivial observer or clock. The staged architecture changes the fundamental research question from selecting better seed operators to deriving the channel and boundary structures from which physically meaningful access seeds can be generated. It does not itself solve S1 or S2, but it converts them into separately falsifiable mathematical targets. H20.8. Status Matrix H21. Consequences of the Staged Architecture Version 13 seed-relative reconstruction remains valid and is used at S3 and S4 once a physically justified generating system is obtained. The Version-17 exact-core theorem is no longer treated as the whole observer-emergence problem. The fundamental derivation target moves to S1 and S2: candidate record channels and operational boundaries must be extracted from interaction structure without presupposing an observer label. The recoverable algebra at S4 is mathematically adjacent to the exact core but does not by itself derive the record channel or boundary. Temporal information at S5 is task-relative. Stable record content and clock information remain distinct. A single integrated model remains desirable but is not assumed by the staged architecture. H22. Interaction-Derived Record Channels, Maximal Channel-Correctable Algebras, and Observer-Readable Recovered Access This section closes two finite-dimensional stages while proving the boundary of what interaction data alone can select. Stage S1a derives the total complementary output channel from an interaction isometry without an observer label. Stage S4 derives the unique largest algebra corrected by one CPTP recovery channel for a fixed selected channel and supported sector. Observer-readable recovered access is a separate intersection with the declared readable or operational algebra and additionally requires physical admissibility of the recovery. A covariance no-go theorem controls unique boundary selection under unresolved output symmetries. The dual-output execution shows that the maximal channel-correctable algebra depends on the physical isometry together with the selected output channel. H22.1. Interaction-Derived Total Record-Channel Theorem Theorem H22.1. Let H_A, H_B, and H_E be finite-dimensional physical Hilbert spaces and let V:H_A -> H_B tensor H_E be an isometry supplied by the physical interaction, including any fixed purified ancillary preparation. Define Phi_B(rho)=Tr_E[V rho V^dagger], R_E(rho)=Tr_B[V rho V^dagger]. Then: (i) Phi_B and R_E are CPTP channels; (ii) R_E is derived from the interaction without an observer label, access seed, detector POVM, or observer boundary; (iii) for a minimal Stinespring dilation of Phi_B, the equivalence class [R_E] is unique up to an isometry on the environment output; and (iv) this uniqueness does not select a proper fragment algebra inside B(H_E). Proof. Conjugation by V is completely positive and partial trace is completely positive. Since V^dagger V=I_A, both reduced maps preserve trace. If V prime:H_A -> H_B tensor H_Eprime is another minimal Stinespring dilation of Phi_B, Stinespring uniqueness supplies an isometry W:H_E -> H_Eprime such that V prime=(I_B tensor W)V. Therefore R_Eprime(rho)=W R_E(rho) W^dagger. Thus the total complementary channel is canonical up to output isometry. Stinespring uniqueness acts on the whole dilation output and supplies no canonical proper subalgebra or tensor fragmentation of B(H_E). QED. Corollary H22.1a. The strongest unconditional observer-free Stage-S1 arrow is interaction isometry -> equivalence class of total complementary record channels. The term record channel here denotes a potential information-bearing output of the interaction. Persistence, common-record survival, redundancy, and operational readability remain separate qualification gates. H22.2. Maximal Channel-Correctable Algebra Theorem Let Phi:B(H_P)->B(H_B) be a finite-dimensional CPTP channel with Kraus operators K_i:H_P->H_B. Equivalently, if a larger input space is used, let P project onto the supported physical code sector and replace K_i by K_i P. Define the compressed Kraus-product algebra and its commutant by N_(Phi,P)=C*({P K_i^dagger K_j P}_{i,j}), A_(Phi,P)^rec=N_(Phi,P) prime intersection B(PH). Theorem H22.2. (i) A_(Phi,P)^rec is a finite-dimensional C*-algebra. (ii) There exists one CPTP recovery channel Q_rec:B(H_B)->B(PH) such that P (Q o Phi)^dagger(X) P = X for every X in A_(Phi,P)^rec. (iii) If B is any unital C*-subalgebra of B(PH) exactly correctable for Phi by one recovery channel, then B is contained in A_(Phi,P)^rec. Hence A_(Phi,P)^rec is the unique largest exactly recoverable observable algebra for (Phi,P). (iv) The algebra is independent of the Kraus representation and transforms covariantly under physical unitary equivalence. Proof. N_(Phi,P) is a finite-dimensional C*-algebra, so its commutant is a finite-dimensional C*-algebra. The operator-algebra Knill-Laflamme theorem states that a unital C*-algebra B on PH is exactly correctable for Phi if and only if [P K_i^dagger K_j P,X]=0 for every X in B and all i,j. Every element of A_(Phi,P)^rec satisfies this condition, so one CPTP recovery channel corrects the whole algebra. Conversely any correctable B satisfies the same commutant condition and is therefore contained in A_(Phi,P)^rec. Equivalent Kraus families are related by an isometry, so they generate the same linear span, and hence the same C*-algebra, of Kraus products. Under a physical unitary U, the Kraus-product algebra is conjugated by U and so is its commutant. QED. This theorem requires one recovery channel for the complete algebra. Elementwise statements of the form "for every X there exists Q_X" are insufficient because incompatible elementwise recoveries need not combine into one physical protocol or a correctable algebra. It also requires the output orientation to be fixed: complementary channels use different Kraus families, and their maximal exactly recoverable source algebras need not coincide. H22.3. Relation between exact and recoverable access Proposition H22.3. Let Phi be a channel on the supported physical algebra and let A_O^exact be an exact stable-readable core for the same protocol. If the identity recovery is admissible on A_O^exact, then A_O^exact subseteq A_(Phi,P)^rec. Proof. Exact stability gives Phi^dagger(X)=X on the exact core. Taking the identity as the recovery satisfies the correction equation, so maximality in H22.2 yields the inclusion. QED. No equality is asserted. Recoverable information can be disturbed by Phi and restored by Q without being pointwise fixed. H22.4. Interaction-Only Boundary Nonuniqueness Theorem Theorem H22.4. Let R_E be the interaction-derived total complementary channel. Let G be a group of output unitaries W_g satisfying Ad_(W_g) o R_E = R_E. Suppose a proposed proper output algebra F is moved by this symmetry, so W_g F W_g^dagger differs from F for some g. Then no covariant rule depending only on the interaction-derived data can uniquely select F. Proof. An intrinsic selector must commute with every automorphism preserving its complete input data. If it uniquely selects F, covariance requires the selected output under the same data to be W_g F W_g^dagger. When these are distinct, uniqueness fails. Additional physical structure breaking or resolving the symmetry is necessary. QED. Corollary H22.4a. An abstract interaction or CPTP channel generally determines a total complementary output channel but not a unique proper fragment decomposition, detector algebra, or observer boundary. H22.5. Observer-Free Candidate Fragment-Record Selection Corollary Assume the physical model supplies a covariant output net F_net={F_xi}_{xi in Xi} from non-observer structure such as interaction events, causal localization, emitted modes, spatial output ports, or gauge-invariant local algebras. Let R_xi denote the restriction of the total complementary channel to F_xi. Let C=W*({P_x}) be a candidate commutative record algebra. Define RecChan(V,F_net) to contain exactly those R_xi satisfying all declared gates: common preservation of the same record algebra across every active hidden channel; nontrivial distinguishability of at least two record sectors on F_xi; exact recoverability of the relevant algebra by one admissible recovery channel; redundant encoding of the same record in the required number of disjoint physical output algebras; covariance under physical equivalence and robustness under the declared perturbation class. Corollary H22.5. If the output net and every qualification gate transform covariantly, then RecChan(V,F_net) transforms covariantly and requires no observer label. It may be empty. It is unique only up to symmetries not resolved by the physical output net. Proof. The total channel is covariant up to output isometry by H22.1. Restriction to a covariantly transported output net commutes with physical equivalence. Each gate is assumed invariant under that transport; therefore membership in RecChan is preserved. The empty set remains permitted, and H22.4 controls unresolved symmetries. QED. H22.6. Collision-model execution The repeated-interaction model supplies a physical output net without an observer label: F_net={B(H_F1),B(H_F2),...}, where each fresh ancilla is distinguished by its interaction event. The interaction induces one-fragment channels R_n:rho_S -> rho_Fn and accumulated-memory channels R_1:n:rho_S -> rho_F1...Fn. Exact backward recovery from M_n to M_(n-1) is the partial trace over F_n, while the strong-record regime provides informative fragment outputs. The null-interaction and thermally unbiased cases fail the distinguishability gate. Thus the model supplies positive and null S1 candidates without first labeling a human or observer subsystem. Given the physically declared repeated-collision event structure, the execution derives record channels directly from the interaction and supplies positive and null candidate outputs. The separate problem of deriving the output-event net is recorded in the S0-S2 status tables. H22.6A. Dual-Output Collision Audit: System Preservation versus Environmental Record Recovery Scope and purpose. This execution applies H22.2 to both outputs of one finite-dimensional interaction isometry. It distinguishes preservation through the reduced final system from record recovery through the retained environment. It is a model-specific exact, zero-error audit. It does not establish approximate recovery, detector-limited accessibility, clock adequacy, or metrological duration. H22.6A.1. Physical parameterization and angle convention Let S, E1, and E2 be qubits. The fragments begin in |0>_E. The collision acting on S and fragment Ek is U_k(theta_k)=P0⊗I+P1⊗R(theta_k), where R(theta)=exp(-i theta sigma_y)=[[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]. This is the full-angle convention; therefore theta=pi/2 produces orthogonal conditional fragment states. Between the first and second collisions the system may undergo Rx(alpha)=exp(-i alpha sigma_x/2). The total isometry is V=U_2(theta_2)[Rx(alpha)⊗I_E1E2]U_1(theta_1)(I_S⊗|00>_E). Convention warning. If standard quantum-computing notation Ry(phi)=exp(-i phi sigma_y/2) is used instead, the same physical perfect-record point is phi=pi. The collision angle and the system self-rotation angle must not be conflated. H22.6A.2. Two channel orientations from one isometry The same V defines Phi_S(rho)=Tr_E1E2[V rho V^dagger] and Phi_E(rho)=Tr_S,out[V rho V^dagger]. For Phi_S, K_ab= and |1> are |e0>=|00> and |e1>=R(theta1)|0>⊗R(theta2)|0>. Their overlap is =cos(theta1)cos(theta2). Exact recovery of the classical source pointer from the environment requires orthogonal supports, attained when this product vanishes. In the tested strong case it vanishes; in the tested weak case it is approximately 0.9779 and exact zero-error recovery fails. Nonorthogonality does not mean no information is present; it marks the handoff from exact algebra correction to quantitative approximate discrimination and recovery. H22.6A.8. Dual-channel theorem consequence Proposition H22.6A. Let V be a fixed interaction isometry and let Phi_S and Phi_E be its reduced and complementary channels. The maximal exactly recoverable source algebras A_S^rec and A_E^rec are separately determined by their channel-specific Kraus-product algebras and need not agree. In the executed two-collision model their dimension vectors are (2,2,1,1) and (2,1,2,1), respectively. Therefore exact recoverability is output-channel relative, and system preservation cannot be substituted for environmental record recovery. Proof. Apply H22.2 separately to the K and L Kraus families and use the audited commutants above. QED. H22.6A.9. Truth boundary and next gates The execution proves a channel-orientation distinction and four exact finite-dimensional classifications. It does not prove that exact relational time is merely idealized, that every realistic network requires approximate recovery, or that the recovered pointer is a clock. The next valid gates are: quantify epsilon-approximate recovery or discrimination in the weak environmental regimes; impose physically admitted detector effects and resource bounds; parameterize an elapsed-time family; compute accessible Fisher information for the selected output; and test G12-G18 calibration, injectivity, stability, and comparison transport. Phase D2 status: the proposed epsilon-approximate extension remains unproved. Any future result must use the fixed approximate-recovery datum and metric declared in H16, report both output orientations, and separate discrimination, state-family recovery, channel recovery, and algebra recovery. H22.6A.10. Reproduction algorithm Construct V in output basis |s_out,e1,e2>. Extract K_e=V[:,e,:] for each environmental basis value e and L_s=V[s,:,:] for each final-system value s. For each channel form all products M_i^dagger M_j. Stack the linear maps X→XG-GX using vec(XG-GX)=(G^T⊗I-I⊗G)vec(X). Compute the singular values and a null-space basis at tolerance 1e-10. Verify sum_i M_i^dagger M_i=I and evaluate max_i,j ||[X,M_i^dagger M_j]||_F for each basis element. This algorithm is Kraus-representation invariant because isometric Kraus changes preserve the span and generated algebra of the products. H22.6B. Three-Fragment S1 Core Qualification and Robustness Audit Scope and status. This finite-dimensional execution extends the declared collision output net to three event-indexed qubit fragments and audits one common physical realization against the Stage-S1 gates. It certifies an exact model-relative S1 core at one strong-collision point and a metrized approximate branch. It does not derive the fragment factorization from an unsegmented S0 datum, prove complete-datum covariance beyond the declared fragment-permutation group, or supply a robust open neighborhood of exact zero-error correction. H22.6B.1. Model and channel family Let S,E_1,E_2,E_3 be qubits, with each fragment initialized in |0>. For k=1,2,3 let U_k(theta_k)=P_0 tensor I + P_1 tensor R_y(2 theta_k) act on S and E_k, with any declared system drift inserted between collisions. The joint isometry is the ordered product of these collisions and drifts on the prepared environmental sector. The record channels R_k are obtained by tracing out S and every fragment except E_k; R_123 is the joint complementary environmental channel. This convention makes theta_k=pi/2 the orthogonal-record point. The executed drift code uses R_y(alpha), not an X-axis rotation. H22.6B.2. Exact common-record and redundancy theorem Theorem H22.6B. At theta_1=theta_2=theta_3=pi/2 and zero system drift, the pointer-conditioned state of every E_k is |0><0| for P_0 and |1><1| for P_1, up to numerical residual below 10^-16. Hence each local trace distance is one. For every k, the compressed Kraus-product commutant is span{I,Z}=W*(P_0,P_1), and the local decoder that measures {|0><0|,|1><1|} and prepares the corresponding pointer sector corrects the complete commutative pointer algebra. The maximum audited decoder residual is 8.66 x 10^-17. Consequently the same common pointer algebra is exactly recoverable from each of three supplied disjoint fragments, giving redundancy multiplicity three in this model. Proof. The strong controlled rotations map the two source pointer sectors to orthogonal computational-basis states in every fragment. Direct nullspace extraction gives zero Z-alignment residual and unit X- and Y-direction residuals for each local commutant. The displayed decoder therefore recovers both minimal pointer projectors, and linearity recovers their generated algebra. QED. H22.6B.3. Conditional persistence and scoped covariance Corollary H22.6B.1. If each emitted fragment undergoes the declared identity continuation T_k=id over the claimed retention interval, the local record and the same decoder persist exactly. This is a conditional persistence witness; the retention law is additional model data. Corollary H22.6B.2. At identical strong couplings and zero drift, the prepared-sector isometry is invariant under every permutation of E_1,E_2,E_3. The maximum audited S_3 covariance residual is zero. No covariance claim beyond this declared fragment-permutation action is imported. H22.6B.4. Exact and approximate robustness boundary Proposition H22.6B.3. Under the symmetric no-drift perturbation theta_k=pi/2-delta, every tested nonzero delta, including 10^-6 radians, collapses each exact local commutant from dimension two to the scalar algebra. Thus no open exact zero-error recovery neighborhood is certified around the strong point. Nevertheless the two local pointer-conditioned states have trace distance D_k(delta)=cos(delta) in the tested branch, so the equal-prior Helstrom error is p_err(delta)=(1-cos(delta))/2. Therefore an approximate redundant-record claim is valid only after fixing a perturbation bound Delta, an accepted error epsilon, and the condition (1-cos(Delta))/2 <= epsilon, together with the selected output, carrier, and recovery metric. H22.6B.5. Drift separation and entanglement boundary In the executed strong-collision drift regime, the joint environmental commutant remains two dimensional while the E_2 and E_3 local commutants are scalar and the E_1 local commutant remains two dimensional. This proves that joint environmental recoverability need not imply threefold fragmentwise redundancy. Environmental negativities for each pointer-conditioned input, across every one-versus-two fragment bipartition, are numerical zero at or below 1.9 x 10^-16. The execution therefore does not certify environmental entanglement. Its supported conclusion is global recovery without threefold local redundancy. H22.6B.6. Full-gate disposition Gate disposition for the exact strong, zero-drift point: provenance is conditional on the supplied event-indexed factorization; distinguishability passes exactly; persistence passes only under declared identity continuation; recoverability passes exactly; threefold redundancy passes exactly; common-record compatibility passes exactly; covariance passes for S_3 fragment permutations; exact generic angle robustness fails, while the declared approximate branch is quantitatively controlled. Regime A is therefore an exact model-relative S1 core witness at the parameter point, with an approximate robust extension. It is not an intrinsically selected or robustly exact S1 family. Stage S1 remains open for intrinsic candidate derivation, independently derived retention, full complete-datum covariance, and broader robustness. H22.6C. Analytic Drift-Scaling Classification for Arbitrary Finite N Scope and status. This section analytically extends H22.6B from the sampled finite-fragment executions to every finite N in the same supplied event-indexed strong-collision architecture. It classifies constant and heterogeneous R_y drifts, identifies the exact recurrence set, and separates local exact recovery from joint-environment recovery. It remains model relative and does not derive the fragment net from an unsegmented S0 datum. H22.6C.1. Drift transition and contrast product After the first strong collision, E_1 stores the initial pointer value exactly. For any later single-fragment marginal, the preceding record fragments reduce the relevant source sector to a classical population vector. The drift R_y(alpha_j) induces B(alpha_j)=[[cos^2(alpha_j/2),sin^2(alpha_j/2)],[sin^2(alpha_j/2),cos^2(alpha_j/2)]], whose population-difference eigenvalue is cos(alpha_j). Hence c_1=1 and c_k=product_(j=1)^(k-1) cos(alpha_j). Theorem H22.6C. For the initial source-pointer alternatives, rho_(E_k|0)=((1+c_k)/2)|0><0|+((1-c_k)/2)|1><1| and rho_(E_k|1)=((1-c_k)/2)|0><0|+((1+c_k)/2)|1><1|. Therefore the local trace distance is D_k=|c_k|=product_(j=1. Thus the exact boundary is pointer-permuting versus non-pointer-permuting drift, not zero versus nonzero drift. H22.6C.4. Heterogeneous first-nonexceptional-drift theorem Corollary H22.6C.3. For arbitrary alpha_1,...,alpha_(N-1), let m=min{j:alpha_j not in pi Z}, with m=infinity if the set is empty. If m is finite, E_1 through E_m are exact local records and E_(m+1) through E_N are inexact; if m=infinity, all N fragments are exact. Hence C(N;alpha_1,...,alpha_(N-1))=min(N,m), with min(N,infinity)=N, and D(N;alpha_1,...,alpha_(N-1))=N+C(N;alpha_1,...,alpha_(N-1)). H22.6C.5. Joint-environment and approximate-record boundaries The joint environmental output remains exactly recoverable for every later drift sequence because it contains E_1, whose pre-drift record is exact. A joint decoder may ignore E_2,...,E_N and decode E_1. This does not prove irreducibly nonlocal storage, environmental entanglement, or collective-access necessity. For constant alpha, approximate local quality follows D_k=|cos(alpha)|^(k-1). It remains exact on alpha in pi Z, vanishes for every k>=2 when cos(alpha)=0, and decays exponentially with k whenever 0<|cos(alpha)|<1. For small alpha, D_k is asymptotic to exp(-(k-1)alpha^2/2). Exact algebraic collapse can therefore coexist with slowly degrading operational distinguishability. H22.6C.6. Claim-status disposition The RC-13 N<=5 scaling curves are promoted to arbitrary-finite-N theorems for the declared model. The phrase any nonzero drift is withdrawn in favor of any drift outside pi Z. Nonzero recurrence angles in pi Z preserve exact records through pointer-label permutations. The count-one law applies when the first drift is nonexceptional. Joint recovery is attributed to the surviving E_1 record and not to a proved distributed ledger. H22.6B remains valid as the alpha=0 exact branch. Stage S1 remains open intrinsically. H22.6D. Task-Relative Approximate Recovery and Operational Redundancy Scope and datum. This subsection solves one model-specific approximate-recovery problem and does not assert a general approximate quantum error-correction theorem. The source task is the commuting binary pointer ensemble {P_0,P_1}; the selected output is one fragment E_k; E_k is retained while S and all other fragments are discarded for the local task; the recovery class is all CPTP measure-and-prepare maps; no ancillary reference is required for this commutative task; the metric is worst-case trace distance over the two pointer inputs; the regime is one-shot at finite N; epsilon belongs to [0,1/2); and ideal fragment access with unrestricted binary POVM and classical repreparation is assumed. Detector noise, finite resources, source coherence, noncommutative reference extensions, and implementation cost are outside this theorem. H22.6D.1. Optimal local pointer recovery Theorem H22.6D. Let c_k=product_(j=1-2epsilon}. Because every factor lies in [0,1], the qualified fragments form an initial prefix. Once one fragment fails the tolerance, no later fragment can re-enter it for this initial-pointer single-fragment task. Corollary H22.6D.2. For constant drift, let r=|cos(alpha)|. If r=1, R_epsilon(N)=N. If r=0 and epsilon<1/2, R_epsilon(N)=1. If 0 X_1 -> ... -> X_N, with X_1=X_0 exactly and transition probabilities Pr(X_(j+1)=X_j)=cos^2(alpha_j/2) and Pr(X_(j+1)!=X_j)=sin^2(alpha_j/2). H22.6E.1. Earliest-retained-fragment sufficiency Theorem H22.6E. Let F be any nonempty fragment subset and let m(F)=min F. Then (1/2)||rho_(F|0)-rho_(F|1)||_1=(1/2)||rho_(E_m|0)-rho_(E_m|1)||_1=product_(j=1-2epsilon. Thus authorization depends only on the earliest retained fragment. There is no nontrivial threshold regime in which every singleton fails while a larger subset succeeds. The stream is not a secret-sharing code for the initial pointer task. H22.6E.4. Joint-environment disposition The full joint environment has zero error because E_1 is exact, and its optimal decoder may ignore every later fragment. Erasing E_1 does not reveal hidden collective information because E_2...E_N is no better than E_2. The earlier distributive-ledger interpretation is therefore rejected for this frozen commuting initial-pointer task. This result neither requires nor establishes environmental entanglement. Collective advantages remain open for different tasks or models, including source-coherence recovery, noncommuting-state discrimination, drift estimation, partial collisions, correlated or interacting fragments, coherent early-record erasure, and non-Markovian process extensions. H22.6E.5. Consolidated disposition RC-15A and RC-16 jointly close the task-relative local and subset recovery problem for the declared commuting initial-pointer stream. Exact algebraic collapse and approximate operational usefulness remain distinct. The earliest retained fragment is a sufficient statistic for every later retained subset, so later fragments supply no collective gain for this task. Stage S1 remains open for intrinsic event-net derivation, independently derived retention, complete-datum covariance, broader perturbation and resource models, and any noncommutative or temporal-information extension. H22.7. Consequences for S0-S5 H22A. Constraint Geometry, Witness Emergence, and Record Minimality Synthesis Version 42 closes Stage S1A by reconstructing candidate record structure without assuming observers, measurement classes, operational boundaries, preselected record channels, or effect systems. The reconstruction proceeds through a hierarchy: interaction structure, constraint geometry, distinction geometry, persistence geometry, transfer geometry, representation geometry, substitutability geometry, and finally record structure. BNP theorem-family label. H22A is certified only as a conditional finite-dimensional reconstruction from the declared S1A interaction, constraint, distinction, persistence, transfer, representation, and substitutability datum. It does not derive physical records, witnesshood, interaction structure, or physical actuality from bare potentiality or from a certified-null provenance. BNP provenance statement. The non-null physical content used by H22A enters through the physical-instantiation bridge, admitted source distinctions, admissible interaction dynamics, supports and transfer relations, persistence conditions, representation relations, substitutability criteria, and the future distinction-relevant consequence class. The theorem family organizes and constrains these inherited data; it does not create their physical status. BNP complete-carrier and operation status. The certified carrier is the complete S1A reconstruction datum actually declared for the finite-dimensional model. Transfer, persistence, representation, substitutability, and stability are mathematically admissible construction relations whose physical realizability and quotient descent are model relative. No manuscript-wide proof establishes that these operations are completely null preserving under every ancillary, correlated, conditional, catalytic, boundary, sector, or completion extension. BNP minimality and witness boundary. Factorization through substitutability geometry is a necessity statement within the declared candidate-record architecture. It is not a proof that substitutability is a unique, least, basement-level, temporally first, causally first, or ontologically simple ground. Failure to identify a deeper independent precursor in the examined reconstruction class is a restricted negative search, not a universal completeness theorem. H22A.1. Constraint-Transfer Reconstruction Theorem Persistent distinctions alone are insufficient for witness formation. A transfer relation exists when admissible interaction dynamics transport distinction structure from one support to another while preserving the relevant distinction class. The resulting transfer geometry is interaction-derived, observer-free, and covariant under datum-preserving automorphisms. Canonical outputs are transfer orbits rather than unique transfer representatives whenever residual symmetry remains. H22A.2. Witness Reconstruction Theorem A witness is not merely a transferred distinction and not merely a representation. Witnesshood emerges when a represented distinction is substitutable for the original distinction with respect to future distinction-relevant consequences under admissible interaction dynamics. Accordingly, witness geometry is identified with substitutable representation geometry. H22A.3. Conditional Record Factorization and Stability Theorem Within the declared finite-dimensional S1A reconstruction class, every candidate record architecture satisfying the stated record conditions factors through substitutability geometry. Persistence, transfer, and representation are individually insufficient for that candidate-record classification. A candidate record is characterized as a stable substitutable representation relative to the declared source distinctions, admissible interactions, supports, consequence class, and physical-instantiation bridge. The result is a conditional factorization and characterization theorem. It does not derive physical recordhood from certified-null inputs, prove that substitutability is a basement-minimal ground, or establish that no deeper precursor exists outside the examined reconstruction class. Claim boundary. H22A.3 proves necessity of the substitutability layer only within the declared candidate-record architecture and fixed finite-dimensional scope. A stronger emergence claim would require a complete physical provenance, complete carrier, classified operations, sound and complete witnesses for physical recordhood, and an independently justified well-founded dependence relation. Those stronger premises are not supplied by H22A.3. H22A.4. Symmetry-Orbit Corollary Transfer, representation, substitutability, witness, and record structures inherit automorphism-orbit nonuniqueness. Record-family nonuniqueness is therefore not primitive; it is inherited from lower-layer symmetry. The canonical outputs of Stage S1A are orbit-valued families rather than uniquely selected records, fully consistent with the symmetry-derived reconstruction philosophy of the manuscript. H22B. S1B Record-Orbit Selection and Realization Program Role in emergence. Stage S1A reconstructs candidate records as stable substitutable representations but terminates at residual automorphism orbits. Stage S1B asks whether one representative can be selected or physically realized without presupposing observers, measurement classes, operational boundaries, effect systems, accessibility domains, selected record channels, or preferred orbit representatives. The stage permits a no-selection result, an orbit-valued realization structure, or selection only after independently physical enrichment. H22B.1. S1B.1A Record-Orbit No-Selection Theorem Theorem H22B.1. Let D_A be the complete S1A reconstruction datum and let G_A=Aut(D_A). Let [R]=G_A R be a residual candidate-record orbit. Let S be a selector defined only from D_A and covariant under every datum-preserving automorphism. If S(D_A) is required to be one member of [R], then S can select a unique representative only if that representative is fixed by every element of G_A. In particular, if [R] is nontrivial and contains no global fixed point, no such covariant selector exists. Proof. Because gD_A=D_A for every g in G_A, covariance gives S(D_A)=S(gD_A)=gS(D_A). Thus the selected representative must be fixed by all g. A representative moved by some datum-preserving automorphism cannot be selected uniquely from the unchanged datum. QED. Corollary H22B.1a. Any unique selector on a nontrivial residual record orbit introduces structure not contained in D_A. Selection therefore implies independent physical enrichment; absence of such enrichment implies orbit-valued output. Claim boundary. The theorem forbids datum-internal asymmetric selection. It does not prove that realization never occurs, that orbit members are ontologically identical, or that later dynamics, access, resources, boundaries, or other independently supplied structure cannot reduce the orbit. H22B.2. Pre-Accessibility Falsification Audit The adversarial audit tested whether S1A already hides a representative selector. Canonicalization by simplicity, support size, persistence depth, transfer robustness, witness multiplicity, or record stability fails whenever the proposed functional is derived solely from D_A, because every invariant of D_A is constant on a G_A orbit. Probability or typicality selection requires an independently justified measure. A metric on the orbit supplies relative separation but no preferred point without a reference, weighting, boundary condition, or optimization rule. Attractor and basin selection require a state space, an evolution law, an asymptotic notion, and, for basin dominance, a measure or typicality rule. If the added dynamics is G_A-covariant, basins are transported with the record orbit and no unique representative is selected. If the dynamics is not G_A-covariant, the symmetry-breaking dynamics is new physical structure rather than a consequence of S1A. Stability ordering and comparison geometry obey the same boundary: if reconstructed from orbit-preserving data, they are orbit invariant; if they distinguish representatives, they encode enrichment not present in D_A. Audit conclusion. No pre-accessibility selector constructed from the Version-42 interaction hierarchy survived the covariance audit. This closes the basin, stability-ordering, and comparison-geometry threats only relative to the complete S1A datum and its residual automorphism group. It does not exclude a selector supplied by later independently physical structure. Anti-aggregation consequence. Orbit averages, typicality assignments, basin weights, and compressed summaries do not select a record-orbit representative unless their measure, boundary data, objective, and physical status are independently supplied. More generally, one admitted exceptional member cannot be discarded merely because it is low weight or atypical when the asserted conclusion is universal, worst case, or provenance complete. H22B.3. Observer-Free Recoverability and the Accessibility Boundary Recoverability is not identified with observer readability. For an encoding or transfer E from a distinction-bearing source structure into a candidate record carrier, recoverability means the existence of an admissible reconstruction map D such that D composed with E preserves the declared distinction class on the supported domain. This definition invokes channels and structural preservation, not a reader, measurement outcome, or first-person observer. Recoverability alone does not uniquely determine an observer or select a record-orbit representative. Multiple reconstructions, capability realizations, or observer algebras may implement the same recovery requirement. Accessibility must therefore be treated as compatibility between recoverability structure and independently admissible access structure. If every residual record automorphism extends to an automorphism of the complete access-recovery datum, accessibility remains orbit valued. Orbit reduction occurs only when some record automorphisms fail to extend because added physical access structure is not preserved. H22B.4. Candidate Invariant Recovery Structure The audit identifies a candidate recovery invariant I_rec = (I_content, G_rec), where I_content is recoverable distinction content and G_rec is recovery geometry: the admissible recovery relations, reachability or connectivity, composition structure, reconstruction dependencies, supported-domain data, and accumulated-memory organization required by the declared recovery problem. Individual recovery channels, capability architectures, and observer realizations may remain nonunique presentations or orbit representatives of this deeper structure. Invariance status. Every explicitly analyzed presentation change, observer realization change, capability realization change, Kraus representation change, and symmetry transport either leaves I_rec unchanged or changes a declared component of it. Pathway, ordering, composition, context, memory, and gluing proposals affect recovery only by changing recoverable content or recovery geometry in the examples examined. Separation status. A difference in recoverable distinction content separates recovery families. A difference in declared recovery geometry also separates them unless the difference is an admitted presentation equivalence. These statements provide necessary classification tests. They do not yet establish that equality of I_rec is sufficient for recovery-family equivalence. H22B.5. Necessity-of-Manifestation and No-Third-Axis Status Provisional Necessity-of-Manifestation Thesis. Every explicitly realized recoverability-relevant effect examined in the S1B program manifests as a change in what can be recovered, a change in how recovery occurs, or both. No explicit recoverability-relevant manifestation channel beyond recoverable distinction content and recovery geometry has been constructed. Third-axis status. A third primitive recoverability axis is not established. Observer, capability, resource, efficiency, implementation, and presentation variations do not define a third recovery axis unless they change recovery content or geometry. Proposed global-obstruction, higher-order, or hidden-gluing invariants remain logical possibilities but presently lack a constructive example that changes recovery-family equivalence while preserving both I_content and G_rec. Truth boundary. Failure to construct a third axis is not a proof that none exists. The universal statement that every possible recoverability effect decomposes into content and geometry remains open. Accordingly, I_rec is the strongest known invariant candidate, not a proven complete invariant. H22B.6. Completeness Program and Falsification Gates The target completeness statement is I_rec(A) = I_rec(B) implies [R_A] = [R_B]. A proof requires four ingredients: invariance of I_rec under every admissible recovery equivalence; separation when either content or geometry differs; exhaustion of all recoverability-relevant properties by the content-geometry dichotomy; and collapse of every minimal counterexample to an admitted presentation symmetry. The present work supplies an extensive adversarial audit and a no-known-modulus result, but not the universal exhaustion step. The hypothesis is falsified by any explicit pair of inequivalent recovery families with identical recoverable distinction content and identical recovery geometry, or by any explicit recoverability-relevant effect that changes neither component. Discovery of such a counterexample must enlarge the invariant package and revise the reconstruction architecture. Until then, the canonical S1B output at this stage is an orbit-valued recovery architecture equipped with the candidate invariant package I_rec and an open completeness obligation. Current status. The Recoverability Completeness Program has advanced from counterexample search toward theorem construction. The leading unresolved targets are the Recovery Decision Exhaustion Principle, the Answer Partition Completeness Theorem, the Recovery Behavior Decomposition Principle, and a formal derivation of the necessity-of-manifestation architecture. No completeness theorem is claimed. Instead, the present record establishes a no-known-third-class result together with a structured foundations program intended either to construct a genuine third decision type or derive its exclusion from operational task theory. Decision-Theoretic Representation Program. The central open question is whether behaviorally relevant recovery decisions admit a third class beyond acceptance decisions and admissibility decisions. Extensive adversarial searches examined provenance, confidence, ranking, explanation, annotation, meta-decision, and context constructions. All realized candidates either modified accepted recoveries, modified admissible recovery realizations, or became operationally irrelevant. Operational Task Foundations. An operational task is represented by T=(Q,A,δ), where Q denotes admissible queries, A admissible answers, and δ the decision rule assigning answers to queries. Recovery tasks specialize this framework by supplying distinction classes, encoding families, recovery families, and recovery criteria. Operational behavior is represented by the collection of answers generated by the task. Role in emergence. The previous S1B.2A audits searched for hidden recoverability moduli through local structure, global obstructions, higher-order organization, self-reference, semantic criteria, and operational reconstruction. No explicit recoverability-relevant manifestation channel beyond recoverable distinction content and recovery geometry was realized. This motivates a foundations program that analyzes operational tasks directly rather than individual recoverability models. H22B.6A. Operational Task Foundations and the Recovery Decision Program Open proof obligation. A complete universal proof remains open. The central unresolved task is to establish that no behaviorally relevant operational consequence can separate states lying within a common joint acceptance-admissibility equivalence class. Resolution of that statement would substantially advance the derivation of Operational Closure and the broader recoverability-completeness program. Current evidence. Independent adversarial construction programs based on ranking structure, provenance structure, explanation structure, decision-valued policies, partition countermodels, same-joint-class separations, operational-output distinctions, query-family extensions, and candidate third primitive recovery objectives all reduced to one of three outcomes: modification of recovery-success structure, modification of realization-admissibility structure, or operational irrelevance. Representation-theorem status. RCEP is not presently asserted as a theorem. Instead it functions as the leading representation-principle candidate underlying Operational Closure, Operational Indistinguishability, Joint-Quotient Universality, Output-Factorization, Operational Output Exhaustion, Decision-Class Exhaustion, and the derivation of the Recovery Semantics Principle. Recovery Consequence Exhaustion Principle (RCEP) (Theorem Candidate). Every operationally meaningful recovery consequence belongs to the recovery-success class or the recovery-realization class. Equivalently, no behaviorally relevant recovery consequence has been identified that manifests through neither acceptance structure nor admissibility structure. Role in emergence. The Operational Task Foundations program reduces the remaining completeness frontier to the classification of operational recovery consequences. Previous adversarial programs examined rankings, provenance, explanations, meta-decisions, operational outputs, partitions, query families, and primitive recovery objectives. No explicit third primitive recovery consequence survived analysis. H22B.6B. Recovery Consequence Exhaustion Principle and Representation Status Open obligation. The central unresolved question remains whether there exists a recovery-behavior determinant that changes recovery behavior while preserving both recoverable distinction content and admissible realization structure. Failure to construct such a determinant strengthens RSCP but does not yet constitute a universal proof. Downstream implications. If RSCP holds, then Recovery Query Exhaustion, Recovery Task Characterization, Recovery Behavioral Completeness, Recovery-Semantic Manifestation, Necessary Objective Dichotomy, Primitive Recovery Objective Completeness, Recovery Consequence Representation, Recovery Consequence Exhaustion, Operational Closure, Operational Indistinguishability, and Joint-Quotient Universality become derivable within a common representation framework. Truth boundary. RSCP is not asserted as a proved theorem. No explicit third recovery-semantic determinant has survived adversarial analysis, but the universal completeness statement remains open. Accordingly, RSCP is adopted as the leading closure-principle candidate governing the remaining completeness program. Motivation and evidential status. The S1B.2A program has examined manifestation channels, outputs, decision classes, partition classes, query families, primitive recovery objectives, recovery consequences, and behavioral determinants. Every explicitly realized recoverability-relevant effect identified during the audit manifested through a change in recoverable distinction content, a change in recovery realization structure, or both. Recovery Semantic Closure Principle (RSCP) (Closure Principle Candidate). For a recovery task T_R, recovery semantics is hypothesized to be completely characterized by the ordered pair (I_content, G_rec), consisting of recoverable distinction content and admissible realization structure. Under RSCP, no additional primitive recovery-semantic component is required to characterize recovery behavior. H22B.6C. Recovery Semantic Closure Principle and Theorem Landscape Truth-first status. The conditional derivation chain does not prove RSCP. Rather, it demonstrates that RSCP generates a coherent, non-contradictory representation framework linking Recovery Query Exhaustion, Recovery Task Characterization, Recovery Behavioral Completeness, Recovery-Semantic Manifestation, and Recovery Consequence Exhaustion. The remaining open question is the universal nonexistence of a third recovery-behavior determinant. Conditional Recovery Consequence Exhaustion. Under RSCP, operational recovery consequences decompose into content-mediated consequences, realization-mediated consequences, or combinations thereof. Recovery Consequence Exhaustion is thereby recast as a downstream consequence of the representation principle rather than an independent primitive assumption. Conditional Recovery Behavioral Completeness and Manifestation. Under RSCP, recovery behavior factors through the recovery-semantic representation. Every recovery-behavior change therefore manifests through a change in recoverable distinction content, a change in admissible realization structure, or both. No additional recovery-semantic determinant is required within the conditional framework. Conditional Recovery Task Characterization. Under RSCP, two recovery architectures possessing identical recoverable distinction content and identical admissible realization structure are recovery-equivalent. Recovery semantics therefore supplies a complete known representation of the recovery task. Conditional derivation chain. Assuming RSCP, recovery semantics is represented by the ordered pair (I_content, G_rec). Under this assumption, every operationally meaningful recovery query derives its answer from recoverable distinction content, admissible realization structure, or both. This yields a conditional Recovery Query Exhaustion principle. Role in the completeness program. The falsification campaign examined local manifestations, global assembly obstructions, hidden gluing data, nonlocal closure invariants, and meta-semantic recovery structures. No explicit third recovery-behavior determinant survived adversarial analysis. Consequently, RSCP is adopted provisionally as the leading representation-principle candidate for conditional development while universal completeness remains open. H22B.6D. Conditional Consequences of the Recovery Semantic Closure Principle Truth boundary and revised limitations. This consolidation strengthens the representation-theoretic framework but does not prove RSCP, does not establish universal completeness, and does not prove the universal nonexistence of a third recovery-behavior determinant. Earlier limitations statements referring to the absence of a complete representation architecture should therefore be interpreted as superseded by the RRDT-A/RRCT-A/RRMT/RRCompT consolidation, while limitations concerning universal completeness remain in force. Recovery Representation Compression Theorem (RRCompT) (Conditional Form). Assume RSCP, RRDT-A, RRCT-A, and RRMT. Let A_R={RQE, RTCT, RBCT, RSMT, RCEP}. Then RRP together with {L1,L2,L3} generates the full currently known recovery-semantic architecture. Equivalently, A_R = Gen(RRP,L1,L2,L3). No additional known recoverability-relevant theorem class is presently required to generate the architecture. Recovery Representation Principle (RRP). Let S_R denote recoverability-relevant structures. The Recovery Representation Principle asserts that every X in S_R factors through (I_content,G_rec). Audit of the architecture shows that RRP together with L1, L2, and L3 generates Recovery Query Exhaustion, Recovery Task Characterization, Recovery Behavioral Completeness, Recovery-Semantic Manifestation, and Recovery Consequence Exhaustion. Recovery Representation Uniqueness (RRUT). Re-audit under RRDT-A, RRCT-A, and RRMT strengthened the uniqueness claim. The framework now supports the conditional statement that (I_content,G_rec) is the unique known minimal supporting representation of the current architecture up to representation-preserving equivalence. Universal uniqueness remains open. Recovery Representation Minimality (RRMT). Content and realization branches were subjected to a necessity and coarsening audit. Removal of recoverable distinction content destroys recoverability characterization; removal of realization structure destroys operational realization analysis. No surviving strictly coarser known representation preserving dependency, closure, and recoverability-relevant distinctions was identified. Accordingly, (I_content,G_rec) functions as the minimal known supporting recovery-semantic representation within the present framework. Closure Domain (RRCT-A). The class S_R of recoverability-relevant structures was defined operationally through behaviorally consequential distinctions. A representable subclass S_R^rep was introduced, and closure was stated conditionally as RSCP ⇒ S_R = S_R^rep. This replaces informal references to recoverability-relevant structure with a defined mathematical domain. Dependency Soundness (RRDT-A). The dependency chain RSCP ⇒ RQE ⇒ RTCT ⇒ RBCT ⇒ RSMT ⇒ RCEP was audited edge-by-edge. Three supporting lemmas were isolated and formalized: L1 Recovery Equivalence, L2 Behavior Factorization, and L3 Consequence Generation. Under these lemmas, every dependency edge possesses an explicit justification rather than relying on narrative implication. Synthesis update. Subsequent analysis formalized the representation-theoretic foundation underlying the Recovery Semantic Closure Principle program. The objective shifted from direct completeness proofs toward rigorous characterization of the conditional recovery-semantic architecture generated by the recovery representation framework. H22B.6E. Representation-Theoretic Consolidation: RRDT-A, RRCT-A, RRMT, RRUT, RRP, and RRCompT H22B.7. Current Answers and Transition Does a recoverability-relevant manifestation channel beyond content and geometry exist? No explicit example is known within the present finite-dimensional audit, but nonexistence is not proved. Does a recoverability-relevant third axis exist? It remains logically possible but unsupported by a constructive realization. The current weight of evidence favors the two-component content-geometry hypothesis without licensing a completeness claim. What is the next truth-first task? Continue the universal decomposition and necessity-of-manifestation program, search explicitly for inequivalent recovery families with identical I_rec, and treat any successful counterexample as a required architectural revision rather than an anomaly to be removed. Completion statement. S1B.1A is complete as a datum-relative no-selection theorem. The pre-accessibility basin, stability-ordering, and comparison-geometry audits are complete as no-selection consequences under orbit-preserving inputs. S1B.2A is advanced through observer-free recoverability, accessibility-extension analysis, and candidate recovery-invariant reconstruction. The recoverability completeness theorem is not complete. H23. Joint Record-Derived Comparison Transducer Theorem Package This section derives the correct comparison object from joint record dynamics. The primitive comparison object is a parameter-independent Markov kernel, or at the quantum level a CPTP degrading channel. A conditional distribution obtained from a single prior-averaged joint state is not sufficient because it may depend on the latent record parameter or preparation distribution. Deterministic and affine comparison maps are derived only as zero-noise and common-coordinate special cases. H23.1. Joint record experiments and regular conditional kernels Let Theta, X_i, and X_j be standard Borel spaces. Let rho_theta be a declared family of physical source states and let R_i and R_j be interaction-derived record channels. For admitted readout POVMs M_i and M_j define p_i(dt_i|theta)=Tr[M_i(dt_i) R_i(rho_theta)], p_j(dt_j|theta)=Tr[M_j(dt_j) R_j(rho_theta)]. If a joint record channel and joint readout produce p_ij(dt_i,dt_j|theta), disintegration yields a regular conditional kernel K_{j<-i}^theta(dt_j|t_i) for each theta, unique p_i(.|theta)-almost everywhere. This existence alone does not define a comparison transducer because K_{j<-i}^theta can depend on theta. Likewise, a kernel obtained after averaging over a sampling measure mu(dtheta) can depend on mu and is not intrinsic to the record channels alone. H23.2. Classical Joint Record-Derived Transducer Theorem Theorem H23.2. A state-independent stochastic comparison transducer from record experiment i to record experiment j exists exactly when there is a Markov kernel K_{j<-i}:X_i rightsquigarrow X_j, independent of theta, such that for every measurable B subseteq X_j and every theta in Theta, p_j(B|theta)=integral_(X_i) K_{j<-i}(B|t_i) p_i(dt_i|theta). Equivalently, the j-record statistical experiment is a Blackwell post-processing of the i-record experiment. Proof. If K_{j<-i} exists, sample t_i according to p_i(.|theta) and then t_j according to K_{j<-i}(.|t_i); the resulting distribution is p_j(.|theta) for every theta. Conversely, any one stochastic protocol that converts every i-record distribution into the corresponding j-record distribution is, by definition, a Markov kernel satisfying the displayed factorization. QED. The universal quantifier over theta is essential. A conditional law fitted to one prior or one preparation does not establish a transducer for the full record family. H23.3. Joint-Dynamics Conditional-Independence Criterion Theorem H23.3. Suppose p_ij(dt_i,dt_j|theta) has standard Borel outcome spaces. If a version of its conditional law satisfies p(dt_j|t_i,theta)=K_{j<-i}(dt_j|t_i) independently of theta, equivalently t_j is conditionally independent of theta given t_i, then K_{j<-i} is a comparison transducer and p_j^theta=p_i^theta K_{j<-i}. Conversely, if a comparison kernel K_{j<-i} exists, the coupling p_tilde_ij(dt_i,dt_j|theta)=p_i(dt_i|theta) K_{j<-i}(dt_j|t_i) has the same marginals and satisfies the conditional-independence relation. Proof. Under conditional independence, disintegration factors the joint law as p_i(dt_i|theta)K_{j<-i}(dt_j|t_i); marginalization gives p_j^theta=p_i^theta K_{j<-i}. Conversely the displayed coupling has the required marginals and a conditional law independent of theta. QED. Thus a record at i is sufficient for transducing to j when the latent record parameter carries no additional information about j once t_i is known. H23.4. Quantum Record-Degradability Transducer Theorem Quantum-transducer map signature. Let A_src be the common physical source algebra, and let B_i and B_j be the record-output algebras. The record channels are Schrödinger-picture CPTP maps R_i:S(A_src) -> S(B_i) and R_j:S(A_src) -> S(B_j). A quantum comparison transducer is a parameter-independent CPTP map D_j<-i:S(B_i) -> S(B_j), with UCP adjoint D_j<-i^dagger:B_j -> B_i. Theorem H23.4. Let R_i and R_j be finite-dimensional interaction-derived CPTP record channels with a common physical input. If there exists a CPTP channel D_{j<-i} such that R_j=D_{j<-i} o R_i, then D_{j<-i} is a state-independent quantum comparison transducer. For any readout POVM M_j, p_j(B|theta)=Tr[D_{j<-i}^dagger(M_j(B)) R_i(rho_theta)]. If the pulled-back POVM is a classical post-processing of M_i, D_{j<-i}^dagger(M_j(B))=integral_(X_i) K_{j<-i}(B|t_i) M_i(dt_i), then p_j^theta=p_i^theta K_{j<-i} for every theta. Proof. Substitute R_j=D_{j<-i} o R_i into the Born rule, pass to the adjoint channel, insert the POVM post-processing identity, and interchange the finite measure integral with the trace. QED. This establishes channel degradability as the strongest state-independent quantum transducer condition used here. A statistical morphism valid only on a restricted experiment may suffice operationally, but the manuscript does not identify it with a globally CPTP comparison channel without an extension theorem. H23.5. Joint-State Insufficiency No-Go Theorem Theorem H23.5. Neither the two marginal record families nor the existence of a compatible joint record distribution is sufficient to establish a parameter-independent comparison transducer. Proof by counterexample. Let theta in {0,1}. Let t_i=0 with probability one for both theta values, while t_j=theta with probability one. The joint law delta_0(dt_i)delta_theta(dt_j) exists for each theta. If one kernel K_{j<-i} satisfied p_j^theta=p_i^theta K_{j<-i} for both theta values, the identical i-distribution would have to map to two distinct j-distributions, which is impossible. QED. At the channel level, common marginalization is compatibility, whereas R_j=D_{j<-i} o R_i is divisibility or degradability. The former does not generally imply the latter. H23.6. Deterministic Graph-Reduction Theorem Theorem H23.6. A stochastic comparison kernel is deterministic exactly when there exists a measurable function Gamma_{j<-i}:X_i->X_j such that K_{j<-i}(B|t_i)=1_B(Gamma_{j<-i}(t_i)) for p_i^theta-almost every t_i across the declared experiment family. Proof. A Dirac kernel clearly defines Gamma_{j<-i}. Conversely, if the conditional measure is supported at one point almost everywhere, its support point defines Gamma_{j<-i}; standard Borel regularity gives a measurable version. QED. Where conditional variance is defined, zero conditional variance plus single-point support is the corresponding diagnostic. Deterministic comparison is therefore a special case, not the primitive assumption. H23.7. Affine Reduction Corollary Corollary H23.7. Suppose the deterministic comparison identifies the same recoverable record coordinate T and the calibrated clocks satisfy t_i(T)=alpha_i T+beta_i, t_j(T)=alpha_j T+beta_j, alpha_i,alpha_j>0. Then Gamma_{j<-i}(t_i)=a_ji t_i+b_ji, a_ji=alpha_j/alpha_i, b_ji=beta_j-(alpha_j/alpha_i)beta_i. If t_k=kappa_k L_k+c_k and L_k(T)=alpha_k T+beta_k, then a_ji=(kappa_j alpha_j)/(kappa_i alpha_i), with the intercept obtained by substitution of the calibrated constants. Proof. Invert the positive-affine relation for t_i to obtain T=(t_i-beta_i)/alpha_i and substitute into t_j(T). QED. Thus the old delta_ji is a derived intercept only after common-coordinate identification, deterministic transduction, and clock calibration have been independently established. Fisher speed alone does not determine the intercept. H23.8. Affine-Noise Kernel and Stochastic Comparison Holonomy For a noisy affine edge let t_(k+1)=a_k t_k+b_k+epsilon_k, K_(k+1,k)(dt_(k+1)|t_k)=nu_k(d(t_(k+1)-a_k t_k-b_k)). For a path gamma of n edges, composition gives t_n=A_gamma t_0+B_gamma+eta_gamma, A_gamma=product_(k=1)^n a_k, B_gamma=b_n+a_n b_(n-1)+...+(product_(l=2)^n a_l)b_1, eta_gamma=epsilon_n+a_n epsilon_(n-1)+...+(product_(l=2)^n a_l)epsilon_1. Theorem H23.8. The composed loop kernel is the identity kernel exactly when A_ell=1, B_ell=0, eta_ell=0 almost surely. Proof. The loop sends t to the law of A_ell t+B_ell+eta_ell. This law is delta_t for every t exactly under the three displayed conditions. QED. Consequently a loop can have A_ell=1 and B_ell=0 yet possess stochastic comparison holonomy through nonzero accumulated noise. The earlier affine holonomy theorem is the zero-noise subcase. H23.9. Collision-Memory Transducer Corollary In the retained-memory collision model, the accumulated record channels satisfy R_(1:n-1)=Tr_(F_n) o R_(1:n). Collision-memory map signature. Let B_1:n=B(H_F1 tensor ... tensor H_Fn) and B_1:n-1=B(H_F1 tensor ... tensor H_F(n-1)). The Schrödinger comparison channel is D_(n-1,n),*=Tr_Fn:S(B_1:n) -> S(B_1:n-1), and its Heisenberg adjoint is D_(n-1,n)^dagger:B_1:n-1 -> B_1:n, D_(n-1,n)^dagger(X)=X tensor I_Fn. Corollary H23.9. The partial trace D_(n-1,n)=Tr_(F_n) is an exact quantum comparison transducer from the later accumulated memory to the earlier accumulated memory. For any earlier-memory POVM, its pullback by D_(n-1,n)^dagger supplies a readout transducer preserving the full earlier record experiment. Proof. The factorization is the exact backward-recovery identity already proved for retained memory. Apply H23.4. QED. This is a genuine record-derived transducer, but it compares nested memories rather than two independently calibrated local clock streams. For fresh fragments F_i and F_j, joint compatibility alone does not establish degradability; the factorization R_j=D_{j<-i} o R_i must be tested. H23.10. Finite-Dimensional Validation Program Construct the joint channel R_ij and verify that its marginals are R_i and R_j. Solve the Choi-matrix feasibility problem for a CPTP D_{j<-i} satisfying R_j=D_{j<-i} o R_i. For selected readout POVMs, solve the linear feasibility problem p_j^theta=p_i^theta K_{j<-i} over the entire parameter family, not a prior average. Test whether the resulting kernel is Dirac almost everywhere; only then infer Gamma_{j<-i}. If deterministic, test positive affinity and compare its slope with the independently calculated Fisher-calibration ratio. Compose physically derived kernels around loops and test A_ell, B_ell, and the law of eta_ell. Failure at any gate is a valid model result and must not be repaired by fitting theta-dependent kernels or manually inserting offsets. H23.11. Completion status and extension frontier V. RECORD ORDER AND LOCAL COORDINATES Role in emergence. Recoverability turns records into an operational preorder and, after quotienting mutual recovery, a partial order. This is the first genuinely temporal layer: before durations are assigned, events become relationally ordered. Exact record recoverability defines a preorder. Quotienting mutual recoverability gives a partial order. Scalar order coordinates belong to observer-accessible chains and are unique only up to increasing transformations. Order alone supplies neither units nor additive duration. VI. OBSERVER-ACCESSIBLE STATISTICAL GEOMETRY Role in emergence. Ordered records acquire local metric content through accessible distinguishability. Fisher geometry measures the rate at which an observer can distinguish neighboring physical states along a relational path. Physical-instantiation boundary. The state path, clock channel, measurement class, parameter domain, and accessible outcome family used in this layer are not generated from bare potentiality by the Fisher construction. They are inherited physical data admitted through the governing bridge J_phys and the corresponding complete carrier and operational protocol. The Fisher line element is an operational property of that declared statistical experiment. It measures accessible distinguishability within the admitted measurement class and is not a fundamental substance, quantum fluid, or basement-level constituent. F_O^acc(lambda)=sup_(Pi in Meas_O) F_cl(lambda;Pi), Meas_O subset of POVM(A_O^op). On constant-support regular intervals, F_O^acc<=F_O^SLD, with equality when an SLD-optimal measurement is admissible. The selected algebra supplies an outer algebraic boundary; physical implementability can impose a stricter Meas_O. Rank-changing points require separate analysis. VII. CALIBRATION, DURATION, AND CLOCK ADEQUACY Role in emergence. Within a J_phys-instantiated clock experiment, integrating accessible Fisher speed constructs a reparameterization-invariant statistical length from operationally available distinguishability. That length is interpreted as local duration only when the declared path, measurement class, ordering, calibration, stability, identifiability, and comparison requirements are satisfied. L_O(lambda)=(1/2) integral sqrt(F_O^acc) d lambda, t_O=kappa_O L_O+b_O. Regular orientation-preserving recalibrations preserving additive interval composition are positive affine maps. Two clocks are affinely equivalent exactly when their Fisher line elements are proportional almost everywhere. A local clock must be identifiable, order-compatible, additively calibrated, and dynamically well posed. VIII. DYNAMICAL-TIME/FISHER-CLOCK CORRESPONDENCE Role in emergence. This section supplies the exact criterion for when a dynamical coordinate becomes operational time: positive constant accessible Fisher speed gives affine calibration, while departures quantify drift, recurrence, saturation, or memory dependence. G0-G3. Exact, homogeneous, and approximate correspondence Physical-provenance qualification for the Fisher-clock correspondence. In this theorem, T maps to rho_C(T) as a declared physically instantiated statistical path under J_phys, not as a basement-level quantum substance or an evolution generated from certified-null potentiality. The clock carrier, target system, channel, preparations, parameter identification, accessible measurement class, and calibration convention are inherited components of the governing physical datum. The theorem determines when the accessible Fisher length of that admitted experiment is affinely correspondent to its declared dynamical coordinate. It does not derive the physical actuality of the clock, the system, the state family, the interaction, or the parameter from the Fisher metric. Theorem G1. Let T map to rho_C(T) on a connected interval, with measurable, locally integrable F_C^acc positive almost everywhere. Define L_C(T)=L_C(T_0)+(1/2) integral_(T_0)^T sqrt(F_C^acc(s)) ds. Then L_C(T)=alpha_C T+beta_C with alpha_C>0 exactly when F_C^acc(T)=4 alpha_C^2 almost everywhere, equivalently dL_C/dT=alpha_C almost everywhere. Proof. L_C is absolutely continuous with almost-everywhere derivative (1/2)sqrt(F_C^acc). Differentiate the affine relation or integrate the constant rate. QED. No ontological converse is licensed: the existence of a differentiable quantum-state path or positive Fisher information does not establish that quantum amplitude, information, Hilbert-space structure, or the Fisher metric is fundamental, nor does it show that physical time has emerged from bare potentiality. After calibration kappa_C=2/sqrt(F_C^0), t_C(T)=T+constant. Statistical homogeneity of increments plus continuous additive interval composition is equivalent to affine L_C and constant Fisher speed. If |(1/2)sqrt(F_C^acc)-alpha_C|<=epsilon, then |L_C(T)-L_C(T_0)-alpha_C(T-T_0)|<=epsilon|T-T_0|, and calibrated clock drift is bounded by (epsilon/alpha_C)|T-T_0|. G4-G6. Positive clocks and dephasing negative test Gaussian location clock: p(y|T) is Gaussian with mean vT and variance sigma^2, F_C=v^2/sigma^2, and kappa_C=2sigma/|v|. For v=1.7, sigma=0.8, F_C=4.515625 and alpha_C=1.0625. Pure unitary clock: F_C=4(Delta H_C)^2/hbar^2 and kappa_C=hbar/Delta H_C. For H_C=(hbar omega/2)sigma_z, |+>, and omega=1.4, F_C=1.96. rho_S(T)=(1/2)[I+exp(-gamma T)sigma_x], F_S(T)=gamma^2 exp(-2gamma T)/(1-exp(-2gamma T)). The dephasing Fisher length is L_S(T)=(1/2) arccos(exp(-gamma T)), is nonaffine, and saturates at pi/4. For gamma=0.7, quadrature agrees with the closed form at tested points with absolute errors below 1.6e-13. G7. Endpoint-Regularized Fisher Duration Theorem G7. For measurable F on (a,b), let v=(1/2)sqrt(F) belong to L1_loc and L(T)=ell_*+integral_(T_*)^T v. L is locally absolutely continuous and nondecreasing. It has a finite endpoint extension exactly when v is one-sided integrable there; the extension is unique, continuous, and absolutely continuous. If the integral diverges, no finite continuous extension preserving the integral law exists. QED by the fundamental theorem for Lebesgue integrals and the monotone improper-integral criterion. If F(a+x) is asymptotic to C x^-alpha, endpoint duration is finite exactly when alpha<2, logarithmically divergent at alpha=2, and power divergent for alpha>2. For dephasing, F(T)=gamma/(2T)-gamma^2/2+O(T), so alpha=1 and L(T) is asymptotic to sqrt(gamma T/2). Rank change alone neither licenses nor forbids an endpoint. G8. Margolus-Levitin audit Theorem G8. For autonomous pure unitary clocks with H bounded below, dL/dT=Delta H/hbar while any orthogonalization time obeys T_perp>=pi hbar/(2 Ebar), Ebar=. Neither Ebar determines Fisher speed nor Delta H determines the ML bound. For a two-level state sqrt(1-p)|0>+sqrt(p)|1>, Ebar=p epsilon and Delta H=epsilon sqrt(p(1-p)), giving explicit fixed-Delta-H and fixed-Ebar separation families. The balanced p=1/2 clock saturates both bounds with L(T_perp)=pi/2. Energy-zero shifts leave both quantities invariant. G9-G10. Consequences and boundary T becomes operational Fisher duration exactly when a designated, implementable channel has positive constant accessible Fisher speed. This criterion distinguishes a genuine relational clock from a merely convenient evolution label and makes clock selection a falsifiable physical result. Questions of fundamental clock selection and absolute scale are recorded in the claim-status ledger. G11-G18. CLOCK ADEQUACY UNDER NONUNITARY AND NON-MARKOVIAN DYNAMICS G11. Clock-audit datum and parameter discipline A clock-audit datum is C_audit=({Lambda_(T,s)},{rho_C(s)},M_C,R_C,H,J,P), where Lambda_(T,s) is the declared two-parameter physical evolution, rho_C is the candidate clock path, M_C is the admitted measurement class, R_C is the admitted memory or recovery architecture, H is the record order, J is the claimed clock interval, and P declares the parameter being estimated. The distinction between elapsed time T, an initial-state parameter theta, and a Hamiltonian or environmental parameter is mandatory. F_T^acc measures neighboring elapsed-time distinguishability; F_theta(T) measures preservation of information about theta. Their flows are not interchangeable. G12. Regime-Independent Fisher Clock Correspondence Theorem Theorem G12. Let T map to rho_C(T) be an absolutely continuous state path generated by a declared physical evolution, including nonunitary or non-Markovian evolution. Let F_T^acc be measurable and locally integrable and define v_C(T)=(1/2)sqrt(F_T^acc(T)) and L_C(T)=L_C(T_0)+integral_(T_0)^T v_C(u)du. Then L_C(T)=alpha T+beta on a connected interval, with alpha>0, exactly when F_T^acc(T)=4alpha^2 almost everywhere. Proof. Absolute continuity gives dL_C/dT=(1/2)sqrt(F_T^acc(T)) almost everywhere. Differentiating the affine relation proves necessity; integrating constant positive Fisher speed proves sufficiency. No unitary, semigroup, Markov, or divisibility hypothesis is used. QED. Consequence. The mathematical G1 criterion was already regime independent. The missing task was explicit execution and failure classification for open and memory-bearing dynamics. G13. Integrated Clock-Drift Bound Theorem G13. Fix alpha>0 and let delta_v(T)=v_C(T)-alpha. Then for every T in the audit interval, |L_C(T)-L_C(T_0)-alpha(T-T_0)| <= integral_(T_0)^T |delta_v(u)|du. If |delta_v|<=epsilon almost everywhere, the earlier linear bound epsilon|T-T_0| follows. Proof. Subtract alpha(T-T_0) from the defining integral for L_C and apply the triangle inequality. QED. This L1 distortion bound is preferable for sharply peaked revivals because it measures accumulated calibration error rather than only the worst local deviation. G14. CP-Divisibility Insufficiency Theorem Theorem G14. CP divisibility is neither sufficient nor necessary for global affine clock adequacy. Sufficiency fails because a CP-divisible amplitude-damping semigroup has elapsed-time Fisher information F_T(T)=gamma^2 exp(-gamma T)/(1-exp(-gamma T)), which is not constant. Necessity fails because affine adequacy is the scalar path condition F_T^acc=4alpha^2 almost everywhere; failure of CP divisibility concerns intermediate-map structure and does not logically negate that scalar condition. Claim boundary. The necessity counterstatement is logical rather than an executed non-CP-divisible constant-speed model. Version 19 does not claim that every non-Markovian process admits an adequate clock. G15. Fisher-Backflow Non-Equivalence Theorem Theorem G15. Let F_theta(T) quantify distinguishability of an encoded parameter theta and let F_T(T) quantify distinguishability with respect to elapsed time. Neither the sign nor monotonicity of dF_theta/dT determines whether F_T is positive and constant. The two quantities depend on distinct tangent vectors, partial_theta rho_theta(T) and partial_T rho_theta(T), respectively. Therefore a non-Markovian revival of parameter distinguishability about theta is not, by itself, a revival of clock adequacy. QED. G16. Reduced-State Recurrence Obstruction Theorem G16. If rho_C(T_1)=rho_C(T_2) for distinct T_1 and T_2 in the claimed global clock domain, no measurement of the instantaneous reduced state defines a globally injective time reading on that domain. Proof. Equal density operators give identical outcome distributions for every POVM, so no instantaneous measurement distinguishes T_1 from T_2. QED. Corollary G16.1. A retained history, branch label, or cycle count can remove the ambiguity only by enlarging the clock from C to CM. It does not restore global adequacy of C alone. G17. Recovery-Adequacy Separation and Memory Extension Theorem G17. Exact recoverability of a record algebra or state family does not imply constant elapsed-time Fisher speed, global injectivity, or order-compatible calibration. If M_C is contained in M_CM, then F_T^acc(C)<=F_T^acc(CM), but adequacy of CM is a property of the enlarged clock-memory system and requires an independent G12 audit. The selected output is part of this audit: recoverability through the final system and through a complementary retained environment may differ, and the Fisher-clock test must be performed on whichever output and measurement class define the claimed clock. Proof. Recovery reconstructs encoded information and does not impose the derivative condition F_T^acc=4alpha^2. Measurement-class inclusion gives Fisher monotonicity by taking a supremum over a larger feasible class. QED. G18. Clock-Adequacy Status Classification Definition G18. A candidate clock is classified on its declared interval and measurement class as follows: Class A, globally affine: F_T^acc=4alpha^2>0 almost everywhere, with global injectivity and compatible dynamics. Class B, approximately affine: a declared integrated drift tolerance is satisfied. Class C, piecewise adequate: identifiable calibrated charts exist, but no single global affine calibration exists. Class D, memory extended: the reduced clock fails, but an enlarged retained-memory system passes the clock gates. Class E, event timer: the candidate is adequate only on a limited task interval and exhausts or saturates distinguishability. Class F, record only: the channel forms or preserves records but lacks adequate elapsed-time Fisher information. Class G, inadequate: zero-speed intervals, recurrence, branch ambiguity, incompatible order, or inaccessible measurements prevent a valid clock. G18.1. Explicit nonunitary and non-Markovian counterexamples Counterexample 1, Markovian dephasing. For rho(T)=(1/2)[I+exp(-gamma T)sigma_x], F_T(T)=gamma^2 exp(-2gamma T)/(1-exp(-2gamma T)) and L(T)=(1/2)arccos(exp(-gamma T)). The evolution is nonunitary and CP divisible, but the speed is nonconstant and the length saturates at pi/4. Thus Markovianity does not imply global clock adequacy; the reduced system is at most a Class E event timer. Counterexample 2, Markovian amplitude damping. For excited-state population p(T)=exp(-gamma T), the energy measurement is optimal for the diagonal family and F_T(T)=gamma^2 exp(-gamma T)/(1-exp(-gamma T)). The endpoint singularity is integrable, but the Fisher speed decays toward zero. The semigroup is therefore not a globally affine clock. Counterexample 3, oscillatory non-Markovian dephasing. Let kappa(T)=exp(-gamma T)cos(omega T) and rho(T)=(1/2)[I+kappa(T)sigma_x]. Then F_T(T)=dot(kappa)(T)^2/(1-kappa(T)^2), with dot(kappa)=-exp(-gamma T)[gamma cos(omega T)+omega sin(omega T)]. Fisher speed vanishes whenever tan(omega T)=-gamma/omega, producing temporal turning points. For gamma=0.12 and omega=1.4, the first audited zeros occur near T=2.183 and T=4.427. The path may be piecewise readable, but instantaneous global calibration fails without a branch or memory label. Numerical audit on 0.1<=T<=5. For gamma=0.7, Markovian amplitude damping gives F_T from approximately 0.0153 to 6.72 and a Fisher-speed coefficient of variation approximately 0.882. For gamma=0.12 and omega=1.4, oscillatory dephasing gives F_T from approximately 4.56e-9 to 2.25 and a Fisher-speed coefficient of variation approximately 0.497. These values diagnose strong inhomogeneity; they are not universal constants. G18.2. Clock curvature and audit interpretation Where F_T^acc>0 is differentiable, define clock curvature chi_C(T)=d[log v_C(T)]/dT=(1/2)d[log F_T^acc(T)]/dT. Homogeneous Fisher ticking gives chi_C=0. Bounded small curvature supports local calibration stability, while zeros or singularities of Fisher speed can make chi_C diverge. Changes in observer-accessible parameter distinguishability can change v_C and chi_C, but Fisher duration L_C remains nondecreasing because its integrand is nonnegative. Audit conclusion. Nonunitarity and non-Markovianity are neither automatic failures nor adequacy certificates. A candidate passes only relative to its declared physical propagator, elapsed-time parameter, observer-accessible measurement class, memory resources, injectivity domain, and drift tolerance. G19-G23. ETP-SPECIFIC NON-MARKOVIAN COLLISION-CLOCK EXECUTION G19. Composite memory-carrier collision datum Let S be a source qubit, M a persistent environmental memory qubit, and F_n a fresh fragment at update n. The memory and all fragments begin in their ground states. The source carries a relational phase |psi_S(T)>=(|0>+exp(-i omega T)|1>)/sqrt(2). At each update apply the excitation-preserving partial swaps U_SM(phi)=exp[-i phi(sigma_+^S sigma_-^M+sigma_-^S sigma_+^M)] and U_MF_n(chi)=exp[-i chi(sigma_+^M sigma_-^F_n+sigma_-^M sigma_+^F_n)]. Chi=pi/2 empties M into each outgoing fragment and gives the memoryless benchmark; 0=s_n|1_S>+m_n|1_M>+sum_(k=1)^n f_(k,n)|1_Fk>. With c_phi=cos(phi), s_phi=sin(phi), c_chi=cos(chi), and s_chi=sin(chi), define s_tilde_n=c_phi s_(n-1)-i s_phi m_(n-1) and m_tilde_n=-i s_phi s_(n-1)+c_phi m_(n-1). Then s_n=s_tilde_n, m_n=c_chi m_tilde_n, and f_(n,n)=-i s_chi m_tilde_n. Earlier emitted amplitudes are unchanged. Consequently |s_n|^2+|m_n|^2+sum_(k=1)^n|f_(k,n)|^2=1. G20. Single-Excitation Fisher Partition Theorem Theorem G20. Let A be any accessible subsystem formed from S, M, and a subset of retained fragments. Let q_A(n) be the sum of the squared conditional excitation amplitudes located in A. For the relational phase T and unrestricted measurements on A, the SLD quantum Fisher information is F_T^A(n)=omega^2 q_A(n). Hence F_T^S=omega^2|s_n|^2, F_T^M=omega^2|m_n|^2, F_T^F_n=omega^2|f_(n,n)|^2, and F_T^F_(1:n)=omega^2 sum_(k=1)^n|f_(k,n)|^2. Proof. After tracing inaccessible modes, the accessible state has support on the vacuum and one-excitation sectors. Its T dependence is only the relative phase exp(-i omega T) multiplying the normalized accessible excitation vector, with total conditional weight q_A. Evaluating the two-dimensional SLD on the vacuum-excitation support gives omega^2 q_A. QED. Corollary G20.1. F_T^S(n)+F_T^M(n)+F_T^F_(1:n)(n)=omega^2. Thus, within the declared orthogonal single-excitation encoding and unrestricted subsystem measurements, the global collision unitary preserves the operational distinguishability of the encoded parameter T across the accessible source, memory, and retained-fragment subsystems. Operational interpretation. The additive identity is a decomposition of a statistical distinguishability measure for one declared parameterized state family. It does not describe transport, flow, redistribution, storage, or conservation of a physical informational or temporal substance. Claim boundary. The additive partition is proved for this orthogonal single-excitation phase encoding. It is not asserted for thermal, multiparticle, or interacting encodings. G21. ETP Memory-Carrier Nondivisibility Corollary Corollary G21. On an interval of updates where the reduced source map is invertible, if F_T^S(n+1)>F_T^S(n), then the reduced source evolution is not CP divisible across that update. Proof. The reduced source channel is an amplitude-damping-type phase-covariant channel with survival amplitude s_n. CP divisibility of the intermediate map requires |s_(n+1)/s_n|<=1. Because F_T^S(n)=omega^2|s_n|^2, a strict revival of source-accessible parameter distinguishability violates that condition. Singular zero-amplitude steps require separate noninvertible analysis. QED. G22. Retained-Memory Fisher Monotonicity Theorem Theorem G22. For retained emitted fragments in the declared model, F_T^F_(1:n+1)=F_T^F_(1:n)+omega^2|f_(n+1,n+1)|^2>=F_T^F_(1:n). Proof. A newly emitted fragment occupies an orthogonal retained mode and earlier emitted amplitudes are unchanged. Apply G20 and add the new nonnegative excitation weight. QED. This theorem concerns the distinguishability of the relational phase T available to the retained-memory observer class. Monotonic growth refers to increasing recoverable parameter distinguishability under that declared access convention. It is a relational measurement outcome, not the accumulation, transport, or storage of a temporal, informational, or Fisher-theoretic fluid. It does not imply constant statistical distance per collision update or monotonic distinguishability in each fresh fragment. G23. Non-Markovian Record-Clock Separation Theorem Theorem G23. In the declared memory-carrier collision model, the parameter distinguishability recoverable from retained memory can grow monotonically while source-accessible parameter distinguishability revives and adjacent retained-memory Bures increments remain nonconstant. Therefore growth of recoverable retained-record distinguishability, exact backward recovery, and revival of source-accessible parameter distinguishability do not imply affine clock adequacy for collision count n. Proof. G21 permits revivals of source-accessible parameter distinguishability, G22 proves monotonic growth of retained-memory parameter distinguishability under the declared access class, and the explicit G23.1 execution gives nonconstant adjacent-step Bures increments. G12 then rules out one affine calibration to update count. QED. G23.1. Numerical execution Memoryless benchmark. For phi=0.6, chi=pi/2, and omega=1, the first source Fisher values are 0.681179, 0.464005, 0.316070, 0.215300, and 0.146658, with no revival. Retained-memory parameter distinguishability rises through 0.318821, 0.535995, 0.683930, 0.784700, and 0.853342. Non-Markovian benchmark. For phi=0.6, chi=0.35, and omega=1, the tuples (F_T^S,F_T^M,F_T^F_n,F_T^F_(1:n)) for n=1,...,5 are (0.681179,0.281334,0.037487,0.037487), (0.145685,0.720787,0.096042,0.133528), (0.027013,0.740756,0.098703,0.232231), (0.386413,0.336516,0.044839,0.277070), and (0.706601,0.014409,0.001920,0.278990). The increase from n=3 to n=5 is an explicit revival of source-accessible parameter distinguishability. By n=10 the retained-memory parameter distinguishability reaches 0.480881. For the same non-Markovian benchmark, adjacent retained-memory Bures angles over the first twenty updates have coefficient of variation approximately 0.524. Equal updates therefore do not yield homogeneous statistical ticks. This statistic is a finite numerical diagnostic, not a universal constant. Numerical interpretation. The listed values and adjacent-angle statistic quantify changing distinguishability of the encoded parameter relative to the declared subsystem and observer-access convention. They are relational measurement outcomes and must not be interpreted as quantities of an informational or temporal fluid accumulated in memory. G23.2. Executed classification Reduced source as a relational-phase clock: Class C, piecewise adequate on injective branches with update-dependent calibration and near-zero failures. Fresh fragment stream as a relational-phase sensor: each fixed fragment can be locally calibrated, but the stream lacks one common calibration; for update count it is Class F or G unless the update label is supplied independently. Accumulated retained memory as a relational-phase clock: Class D, memory extended, and locally Class A on a fixed injective phase branch when the optimal measurement is admitted. Accumulated retained memory as an update-count clock: not globally affine; at best Class E on a declared finite task interval. G23.3. Scope and process-tensor frontier The finite-dimensional ETP-specific non-Markovian audit is complete within the declared model. It demonstrates that the observer-accessible distinguishability of the encoded parameter can change across declared subsystems and access classes, including revival in the reduced source and monotonic increase for accumulated retained memory, while update-count duration remains nonaffine. These changes do not describe movement, storage, backflow, or accumulation of a physical informational or temporal substance. The thermal, restricted-measurement, process-tensor, continuum, and battery extensions are listed in the scope matrix. IX. LOCAL M1 MODEL Role in emergence. The M1 model demonstrates that systems can agree on record order while assigning different local temporal metrics. It therefore isolates local duration from any premature assumption of global synchronization. F_i(lambda)=4g_i^2w_i^2+2g_i^2w_i(1-w_i)[1+cos(2g_i lambda)]. M1 establishes ordinal consensus with local Fisher-metric discord for generic parameter mismatch. It therefore proves that shared event order can precede and coexist with disagreement in local duration; synchronization and direction are developed in the subsequent layers. X. CLOCK GLUING BY OPERATIONAL COMPARISON TRANSPORT Role in emergence. Local clocks become a shared temporal structure only through physical comparison maps. The transport and holonomy theorems determine when local calibrations glue consistently and when irreducible global disagreement remains. Version 17 supplies the record-dynamical origin criterion for comparison edges in H23. The affine groupoid below remains the deterministic zero-noise subtheory. In general, comparison edges are Markov kernels or degrading channels, and a loop can retain stochastic holonomy even when its affine coefficients close. D0-D8. Groupoid, holonomy, and global calibration H_ell=Gamma_gamma_n ... Gamma_gamma_1, H_ell(t)=A_ell t+B_ell. Affine comparison arrows transform by local clock-chart conjugation. A connected reversible comparison network admits a global parallel calibration exactly when every closed loop has identity holonomy. Local fragmentation is a missing required edge; comparison curvature is disagreement between direct and sequential comparisons; global holonomy is a nonidentity loop despite local edge existence. D9. Positive Gaussian M3 comparison-holonomy model For three translated minimum-uncertainty Gaussians with (s_1,s_2,s_3)=(1,1.25,0.8) and (sigma_1,sigma_2,sigma_3)=(0.5,0.625,0.4), each position measurement is optimal and F_i^acc=4. With offsets delta_21=0.30, delta_32=-0.10, delta_13=0.25, the loop has A_ell=1 and B_ell=0.45. Numerical quadrature verifies F_i=4 at lambda=-0.4,0,0.7; gauge rescalings alpha=2 and 0.7 give B_ell prime=0.90 and 0.315. The offsets remain independent comparison data and are not derived from the collision mechanism. Version 17 further requires that each edge protocol be admitted by the selected observer measurement structures. XI. CONDITIONAL RECORD-TO-THERMODYNAMIC ARROW Role in emergence. Metric time does not by itself select a direction. This section identifies the additional record, memory, reservoir, and entropy-production conditions under which a local temporal arrow can be supported. I(X:M_(n+1))>=I(X:M_n), Sigma=Delta S_A-beta Q_A=I(A:B)+D(rho_B prime||tau_B)>=0. Backward recoverability, threshold redundancy, append-only memory entropy, and thermodynamic entropy production form separate gates. Their conjunction, rather than record growth alone, is what supports a local temporal arrow. XII. EXPLICIT REPEATED-INTERACTION OPEN DYNAMICS Role in emergence. The collision model realizes the abstract sequence dynamically: interactions create records, retained memories order them, operational distinguishability and entropy balances can be computed, and a continuum limit connects discrete updates with effective evolution. F0-F1. Microscopic architecture and recovery U_theta=P_0 tensor I+P_1 tensor R_theta, Phi_theta(rho_S)=Tr_F[U_theta(rho_S tensor tau_F)U_theta^dagger]. Fresh independent ancillas give rho_S(n)=Phi_theta^n[rho_S(0)]. Retained memory M_n=F_1...F_n exactly recovers M_(n-1) by tracing out F_n, yielding monotone I(X:M_n) by data processing. F2. Strong classical-record regime For theta=pi/2, p=0.3, and beta epsilon=4, r=0.9820137900, H(X)=0.8812908992 bits, and one-fragment accessible information is 0.7599291097 bits, or 86.2291 percent of H(X). For n=1,...,8, memory mutual information is 0.759929, 0.846117, 0.874754, 0.879353, 0.880903, 0.881175, 0.881267, 0.881284 bits. Memory entropy is 0.889908, 1.106075, 1.264692, 1.399270, 1.530799, 1.661050, 1.791122, 1.921118 bits. F3-F5. Thermodynamics, weak channel, and continuum limit For the strong diagonal collision, Q_S=-0.2892082740 epsilon and Sigma=1.1568330961 nats, with I(S:F)=0.5267427198 and D(rho_F prime||tau_F)=0.6300903763 nats. For theta=0.2, coherence multiplies by cos(theta)=0.9800665778, Delta S_S=0.0558485117, Q_S=-0.0190248447 epsilon, and Sigma=0.1319478907 nats. Under theta^2=2 gamma Delta T, the reduced map converges to L(rho)=(gamma/2)(sigma_z rho sigma_z-rho). For gamma=0.7 and T=1.3, coherence errors are 8.783e-3, 4.331e-3, 1.719e-3, 8.570e-4, and 4.279e-4 for Delta T=0.1,0.05,0.02,0.01,0.005. F6-F7. Model achievements and extension frontier The CPTP map, backward recovery, strong-record information growth, finite-step entropy production, GKSL limit, and update-count coarse graining remain explicit. Version 20 additionally executes collision-record Fisher information in a separate composite memory-carrier model: revivals of reduced-source parameter distinguishability diagnose reduced nondivisibility on invertible steps, recoverable parameter distinguishability from retained fragments grows monotonically for the encoded phase, and update-count ticking remains nonaffine. The original driven collision, autonomous battery dilation, and memory-carrier audit remain distinct declared models; no single model is claimed to realize every layer. XIII. EFFECT ON ETP AS A WHOLE Synthesis. The preceding layers jointly explain why relational time can emerge without collapsing order, duration, synchronization, and direction into one primitive. Each theorem contributes one necessary structural transition while preserving the independence of the others. Local Fisher geometry and global synchronization remain independent; equal local metrics can coexist with nontrivial comparison holonomy. The master constraint is controlled in explicit compact and noncompact representations but does not select the fundamental ETP symmetry. Version 21 converts symmetry selection from a single undifferentiated gap into SYM-1 through SYM-5. The immediate finite-dimensional target reconstructs only the maximal effective symmetry of a complete physical datum and explicitly does not identify it with the unique pre-reduction gauge group. Version 22 proves SYM-1: a complete closed finite-dimensional physical datum determines a unique maximal effective compact Lie action, while unitary lifts, covers, central extensions, and pre-reduction gauge interpretations remain nonunique. Version 23 proves SYM-2: proper gauge lies in K_null, but K_null is not a gauge certificate; faithful G_eff actions are non-gauge, and boundary, dynamical, operational, and superselection roles require separate tests. Version 24 formulates SYM-3 without presuming recoverability: effective derivations are physical by default, generator representatives are generally defined only modulo the commutant, and C1 is available only after independent gauge-lift certification. Version 25 proves the first SYM-3 block: (g_eff,delta,pi_0) is reconstructible; physical generator lifts are defined modulo the commutant; nonzero effective derivations are not gauge; and a fixed certified constraint family admits many positive master metrics with one common physical kernel. Version 26 completes finite-dimensional SYM-3B relative to a declared admissibility specification: reduced, gauge-orbit, and strong master equivalence are separated; covariance and occupancy are proved; and explicit inequivalent lifts establish genuine compatibility-class multiplicity without altering the reduced physics. Version 27 synchronizes all finite-dimensional SYM-3 status statements with C13. Distinguished-representative selection is treated as a new research question, not as evidence that the reduced datum already selects a preferred lift or master metric. Version 28 makes the emergence logic explicit across the manuscript: physical reduction provides the arena, observer-access and record the operational substrate, recoverability the order, Fisher geometry the duration, clock adequacy the temporal coordinate, comparison transport the global structure, and autonomous record thermodynamics the conditional local arrow. Superseded limitations are updated and remaining extensions are concentrated in status tables. Version 29 clarifies that unreduced gauge nonuniqueness is an expected consequence of reduction when reduced observables cannot distinguish the lifts, and that finite-clock backreaction and degradation in ARF2 are explicit open S5 requirements rather than ignored effects. Version 30 proves SYM-3C.1: observer, record, Fisher, clock, comparison, and holonomy structures covariantly derived from a complete reduced datum cannot distinguish reduced-equivalent gauge lifts or select an on-shell-equivalent master metric. Only independently justified gauge-invariant enrichment with a separating physical outcome can split the compatibility class. Version 31 proves SYM-3C.2: independently justified boundary conditions, physical boundary observables, charges, channels, and gluing relations may separate bulk-equivalent lifts, but auxiliary edge presentations cannot. The canonical result is a boundary compatibility hierarchy rather than a unique edge ontology or boundary completion; any temporal consequence remains downstream of record, recovery, Fisher, clock, and comparison gates. Version 31 also closes the bounded finite-dimensional SYM-3 architecture. Remaining items are explicitly classified as broader mathematical extensions or ETP-specific executions rather than missing conceptual stages. The next principal symmetry investigation is SYM-4 sector and commutant decomposition. The open coordinate T becomes physical duration only relative to an adequate selected clock channel. Endpoint duration depends on integrability of the selected accessible Fisher speed, not pointwise QFI finiteness. Fisher duration and energetic resources remain distinct. Version 17 replaces a manually listed seed family, within a declared record model, by the exact stable-readable algebra. It does not derive the record model from bare potentiality. A selected algebra can lower accessible Fisher information, invalidate a candidate clock, or remove a comparison edge. These are model findings, not contradictions of the general theorems. A trivial selected algebra is a valid null result and must not be adjusted away. Version 19 shows that nonunitarity, CP divisibility, recovery, record formation, and revival of encoded-parameter distinguishability is individually insufficient for clock adequacy; the elapsed-time Fisher path must pass the G12-G18 audit. Version 20 executes the audit on an ETP-compatible memory-bearing collision model. Non-Markovian memory changes which declared subsystem permits recovery of encoded-parameter distinguishability, but monotonic growth of retained-memory distinguishability still does not make collision count an affine duration coordinate. Version 18 closes exact finite-horizon energy-conserving record formation and the inclusive battery-aware entropy ledger under declared hypotheses; it does not identify energy, work, clock coordinate, Fisher duration, and thermodynamic direction. Version 34 completes the bounded finite-dimensional SYM-4 architecture after sanitation: the implemented isotypic decomposition, joint algebra and commutant, central Wedderburn sectors, observer preservation/readability/recordability/recovery hierarchy, fixed-sector SLD Fisher decomposition, clock-adequacy gates, and comparison-transport no-assembly boundary are synchronized. Recovery of the complete statistical model is sufficient for SLD-Fisher preservation, while equality for one tangent is not used as a recoverability certificate. Version 35 establishes the finite-dimensional SYM-5 foundation. The joint algebra and its sectors define an arena of possible distinctions, not a preferred observer. Physical interactions and output structures generate candidate access seeds; admissibility then requires provenance, distinguishability, sector classification, persistence, recordability, recoverability, covariance, robustness, and compositional closure. The theory therefore permits empty, symmetry-orbit, nonunique, Pareto, and conditionally unique observer families. Observer selection affects relational time only through the resulting accessible statistical experiments and does not bypass the Fisher, clock, comparison, or holonomy gates. Version 36 closes the bounded finite-dimensional interaction-relative physical-output-net step and establishes physical effect-system reconstruction. ETP now distinguishes the minimal event-grounded output net, record-qualified fragments, observer-qualified domains, and physically implementable effects. Record degradability supplies an operational preorder but does not replace causal order or duration. Effect-system pullback supplies the missing bridge from physical outputs to observer-access seeds. The theory therefore advances without treating tensor factors, all mathematical effects, or update labels as physically selected by convention. - The Version-54 dual-output collision audit executes H22.2 on both outputs of one isometry. It separates final-system preservation from environmental-record recovery, verifies dimension vectors (2,2,1,1) and (2,1,2,1), and shows that a perfect retained early record can survive later system rotation even when final-system exact access becomes scalar. This strengthens the complete-carrier principle without promoting exact recovery to clock adequacy. Cross-program synthesis update. The manuscript now links ground-frontier provenance discipline with causal-sector diagnostics through a common anti-aggregation rule while preserving their different mathematical types. It distinguishes C_infinity, C_1, and the cancellation-sensitive C_bar; uses the 1 percent star sector only as a bounded structural illustration; and forbids importing a probability measure or unique selector into a minimal-ground frontier without independent justification. XIV. REQUIRED WORK - Extend the dual-output collision audit to epsilon-approximate recovery and Helstrom or channel-distance bounds in the weak nonorthogonal environmental regimes. - Compute detector-restricted accessible Fisher information separately for the reduced-system and complementary-environment outputs; test G12-G18 before making a duration or synchronization claim. - Extend the dual-output construction to thermal fragments, initially correlated fragments, process tensors, and resource-constrained recovery while preserving channel orientation explicitly. Execute Choi-matrix degradability tests between physically derived fragment and memory channels. Compute parameter-independent readout kernels over the entire collision-coordinate family and reject prior-dependent conditional fits. Derive a nontrivial multi-edge stochastic loop from one record-forming model and compare its affine and noise holonomies. Identify whether any collision-record streams satisfy the common-coordinate constant-speed conditions needed for affine reduction. Extend the completed bounded interaction-relative output-net theorem to gauge-constrained interfaces, continuous event domains, indefinite causal structure, approximate stability, and nontrivial composition. Extend the maximal recoverable-algebra theorem to approximate recovery with quantitative error bounds and to normal infinite-dimensional channels. Extend the executed fragment and accumulated-memory Fisher analysis to mixed thermal inputs, restricted POVMs, and general multiparticle sectors. Derive operational boundaries only after adding the physical symmetry-breaking data required by the boundary nonuniqueness theorem. Stage S1 priority: derive a covariant, falsifiable candidate-record-channel family from physical interaction structure, permitting an empty result. Stage S2 priority: derive operational boundaries and closure rules from composable channels, controls, memories, instruments, locality, and resource constraints. Stage S4 mathematical channel-correction construction completed for a fixed selected channel and supported sector: the unique largest exactly correctable algebra is the commutant of the compressed Kraus-product algebra, with one recovery channel correcting the whole algebra. Observer-relative S4 access additionally requires intersection with the declared readable or operational algebra, physical admissibility of the recovery, and proof of any S3-to-S4 inclusion. Stage S5 finite-dimensional execution completed for one memory-carrier phase model; extend to restricted exact, recoverable, and operational measurement classes. SYM-1 completed in the closed finite-dimensional scope: extend physical-datum reconstruction only when moving to infinite-dimensional, nonclosed, or categorical settings. SYM-3 is formally frozen as complete in the bounded finite-dimensional, declared-domain, declared-boundary, and declared-protocol scope through SYM-3C.2. Bounded finite-dimensional SYM-4 is also complete within its declared implemented-symmetry, sector, recovery, Fisher, clock, and comparison scope and is the immediate input to active SYM-5. Remaining unbounded, noncompact, BRST/BFV, structure-function, infinite-family, continuum-boundary, asymptotic, gravitational-corner, and ETP-specific boundary-model items are extensions or executions. Extend completed SYM-3B beyond bounded finite dimension: unbounded domains, noncompact rigging maps, BRST/BFV stable equivalence, structure functions, and explicit edge-charge models. Derive or independently justify an ETP-specific boundary algebra, charge family, boundary condition, interface channel, and gluing law; then test the C15 separation and recoverability gates. Continue material-reference, domain-sensitive, and admitted off-shell enrichment tests separately. Do not use observer, record, Fisher, clock, comparison, or holonomy data derived only from the common reduced datum to claim a distinguished lift or master metric. Execute model-specific SYM-4 applications: compute J_joint, Z(J_joint), and J_joint', observer-readable and recoverable sector information, sector-resolved Fisher rates, G18 clock class, and comparison transports in explicit ETP models. Execute the eight SYM-5 falsification benchmarks, including symmetry-exchanged candidates, null complementary output, readable-but-unrecoverable records, recoverable-but-sector-blind access, incompatible redundant records, noiseless multiplicity candidates, and sector-swapping recovery. Execute the established physical effect-system construction on the repeated-collision output net, including restricted detectors, null outputs, energy and superselection constraints, approximate resolution, and inclusion compatibility. Treat broader output-net settings as extensions of the completed bounded interaction-relative step. Phase C4 synchronized SYM-5 work program: reconstruct model-level robustness theorems with explicit metrics and perturbation classes; define a complete resource theory before cost or selection claims; re-audit the C17.3F benchmark suite after its gate dependencies are proved; define the physical composition datum and test correlated, shared-environment, feedback, boundary, memory, resource-conflict, and postselection countermodels before any closure theorem is restored. Construct one integrated finite-dimensional ETP execution in which the admissible observer family, observer-sector incidence, accessible Fisher geometry, G18 clock class, and comparison transports are calculated together. Use SYM-6 categorical or Tannaka escalation only if the reconstructed group action and joint algebra fail to capture independently supplied physical sector-category, intertwiner, composition, duality, and faithful-fiber-functor data. Extend stable-readable selection to recoverable and approximate algebras with perturbation control. Extend observer selection and factorization to normal infinite-dimensional type-II/type-III settings and edge modes. - Phase D3 nonfinite program: formulate normal-algebraic replacements for finite matrix decompositions; declare preduals, normal channels, conditional expectations, domains and cores, direct-integral sectors, and convergence modes; then test Type II/III observer access, unbounded recovery, noncompact symmetry, continuum output nets, and process tensors without importing finite-dimensional compactness or attainment arguments. Compute accessible Fisher rates under physically restricted detectors and auxiliary autonomous clock channels. Execute the parameter-independent comparison-kernel and degradability tests on multiple collision-record streams and test intrinsic stochastic loop holonomy. Quantify finite-clock timing error and degradation, finite-battery capacity and boundary risk, reusable-battery disturbance and correlation error, and the autonomous continuum limit for the energy-conserving collision family. Extend the completed composite-collision audit to process tensors, thermal or initially correlated fragments, interventions, and a non-Markovian continuum limit. Derive semiclassical and relativistic proper-time limits and reconcile companion ETP manuscripts. Integrate and bibliographically verify the external records, causality, process-tensor, channel-divisibility, network-discrimination, compatibility, and tomography literature before publication citation. Prove witness completeness only for a fixed candidate class with complete carriers, preparations, effects, context testers, adaptive strategies, resources, equivalence, and falsifiers; otherwise retain restricted-null or undetermined status. Test whether the RC cursor-detector-memory compatibility complex is flag-like, so pairwise compatibility determines all finite contexts, or genuinely higher order. Construct an ETP-specific pair with identical support, action, record, exchange, and resource signatures but different joint compatibility, or prove that the declared autonomous axioms forbid such a pair. S1/S2 post-Stage-8 frontier: test whether a complete ETP-specific physical model derives the response law, process carrier, compatibility and composition structure, record channel, and operational boundary without inserting any of them by convention. Preserve Stages 8H through 8R as completed only in their declared finite-dimensional theorem, countermodel, and model-execution scopes. M8-R1 publication gate: release the preserved Stage 8 scripts, machine-readable outputs, manifests, and checksums; verify every new reference at point of use; and retain noisy nearest-causal projection, minimal Markov order, minimal bond dimension, process-valid CMI recovery, intrinsic S1/S2 derivation, and nonfinite extension as open. - M9 synchronization completed: Stage 8S-8Z is integrated as a unified finite audit block, and Stage 8 is closed programmatically within its declared scopes. - Stage 9 priority: pursue intrinsic S1 candidate-record and S2 operational-boundary derivation from complete physical structure without importing event nets, response laws, controller semantics, calibration anchors, tester classes, representative choices, or observer cuts by convention. - Preserve physical-anchor selection, thermodynamic orientation, first-event selection, device-independent certification, and nonfinite extension as open unless separately proved. XV. CLAIM-STATUS LEDGER XVI. THEOREM-DEPENDENCY MAP AND SCOPE OF SYNTHESIS Narrative function. This dependency map is the logical spine of the theory. It shows which established structures license the next temporal layer and prevents conclusions from being imported before their hypotheses are met. The manuscript is a layered emergence framework supported by complementary benchmarks. Each implication below has explicit hypotheses, so the theory identifies exactly when one established structure licenses the next temporal layer. The integrated-model frontier is recorded in the scope matrix rather than used to interrupt the constructive narrative. XVI.A. CROSS-PROGRAM FAMILY-PRESERVATION AND ANTI-AGGREGATION AUDIT Methodological status. This subsection is a cross-program synthesis, not a new equivalence theorem. The minimal-ground frontier is order theoretic, whereas the causal-locality diagnostics are norms of a measurable sector family. They instantiate the same audit rule under different mathematical realizations: a reduction with weaker information content than the quantifier of the claim cannot certify the stronger claim. For a declared measurable sector family, let c_x(t)=||[A_x(t),B_x]||, C_infinity(t)=ess sup_x c_x(t), C_1(t)=integral c_x(t) dmu(x), and C_bar(t)=||integral [A_x(t),B_x] dmu(x)||. C_infinity is the worst-positive-measure diagnostic; C_1 is the sector-weighted mean of the norms; and C_bar is cancellation sensitive because averaging occurs before the norm. These quantities answer different questions and are not interchangeable. Rare-sector illustration. In the finite chain-ring-star diagnostic, chain-like and ring-like sectors carry 0.49 and 0.50 of the declared weight while a faster star-like sector carries 0.01. C_1 records the star contribution only in proportion to that weight, whereas C_infinity registers it because it remains a positive-measure sector. This does not assign probabilities to minimal grounds. It provides a structural warning: one admitted but unreported boundary condition, sector, bridge component, or incomparable minimal ancestor can control a downstream emergence claim even when it disappears from a typical, weighted, or compressed narrative. Audit rule. Averaging is not itself the error. The error is applying an aggregate whose information order is weaker than the claim quantifier and then presenting that aggregate as if it preserved the complete family. Any universal causal, emergence, uniqueness, or provenance-completeness statement must therefore survive the strongest admitted family-level witness relevant to its scope. constraint data -> (H_phys,A_phys) -> operational observer algebra -> accessible Fisher geometry -> local duration local calibrated clocks -> process-tensor operational history structure -> comparison protocols -> global parallel calibration or comparison holonomy recoverable records + objective memory + reservoir assumptions -> local arrow statements The compact SU(2) and noncompact translation models test physical-sector construction. Gaussian and unitary models test constant-speed clocks. The dephasing model tests a nonuniform clock and an integrable rank-changing endpoint. The Gaussian M3 model tests comparison holonomy. The collision model tests explicit record dynamics, recovery, entropy production, and a continuum limit. The Version-17 staged observer program tests the exact stable-readable core relative to a declared record protocol. These examples are mutually compatible but are not claimed to be one jointly realized physical system. TABLE XVI.1. ESTABLISHED LAYERS AND CONSOLIDATED EXTENSION FRONTIER XVII. RESTORED OBSERVER-ALGEBRA STATEMENTS AND SELECTOR SYNTHESIS Role in emergence. These restored statements ensure that temporal accessibility is representation stable: the observer state is an algebraic restriction, access is organized by exact, recoverable, and operational classes, and Fisher information respects that hierarchy. A. Complete factorization formulas H_phys ~= direct-sum_z (H_O,z tensor H_Oc,z) A_O ~= direct-sum_z (B(H_O,z) tensor I_Oc,z) A_O' ~= direct-sum_z (I_O,z tensor B(H_Oc,z)) Z_O ~= direct-sum_z C I_z The sector label z is central. A tensor factor for O exists canonically only within a central sector and only up to local unitaries, multiplicity-space relabeling, and permutations of isomorphic sectors. The representation-independent observer state is the restriction omega|A_O; partial trace is a derived coordinate realization. B. Smallest-generated versus largest-admissible reconciliation Corollary H12.1. Let A_O^exact be the unique largest C*-subalgebra satisfying exact stability and readability. Choose any unital self-adjoint generating set S_O^exact with W*(S_O^exact)=A_O^exact. Theorem H1 returns the unique smallest von Neumann algebra containing S_O^exact, namely A_O^exact itself. Thus maximality refers to satisfaction of physical gates, while minimality refers to algebraic generation from a chosen presentation; the two statements are compatible. Proof. H12 gives maximality among algebras satisfying the gates. H1 applied to S_O^exact gives the smallest algebra containing that generating family. By definition that generated algebra equals A_O^exact. QED. C. Selector datum with closure rule D_O=(A_phys,Omega,{Phi_a,t^dagger},E_O^dagger,Eff_O,Cl_O) Cl_O is the operational closure rule specifying which compositions, repeatable experiments, adjoints, and classical postprocessings are physically admitted. The selector is intrinsic only relative to the complete datum D_O, including Cl_O. D. Access hierarchy A_O^exact subseteq A_O^rec subseteq A_O^op A_O^exact is the fixed multiplicative, directly readable core proved in Version 17. A_O^rec contains information restored by admitted recovery channels. A_O^op is the full algebra represented by all admitted finite-time operational protocols. Equality among these algebras requires a separate theorem and is not assumed. E. Fisher hierarchy and monotonicity F_O^exact <= F_O^recoverable <= F_O^op Proposition H18.1. If Meas_O^exact subseteq Meas_O^recoverable subseteq Meas_O^op, then the corresponding accessible classical Fisher suprema satisfy the displayed inequalities pointwise on every regular parameter interval. Proof. Each quantity is a supremum over a nested feasible set. QED. The clock task must state which measurement class it uses. Exact record stability is not synonymous with temporal identifiability: a perfectly preserved pointer value may contain no continuing information about the elapsed dynamical coordinate. XVIII. RESTORED ENDPOINT-REGULARIZED FISHER DURATION Role in emergence. Relational duration remains meaningful at rank-changing boundaries exactly when accessible Fisher speed is integrable. This gives the emergent temporal metric a mathematically controlled completion. A. Operational endpoint convention Let I=(a,b) be a regular interval and F_C^acc:I->[0,infinity] be measurable. Define v_C(T)=(1/2)sqrt(F_C^acc(T)). At a rank-changing endpoint, the point value of SLD QFI need not equal its interior limit and is not the duration datum. Duration is the one-sided metric integral. For restricted observers this completion is formed from the declared accessible Fisher speed and need not equal unrestricted Bures completion. L_C(T)=L_C(T_*)+integral_(T_*)^T v_C(s) ds A finite endpoint a is duration-admissible from the right exactly when integral_a^(T_*) v_C(s) ds is finite. The definition depends only on the almost-everywhere equivalence class of F_C^acc. B. Endpoint-Regularized Fisher Duration Theorem Theorem G7. Let -infinity[0,infinity] be measurable, and v=(1/2)sqrt(F) in L1_loc. Then: (i) L is locally absolutely continuous and nondecreasing with L prime=v almost everywhere; (ii) L has a finite right limit at a iff v is integrable on (a,T_*), and the resulting extension is unique, continuous, and absolutely continuous; (iii) the analogous result holds at b, and if both conditions hold, L(w)-L(u)=(1/2) integral_u^w sqrt(F(s)) ds on the closed interval; (iv) if the one-sided integral diverges, no finite continuous extension preserving the integral law exists. Proof. Local absolute continuity and the almost-everywhere derivative follow from the fundamental theorem for Lebesgue integrals. Since v is nonnegative, L is nondecreasing. For T0, then v(a+x) is asymptotic to (sqrt(C)/2)x^(-alpha/2). Duration is finite exactly when alpha<2. For alpha<2, L(a+x)-L(a) is asymptotic to [sqrt(C)/(2-alpha)]x^(1-alpha/2); alpha=2 gives logarithmic divergence (sqrt(C)/2)|log x|; alpha>2 gives power-law divergence. Essentially bounded and vanishing endpoints are duration-admissible. For the dephasing model, F(T)=gamma/(2T)-gamma^2/2+O(T), so alpha=1 and L(T) is asymptotic to sqrt(gamma T/2). The endpoint is duration-admissible despite divergent interior SLD speed. XIX. RESTORED MARGOLUS-LEVITIN AUDIT Role in emergence. Temporal distinguishability and energetic resources are complementary rather than interchangeable. The audit clarifies which resource controls Fisher speed and which constrains orthogonalization. A. Scope The Margolus-Levitin theorem is used only for autonomous unitary evolution generated by a time-independent Hamiltonian bounded below. With E0=inf spectrum(H) and Ebar=, orthogonalization obeys T_perp>=pi hbar/(2Ebar). It is not applied to the driven collision channel or arbitrary open dynamics without a separately proved open-system speed limit. B. Metric-Resource Separation Theorem Theorem G8. For a pure autonomous unitary clock with finite Ebar and Delta H: (i) dL/dT=Delta H/hbar; (ii) T_perp>=pi hbar/(2Ebar); (iii) neither Ebar determines Fisher speed nor Delta H determines the ML bound. If the orbit reaches orthogonality, L(T_perp)>=pi/2. Simultaneous equality is a special saturation condition. Proof. The pure-state QFI is 4(Delta H)^2/hbar^2. For energies 0 and epsilon and state sqrt(1-p)|0>+sqrt(p)|1>, Ebar=p epsilon and Delta H=epsilon sqrt(p(1-p)). At fixed Delta H=D, epsilon=D/sqrt(p(1-p)) gives Ebar=D sqrt(p/(1-p)); at fixed Ebar=M, epsilon=M/p gives Delta H=M sqrt((1-p)/p). Hence neither moment determines the other. Orthogonal pure states have Bures angle pi/2, so any Fisher path joining them has length at least pi/2. QED. Balanced saturation. For p=1/2, Ebar=Delta H=epsilon/2, T_perp=pi hbar/epsilon, the ML bound is saturated, and L(T_perp)=pi/2. At fixed Ebar=1, p=0.20,0.50,0.80 give Delta H=2.0,1.0,0.5. Shifting H by cI leaves both Ebar and Delta H invariant. These are resource-separation tests, not universal open-system speed claims. XX. RESTORED GAUSSIAN M3 COMPARISON-HOLONOMY MODEL Role in emergence. This model provides a positive demonstration that individually valid local clocks can possess nontrivial comparison holonomy, making global temporal structure an empirical relational question rather than an assumed background. |psi_i(lambda)>=exp(-i s_i lambda P_i/hbar)|0_sigma_i> p_i(q|lambda)=[1/sqrt(2 pi sigma_i^2)] exp[-(q-s_i lambda)^2/(2 sigma_i^2)] Choose s=(1,1.25,0.8) and sigma=(0.5,0.625,0.4). Then s_i/sigma_i=2. Position measurement has classical Fisher information s_i^2/sigma_i^2, equal to the pure-state translation QFI, so F_i^acc=4 and L_i(lambda)=lambda. q_j=(s_j/s_i)q_i+s_j delta_ji, Gamma_ji(t)=t+delta_ji With delta_21=0.30, delta_32=-0.10, delta_13=0.25, the direct 1-to-3 map is t-0.25 and the relay through 2 is t+0.20. Their discrepancy is 0.45. H_ell=Gamma_13 Gamma_32 Gamma_21=[[1,0.45],[0,1]], H_ell(t)=t+0.45 Thus A_ell=1 and B_ell=0.45 !=0. Under h(t)=alpha t+beta, B_ell changes to alpha B_ell, so nonvanishing is invariant. Numerical quadrature gives F_i=4 at lambda=-0.4,0,0.7; alpha=2 and 0.7 give B prime=0.90 and 0.315. The offsets are independent comparison data, not derived from the reduced clock states or collision dynamics. XXI. RESTORED REPEATED-INTERACTION EXECUTION Role in emergence. The restored execution supplies a concrete bridge from microscopic interaction to records, recoverability, entropy production, and effective open dynamics. A. Microscopic architecture and recovery tau_F=exp(-beta H_F)/Z_F, H_F=epsilon|1><1|, U_theta=P_0 tensor I+P_1 tensor R_theta Phi_theta(rho_S)=Tr_F[U_theta(rho_S tensor tau_F)U_theta^dagger] Fresh fragments give rho_S(n)=Phi_theta^n[rho_S(0)] and retained memory M_n=F_1...F_n. Exact recovery is Tr_(F_n)[rho_(X M_n)]=rho_(X M_(n-1)), hence I(X:M_n)>=I(X:M_(n-1)). B. Strong record regime At theta=pi/2, p=0.3, beta epsilon=4, r=0.9820137900, H(X)=0.8812908992 bits, and I_acc(X:F_1)=0.7599291097 bits. One fragment carries 86.2291 percent of source information. For threshold delta>=0.137709, each retained fragment qualifies individually. The mutual-information and memory-entropy sequences reported in F2 are retained exactly. C. Thermodynamic identity Sigma=Delta S_S-beta Q_S=I(S:F)+D(rho_F prime||tau_F) Strong collision: Q_S=-0.2892082740 epsilon, Sigma=1.1568330961 nats, I(S:F)=0.5267427198 nats, and D=0.6300903763 nats. Weak tested collision: Delta S_S=0.0558485117 nats, Q_S=-0.0190248447 epsilon, Sigma=0.1319478907 nats. The interaction is externally driven, so the identity is an entropy balance under the declared heat convention, not an autonomous undriven thermalization theorem; implementation work remains unmodeled. D. Weak channel and continuum limit rho_01 prime=cos(theta)rho_01, theta^2=2 gamma Delta T Phi_(Delta T)=I+Delta T L+O(Delta T^2), L(rho)=(gamma/2)(sigma_z rho sigma_z-rho) At fixed T=n Delta T, Phi_(Delta T)^n approaches exp(TL) and rho_01(T)=exp(-gamma T)rho_01(0). For gamma=0.7 and T=1.3, absolute coherence errors are 8.783e-3, 4.331e-3, 1.719e-3, 8.570e-4, and 4.279e-4 for Delta T=0.1,0.05,0.02,0.01,0.005. E. Exact-core selector in the collision channel For the reduced dephasing channel with lambda=cos(theta), the pointer algebra W*(P_0,P_1) is fixed and lies in the multiplicative domain for every |lambda|<1, including the strong-record point lambda=0. At lambda=1 the channel is identity and the full matrix algebra is fixed. At lambda=-1 the channel is unitary conjugation and the one-step fixed algebra is diagonal, while the full map is multiplicative. The positive selector benchmark therefore uses |cos(theta)|<1, not 0<|cos(theta)|<1. This exact core does not by itself estimate elapsed T. The outgoing-fragment or retained-memory Fisher rate for T remains a separate calculation. F. Dual-output preservation and record-recovery audit The Version-54 execution applies the exact recovery theorem to both outputs of the two-fragment isometry. For the reduced final-system channel, the recoverable-algebra dimensions in aligned strong, aligned weak, rotated strong, and rotated weak regimes are 2, 2, 1, and 1. For the complementary retained-environment channel they are 2, 1, 2, and 1. The environmental strong-record algebra remains nontrivial after the intermediate system rotation because the orthogonal record already deposited in E1 remains part of the retained carrier. The weak environmental outputs are nonorthogonal and therefore fail nontrivial zero-error recovery, while still permitting a later approximate-information analysis. The complete matrices, spectra, residuals, convention audit, and proof boundary appear in H22.6A. XXII. AUTONOMOUS RECORD-FORMATION AND ENERGY-CONSERVING WORK STORAGE Role in emergence. Autonomous activation and explicit energy conservation show how record-forming temporal order can be implemented without an externally scheduled interaction, while keeping clock quality, work quality, and direction as independently audited properties. The driven collision and the autonomous dilation are complementary benchmarks. The autonomous construction adds explicit work storage, exact total-energy conservation on a declared supported sector, retained-memory recoverability, and an inclusive entropy-production identity. Together they show how interaction order, records, energetic accounting, clock adequacy, and temporal direction can be tested as linked but distinct layers. ARF0. Autonomous record-formation datum An autonomous record-formation datum is A_RF=(S,{F_n},W,C,H_tot,Omega,R,M), where S is the source, F_n are record fragments, W is a work store, C is an internal clock or transport degree of freedom, H_tot is time independent, Omega is the admitted state family, R=W*({P_x}) is a nontrivial pointer algebra, and M is the admitted fragment measurement class. The datum is admissible only when interaction ordering requires no externally imposed time-dependent Hamiltonian, all endpoint energy terms are declared, and the work-quality convention for W is stated. ARF1. Exact Energy-Conserving Record-Collision Realization Theorem Theorem ARF1. Let {P_x} be a finite pointer decomposition on S with [P_x,H_S]=0. Let F and W have finite-dimensional supported energy sectors. Suppose every fragment transition used by the desired controlled record interaction has a battery transition with the opposite energy gap. Then there exists a unitary U_ec on S tensor F tensor W, block diagonal in total bare energy, such that [U_ec,H_S+H_F+H_W]=0, and, on the declared supported input sector, the conditional fragment transformation implements the record interaction while W absorbs the compensating energy change. Proof. Decompose H_S+H_F+H_W into total-energy eigenspaces. The gap-matching hypothesis places each desired fragment transition and its compensating battery transition in one common energy eigenspace. Define the required controlled unitary rotation inside each matched block and the identity on unmatched blocks. Their orthogonal direct sum is unitary and preserves every total-energy eigenspace, hence commutes with the total bare Hamiltonian. Because the pointer projectors commute with H_S, pointer control introduces no source-energy mismatch. QED. Claim boundary. Exactness is asserted on the declared supported sector. The theorem does not imply indefinite reuse, an exact realization of every non-energy-conserving reduced unitary with a finite normalizable battery, or derivation of the interaction from bare ETP potentiality. ARF1.1. Finite qubit-ladder realization H_F=epsilon |1><1|, H_W=epsilon sum_(m=0)^M m |m><0,m+1|+|0,m+1><1,m|, H_RF=g P_1 tensor sum_(m=0)^(M-1) G_m. Every G_m acts inside a fixed total-energy sector, so [H_RF,H_S+H_F+H_W]=0 and U_ec(theta)=exp(-i theta H_RF/g) is exactly energy conserving. A finite horizon and an interior battery-support condition prevent boundary failure for the admitted collision histories. ARF2. Clock-Activated Autonomous Collision Theorem Theorem ARF2. Let E=SFW be the energy-conserving engine of ARF1. Suppose an internal clock C and a localized interaction Omega define one time-independent Hamiltonian H_aut=H_E+H_C+Omega. If the clock trajectory encounters the declared interaction region during the operational interval and the engine interaction commutes with H_E at every clock position, then the induced engine scattering transformation commutes with H_E and implements the declared collision without an externally time-dependent Hamiltonian. Proof. The complete evolution is generated by the single time-independent H_aut. Interaction activation is supplied by the clock degree of freedom entering the interaction region, rather than by an external schedule. Pointwise commutation with H_E makes the induced time-ordered engine transformation energy preserving. Under the declared scattering or endpoint conditions, that transformation is U_ec. QED. Claim boundary. This is an ideal-clock conditional theorem. Finite-clock localization, backreaction, normalization, timing error, and degradation require a separate quantitative analysis. The internal clock coordinate is not automatically operational duration; G1 still requires constant nonzero accessible Fisher speed for that identification. Ledger cross-reference. Finite-clock localization error, backreaction, normalization, timing uncertainty, and degradation are tracked under Autonomous collision activation in the Claim-Status Ledgers and remain designated open S5 tasks; they are not treated as absent or negligible in ARF2. ARF3. Autonomous Record-Distinguishability Theorem Theorem ARF3. Let the binary source pointer be {P_0,P_1}. Let a fragment begin in tau_F=r|0><0|+(1-r)|1><1|. On an interior battery sector, let the P_0 branch act trivially and the P_1 branch perform a complete resonant exchange that reverses the fragment populations while transferring the compensating energy to W. Then rho_(F|0)=tau_F, rho_(F|1)=(1-r)|0><0|+r|1><1|, and (1/2)||rho_(F|0)-rho_(F|1)||_1=|2r-1|. Hence the autonomous collision creates a nontrivial readable fragment record exactly when r!=1/2. Proof. The P_0 branch preserves tau_F. The complete resonant exchange reverses its two populations in the P_1 branch. The difference of the diagonal conditional states has eigenvalues plus or minus (2r-1), giving the trace-distance formula. QED. Corollary ARF3.1. For the admitted ground-state effect Q_0, E_F^dagger(Q_0)=rP_0+(1-r)P_1. If r!=1/2, this effect and the identity generate W*(P_0,P_1). If r=1/2, the output is preparation independent and the record selector correctly returns no nontrivial pointer information. RC-11 integrated positive-witness audit. The frozen autonomous binary collision model supplies one fragment. For r != 1/2, the ARF3 distinguishability witness passes; for r = 1/2, it fails. On the frozen one-fragment datum, the redundancy gate fails. Persistence, candidate-specific integrated recovery, common-record compatibility, and covariance remain untyped, while robustness is supported only by the formula-level separation margin |2r-1|. Accordingly, this frozen model is not S1-qualified. This is a model-scoped integrated-witness failure, not a universal impossibility theorem and not a proof that the fully qualified S1 family is empty. ARF3, its proof, and its formula are unchanged. RC-12 successor qualification. The RC-11 one-fragment negative result remains correct for its frozen datum but is superseded for the enriched N=3 model by the exact-point three-fragment theorem package in H22.6B. That package does not modify ARF3. It establishes model-relative common-record recovery and redundancy at the strong, zero-drift point, subject to conditional persistence, scoped S_3 covariance, failure of exact perturbative robustness, and continuing intrinsic-provenance limits. ARF4. Autonomous Retained-Memory Recoverability Theorem Theorem ARF4. Let M_n=F_1...F_n retain all outgoing fragments from the autonomous collision chain. Then rho_(X M_n)=Tr_(F_(n+1))[rho_(X M_(n+1))] and I(X:M_(n+1))>=I(X:M_n). Proof. The first identity is the defining marginal relation for M_(n+1)=M_n F_(n+1). Partial trace is CPTP, so data processing for quantum mutual information gives the inequality. QED. Claim boundary. The result concerns retained total memory. It does not imply monotonic information in every individual fragment, independent fragments, unlimited growth in a finite recurrent model, or absence of battery-mediated backflow in reduced subsystems. ARF5. Autonomous Collision Energy-Ledger Theorem Theorem ARF5. If [U,H_S+H_F+H_W]=0 and the interaction energy is zero at the accounting endpoints, then Delta E_S+Delta E_F+Delta E_W=0. With Q_S=-Delta E_F and W_in=-Delta E_W, Delta E_S=Q_S+W_in. Proof. Commutation preserves the expectation of the total bare Hamiltonian. Substitution of the heat and work-transfer conventions gives the first-law identity. QED. Endpoint qualification. If interaction energy does not vanish at the endpoints, the exact ledger includes Delta E_int. It may not be silently discarded. ARF6. Inclusive Autonomous Entropy-Production Identity Theorem ARF6. Let the initial collision state be rho_S tensor tau_F tensor omega_W, with tau_F=exp(-beta H_F)/Z_F, and let the final state be rho_prime_SFW=U(rho_S tensor tau_F tensor omega_W)U^dagger. Define I_prime_tot=D(rho_prime_SFW || rho_prime_S tensor rho_prime_F tensor rho_prime_W). Then, for Q_S=-Delta E_F, Delta S_S+Delta S_W-beta Q_S=I_prime_tot+D(rho_prime_F||tau_F)>=0. Proof. Product preparation and global unitarity imply I_prime_tot=Delta S_S+Delta S_F+Delta S_W. Thermal reference algebra gives D(rho_prime_F||tau_F)=-Delta S_F+beta Delta E_F. Adding and using Q_S=-Delta E_F proves the identity. QED. Corollary ARF6.1. If Delta S_W=0, the familiar form Delta S_S-beta Q_S>=0 is recovered. If Delta S_W>0, the battery has absorbed entropy as well as energy, and -Delta E_W must not automatically be described as pure work. The battery free-energy change Delta F_W^(beta)=Delta E_W-beta^(-1)Delta S_W supplies a work-quality diagnostic. Reusability additionally requires bounds on battery disturbance and final battery-engine correlations. ARF7. Finite-Capacity and Exactness Boundaries A finite battery ladder cannot support every unbounded collision history indefinitely; Version 18 therefore asserts exactness only over a declared finite horizon and supported energy window. Exact global energy conservation does not imply exact reproduction of every reduced driven unitary for a finite normalizable battery. General reduced-unitary simulation may require an ideal translation-covariant weight or an explicit approximation bound. Energy conservation does not imply a readable record. ARF3 may return the unbiased null result. Record formation does not imply entropy-neutral work storage. ARF6 requires battery entropy and correlations to be retained in the ledger. Autonomous scheduling does not itself define physical duration. The accessible-Fisher clock correspondence remains an independent gate. ARF8. Autonomous Record-Formation and Inclusive Local-Arrow Theorem Theorem ARF8. Consider a finite sequence of internally ordered collisions satisfying ARF1-ARF6. Assume pointer-energy compatibility, exact total-energy conservation, an internal time-independent activation mechanism, initially thermal and uncorrelated incoming fragments, nontrivial fragment distinguishability, retained memory, and a consistent endpoint heat-work convention. Then the model simultaneously realizes autonomous collision ordering, explicit work storage, readable record formation, exact backward memory recovery, monotonic retained-memory information, and nonnegative inclusive entropy production: I(X:M_(n+1))>=I(X:M_n) and Delta S_S+Delta S_W-beta Q_S=I_prime_tot+D(rho_prime_F||tau_F)>=0. Proof. Apply ARF2 for autonomy, ARF1 and ARF5 for exact energy conservation and first-law accounting, ARF3 for nontrivial record formation, ARF4 for retained-memory monotonicity, and ARF6 for the inclusive entropy-production identity. The hypotheses remain separate and no conclusion is substituted for another. QED. Claim boundary. ARF8 proves a conditional finite-step local-arrow architecture. It does not prove a universal or cosmological arrow, derive H_aut from bare potentiality, establish indefinite operation with a finite battery, make the ideal clock costless, or show that every subsystem entropy is monotonic. XXIII. COMPLETE CLAIM-STATUS LEDGER The ledger preserves truth-first accountability. It records exactly which links in the emergence chain are proved, conditional, model-executed, falsified, or still open, without interrupting the main explanatory arc. Cumulative disposition after Phase E8. No principal theorem is withdrawn solely by Phases E2 through E8. The corrected manuscript narrows summary and status language to proved theorem scopes, records the remaining open extensions, preserves historical text only as non-governing genealogy under the current-authority rule, and applies the E8 definition-to-use, symbol-scope, and escalation rules to every current scientific assertion. Both current claim-status ledgers contain separate Phase E2 through E8 entries and the cumulative Version 55 E2-E8 status. E5 checkpoint. Quantifier, modality, and theorem-strength audit: CONDITIONAL PASS. Mandatory repairs applied in this revision include candidate-class and equivalence indexing for uniqueness and maximality; operational limits on biconditionals; normalization of unsupported 'generic' language; distinction among exact stability, persistence, robustness, and approximate recovery; scoped meanings of canonicality and completion; and preservation of almost-everywhere, selected-output, support, and regularity qualifiers. E4 checkpoint. Cross-reference, dependency, and proof-import audit: CONDITIONAL PASS. Mandatory repairs applied in this revision include the H20.5 channel-correction versus observer-readable distinction; complete C14.3 premise carriage; graph-level holonomy transport; current SYM-4-to-SYM-5 roadmap synchronization; separation of C17.3B abstract completion from C17.3C model execution; Table XVI.1 traceability; and a firewall against theorem-style imports from reclassified C17.3D-G programs. PHASE E4-E5 CONSISTENCY-AUDIT LEDGER E7 historical claim boundary. The E7 edit changed notation and cross-reference clarity, not theorem content. Historical formulas remain genealogy. Phase E8 subsequently completed the definition-to-use and symbol-scope audit with controlled binary corrections. E8 claim boundary. The audit and binary correction synchronize current definitions, uses, symbol categories, and theorem-local channel pictures without enlarging theorem claims. Historical formulas remain genealogy. A replacement is authoritative only where the intended mathematical category is recoverable from the local definition, theorem premises, proof, and downstream uses; otherwise the location is escalated to substantive theorem-scope review. Applied distinctions: D_anchor versus A_phys; J_phys and master J or J_M versus J_joint; A_out(D) and S_out(D) versus fragment labels; Instr(D) versus Eff_phys(D); and direction-bearing K_{j<-i}, D_{j<-i}, and Gamma_{j<-i}. E7 binary checkpoint. Controlled E3 notation-normalization execution: CONDITIONAL PASS WITH CORRECTIONS APPLIED. E8 binary result. Definition-to-use and symbol-scope audit: CONDITIONAL PASS WITH CORRECTIONS APPLIED. No principal theorem is withdrawn solely by E8. Remaining manuscript-correction gates continue after this binary checkpoint. E8 escalation boundary. A symbol replacement is authoritative only where its intended mathematical category is recoverable from the local definition, theorem premises, proof, and downstream uses. Any future conflict among those sources must be treated as a substantive theorem-scope defect rather than silently normalized. E8 authority rule. OT denotes the observer-temporal constructor; T remains a temporal parameter only in a declared clock model. Rel_op denotes operational relations; record channels retain direction- or output-bearing R subscripts. Q_chg denotes charge data, Q_rec denotes recovery maps, and ResTol denotes resource and tolerance data. Meas denotes measurement classes. J_joint denotes the joint symmetry-observable algebra, while J and J_M remain master-constraint operators. Phase E8 disposition. CONDITIONAL PASS WITH CONTROLLED BINARY CORRECTIONS APPLIED. The edit resolves the confirmed current-scope notation collisions identified by the E8 audit, adds theorem-local channel-picture signatures in C16.3 and C17.3, and preserves theorem statements and claim ceilings except for correction of the C17.3 joint-algebra intersection target. PHASE E8 DEFINITION-TO-USE AND SYMBOL-SCOPE BINARY CHECKPOINT Truth boundary. No theorem is withdrawn or enlarged solely by this correction. The H9 selector datum records Eff_O as the admitted output-effect system, while Meas_O remains the distinct implementable measurement class in Fisher statements. Residual map-signature verification and final integrated certification remain separate gates. Result. CONDITIONAL PASS WITH CONTROLLED RESIDUAL-ALIAS CORRECTIONS APPLIED. The correction reconciles the C9.1 operational-relation datum; physical- and observer-algebra central projectors; quantum comparison-channel composition; measurement-class and effect-system notation; joint-algebra references; and theorem-local recovery-map aliases. POST-E8 RESIDUAL ALIAS BINARY CHECKPOINT Truth boundary. No theorem is withdrawn or enlarged beyond its declared map class by this correction. A later map may inherit a signature only from a uniquely named controlling definition with compatible carriers, picture, support, parameter convention, and exactness status. C15.7 scope boundary. The present theorem certifies only isometric gluing maps G_glue:H_comp -> H_glue. CPTP interface channels, partial isometries with smaller initial support, and algebraic quotient maps require separate theorems and are not imported by the isometric statement. Result. CONDITIONAL PASS WITH CONTROLLED MAP-SIGNATURE COMPLETIONS APPLIED. The revision installs the governing Schrödinger/Heisenberg dual template; types the C14 physical transport and C15 boundary isomorphism; resolves C15.7 to an isometric gluing class on a declared composite input carrier; types the C17 event-output embeddings and degrading channel; assembles the H9-H12 active-map block; and declares the H23 quantum-transducer and partial-trace carrier signatures. THEOREM-LOCAL MAP-SIGNATURE BINARY CHECKPOINT REFERENCES [1] D. N. Page and W. K. Wootters, Physical Review D 27, 2885 (1983). [2] S. L. Braunstein and C. M. Caves, Physical Review Letters 72, 3439 (1994). [3] D. Petz, Linear Algebra and its Applications 244, 81 (1996). [4] H. Spohn, Journal of Mathematical Physics 19, 1227 (1978). [5] C. Weedbrook et al., Reviews of Modern Physics 84, 621 (2012). [6] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (1982). [7] R. Bott and L. W. Tu, Differential Forms in Algebraic Topology (1982). [8] F. Ciccarello et al., Physics Reports 954, 1 (2022). [9] G. Lindblad, Communications in Mathematical Physics 48, 119 (1976). [10] E. Nelson, Annals of Mathematics 70, 572 (1959). [11] T. Thiemann, Classical and Quantum Gravity 23, 2249 (2006). [12] B. Dittrich and T. Thiemann, Classical and Quantum Gravity 23, 1025 (2006). [13] T. Nishiyama and Y. Hasegawa, Physical Review E 114, 014120 (2026). [14] L. P. Bettmann and J. Goold, arXiv:2409.06083. [15] N. Margolus and L. B. Levitin, Physica D 120, 188 (1998). [16] N. Hörnedal and O. Sönnerborn, Physical Review Research 5, 043234 (2023). [17] D. Safranek, Physical Review A 95, 052320 (2017). [18] L. Seveso et al., Journal of Physics A 53, 02LT01 (2020). [19] J. von Neumann, Mathematische Annalen 102, 370 (1930). [20] E. Knill, R. Laflamme, and L. Viola, Physical Review Letters 84, 2525 (2000). [21] D. Kribs et al., Quantum Information and Computation 6, 382 (2006). [22] H. Casini, M. Huerta, and J. A. Rosabal, Physical Review D 89, 085012 (2014). [23] S. Moudgalya and O. I. Motrunich, Physical Review X 12, 011050 (2022). [24] M.-D. Choi, Illinois Journal of Mathematics 18, 565 (1974). [25] V. Paulsen, Completely Bounded Maps and Operator Algebras (2002). [26] M. Rahaman, Multiplicative Properties of Quantum Channels, Journal of Physics A: Mathematical and Theoretical 50, 345302 (2017). [27] M.-D. Choi, N. Johnston, and D. W. Kribs, The multiplicative domain in quantum error correction, Journal of Physics A: Mathematical and Theoretical 42, 245303 (2009). [28] A. S. Holevo, On complementary channels and the additivity problem, Probability Theory and Applications 51, 133 (2006). [29] D. Kretschmann, D. Schlingemann, and R. F. Werner, The Information-Disturbance Tradeoff and the Continuity of Stinespring Representation, IEEE Transactions on Information Theory 54, 1708 (2008). [30] C. Beny, A. Kempf, and D. W. Kribs, Generalization of Quantum Error Correction via the Heisenberg Picture, Physical Review Letters 98, 100502 (2007). [31] C. Beny, A. Kempf, and D. W. Kribs, Quantum Error Correction of Observables, Physical Review A 76, 042303 (2007). [32] J. K. Korbicz, Roads to Objectivity: Quantum Darwinism, Spectrum Broadcast Structures, and Strong Quantum Darwinism, Quantum 5, 571 (2021). [33] D. Blackwell, Comparison of Experiments, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 93-102 (1951). [34] D. Blackwell, Equivalent Comparisons of Experiments, Annals of Mathematical Statistics 24, 265-272 (1953). [35] J. T. Chang and D. Pollard, Conditioning as Disintegration, Statistica Neerlandica 51, 287-317 (1997). [36] F. Buscemi, Degradable Channels, Less Noisy Channels, and Quantum Statistical Morphisms: An Equivalence Relation, Problems of Information Transmission 52, 201-213 (2016). [37] C. Duarte, L. Catani, and R. C. Drumond, Relating Compatibility and Divisibility of Quantum Channels, International Journal of Theoretical Physics 61, 169 (2022). [38] P. Strasberg, G. Schaller, T. Brandes, and M. Esposito, Quantum and Information Thermodynamics: A Unifying Framework Based on Repeated Interactions, Physical Review X 7, 021003 (2017). [39] A. S. L. Malabarba, A. J. Short, and P. Kammerlander, Clock-Driven Quantum Thermal Engines, New Journal of Physics 17, 045027 (2015). [40] S.-Y. Wang, Informational Work Storage in Quantum Thermodynamics, Quantum Reports 1, 37-49 (2019). Version 18 | Autonomous Record Formation and Energy-Conserving Work Storage [41] X.-M. Lu, X. Wang, and C. P. Sun, Quantum Fisher Information Flow and Non-Markovian Processes of Open Systems, Physical Review A 82, 042103 (2010). [42] H. Song, S. Luo, and Y. Hong, Quantum Non-Markovianity Based on the Fisher-Information Matrix, Physical Review A 91, 042110 (2015). [43] P. Abiuso, M. Scandi, D. De Santis, and J. Surace, Characterizing (Non-)Markovianity through Fisher Information, SciPost Physics 15, 014 (2023), with corrected manuscript arXiv:2204.04072v6. [44] X.-Y. Chen et al., Global Correlation and Local Information Flows in Controllable Non-Markovian Open Quantum Dynamics, npj Quantum Information 8, 22 (2022). [45] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Quantum Speed Limits in Open System Dynamics, Physical Review Letters 110, 050403 (2013). Version 19 | Clock Adequacy under Nonunitary and Non-Markovian Dynamics [46] F. Ciccarello, G. M. Palma, and V. Giovannetti, Collision-Model-Based Approach to Non-Markovian Quantum Dynamics, Physical Review A 87, 040103(R) (2013). [47] F. Ciccarello and V. Giovannetti, A Quantum Non-Markovian Collision Model: Incoherent Swap Case, Physica Scripta T153, 014010 (2013). [48] K. Beyer, K. Luoma, T. Lenz, and W. T. Strunz, Measured Composite Collision Models: Quantum Trajectory Purities and Channel Divisibility, Entropy 24, 715 (2022). [49] J. Keeling, E. M. Stoudenmire, M.-C. Banuls, and D. R. Reichman, Process Tensor Approaches to Non-Markovian Quantum Dynamics, Physical Review X 16, 020502 (2026). Version 20 | ETP-Specific Non-Markovian Collision-Clock Execution [50] B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer (2015). [51] S. Doplicher and J. E. Roberts, A New Duality Theory for Compact Groups, Inventiones Mathematicae 98, 157-218 (1989). [52] S. L. Woronowicz, Tannaka-Krein Duality for Compact Matrix Pseudogroups: Twisted SU(N) Groups, Inventiones Mathematicae 93, 35-76 (1988). Version 21 | Staged Physical-Datum Symmetry-Reconstruction Program [53] J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer (2013), closed subgroup and quotient Lie-group results. [54] G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press (1979). Version 22 | Finite-Dimensional Physical-Datum Symmetry Reconstruction and Nonuniqueness [55] T. Thiemann, Reduced Phase Space Quantization and Dirac Observables, Classical and Quantum Gravity 23, 1163-1180 (2006). [56] A. P. Balachandran et al., Superselection, Boundary Algebras, and Duality in Gauge Theories, Physical Review D 106, 025001 (2022). Version 23 | Gauge-versus-Physical Symmetry Classification [57] B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer (2015), one-parameter subgroups and derived representations. [58] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton University Press (1992). [59] T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press (2007), master-constraint and constrained-system framework. Version 24 | SYM-3 Generator Reconstruction and Master-Constraint Recoverability Program [60] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer (1987), derivations and automorphism groups. [61] G. K. Pedersen, Analysis Now, Springer (1989), finite-dimensional operator-algebra preliminaries and orthogonal projection. Version 25 | Effective Lie-Algebra Reconstruction and Master-Constraint Nonuniqueness [62] D. Giulini and D. Marolf, On the Generality of Refined Algebraic Quantization, Classical and Quantum Gravity 16, 2479-2488 (1999). [63] E. Martinez-Pascual, Constraint Rescaling in Refined Algebraic Quantisation: Momentum Constraint, Journal of Mathematical Physics 52, 122104 (2011). [64] G. Barnich and M. Grigoriev, Hamiltonian BRST and Batalin-Vilkovisky Formalisms for Second Quantization of Gauge Theories, Communications in Mathematical Physics 254, 581-601 (2005). [65] P. Xu, Momentum Maps and Morita Equivalence, Journal of Differential Geometry 67, 289-333 (2004). Version 26 | Gauge-Lift Equivalence and Constraint-Compatibility Classification Version 27 | Manuscript-Wide Synchronization of Finite-Dimensional SYM-3 Status Version 28 | Theory-Level Narrative Audit and Relational-Time Emergence Roadmap Version 29 | Symmetry-Nonuniqueness and Finite-Clock Clarification [66] T. Carette, J. Glowacki, and L. Loveridge, Operational Quantum Reference Frame Transformations, Quantum 9, 1680 (2025). [67] T. Thiemann, (Quantum) Reference Frames, Relational Observables, Gauge Reduction and Physical Interpretation, arXiv:2603.04072 (2026). [68] L. Chataignier, P. A. Hoehn, M. P. E. Lock, and F. M. Mele, Relational Dynamics with Periodic Clocks, New Journal of Physics 28, 034504 (2026). Version 30 | SYM-3C.1 Observer-Temporal Descent Invariance and Enriched Gauge-Lift Separation [69] W. Donnelly and L. Freidel, Local Subsystems in Gauge Theory and Gravity, Journal of High Energy Physics 09, 102 (2016). [70] A. Riello, Edge Modes without Edge Modes, arXiv:2104.10182 (2021). [71] A. Riello, Symplectic Reduction of Yang-Mills Theory with Boundaries: From Superselection Sectors to Edge Modes, and Back, SciPost Physics 10, 125 (2021). [72] L. Freidel, M. Geiller, and D. Pranzetti, Edge Modes of Gravity. Part I. Corner Potentials and Charges, Journal of High Energy Physics 11, 026 (2020). Version 31 | SYM-3C.2 Boundary-Charge, Edge-Presentation, and Interface Separation Version 31.2 | Formal Closure of the Bounded Finite-Dimensional SYM-3 Program [73] B. Simon, Representations of Finite and Compact Groups, American Mathematical Society (1996). [74] M. Takesaki, Theory of Operator Algebras I, Springer (2002), finite-dimensional von Neumann algebra and commutant structure. [75] J. Liu, J. Chen, X.-X. Jing, and X. Wang, Quantum Fisher Information and Symmetric Logarithmic Derivative via Anti-Commutators, Journal of Physics A: Mathematical and Theoretical 49, 275302 (2016). [76] L. Gao, H. Li, I. Marvian, and C. Rouzé, Sufficient Statistic and Recoverability via Quantum Fisher Information, Communications in Mathematical Physics 405, 173 (2024). [77] C. Ferrie, The Best Fisher Is Upstream: Data Processing Inequalities for Quantum Metrology, Quantum Information Processing 13, 1435-1446 (2014). Version 32-workaround | Implemented Symmetry Sectors, Joint Observable Decomposition, and Recoverable Sector Structure Version 33 | Complete SYM-4 Sector Architecture, Fisher Geometry, Clock Adequacy, and Comparison Transport Version 34 | Controlled Sanitation of the Complete SYM-4 Architecture [78] E. Witten, Algebras, Regions, and Observers, Proceedings of Symposia in Pure Mathematics 107, 247-276 (2024); arXiv:2303.02837. [79] M.-D. Choi and D. W. Kribs, A Method to Find Quantum Noiseless Subsystems, Physical Review Letters 96, 050501 (2006). [80] T. P. Le and A. Olaya-Castro, Strong Quantum Darwinism and Strong Independence Are Equivalent to Spectrum Broadcast Structure, Physical Review Letters 122, 010403 (2019). [81] J. K. Korbicz, Roads to Objectivity: Quantum Darwinism, Spectrum Broadcast Structures, and Strong Quantum Darwinism, Quantum 5, 571 (2021). Version 35 | Observer-Sector Selection and Accessibility Reconstruction Foundation [82] J. Barrett, R. Lorenz, and O. Oreshkov, Quantum Causal Models, arXiv:1906.10726 (2020). [83] C. Portmann, C. Matt, U. Maurer, R. Renner, and B. Tackmann, Causal Boxes: Quantum Information-Processing Systems Closed under Composition, IEEE Transactions on Information Theory 63, 3277-3305 (2017). [84] C. J. Fewster and R. Verch, Quantum Fields and Local Measurements, Communications in Mathematical Physics 378, 851-889 (2020). [85] F. A. Pollock et al., Non-Markovian Quantum Processes: Complete Framework and Efficient Characterization, Physical Review A 97, 012127 (2018). [90] J. Keeling, E. M. Stoudenmire, M.-C. Bañuls, and D. R. Reichman, Process Tensor Approaches to Non-Markovian Quantum Dynamics, Physical Review X 16, 020502 (2026). [89] O. Fawzi and R. Renner, Quantum Conditional Mutual Information and Approximate Markov Chains, Communications in Mathematical Physics 340, 575-611 (2015). [88] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Operational Markov Condition for Quantum Processes, Physical Review Letters 120, 040405 (2018). [87] P. Taranto, F. A. Pollock, S. Milz, M. Tomamichel, and K. Modi, Quantum Markov Order, Physical Review Letters 122, 140401 (2019). [86] F. A. Pollock et al., Non-Markovian Quantum Processes: Complete Framework and Efficient Characterization, Physical Review A 97, 012127 (2018). Version 36 | ETP-Grounded Physical Output Nets and Effect-System Reconstruction MANUSCRIPT SYNCHRONIZATION AUDIT AND VERSION 49 STATUS Version 49 synchronization objective This audit synchronizes the completed SYM theorem program, FIT compression program, and process-tensor extension program with the manuscript claim-status architecture. SYM status SYM-1 through SYM-4 finite-dimensional stages are treated as closed within their declared domains. Observer-sector selection, accessibility reconstruction, sector decomposition, commutant structure, recoverability, and operational boundary emergence remain synchronized with the theorem hierarchy. FIT status Phase D1 current-status correction. FIT-E and FIT-F provide a conditional recovery-representation and invariant-packaging architecture in their declared categories. They do not prove universal recovery-semantic completeness, sufficiency of equality of I_rec for every recovery-family equivalence, or exclusion of a third recovery determinant. FIT-G supplies conditional compression results for explicitly declared event-generated and collision-generated recovery classes. Broader non-Markovian, process-level, approximate, compositional, and nonfinite compression remain open. Process-tensor status PTE-A and PTE-B supply finite-dimensional process-tensor constructions and persistent-memory benchmarks within declared intervention classes. Enlarged-carrier Markovity, memory emptying, source-only causal-break insufficiency, and bounded-memory compression are model and instrument relative. Minimal Markov order, minimal bond dimension, continuum limits, and instrument-independent completion remain open. Open problems Remaining open questions include minimal reduced-source Markov order, minimal process-tensor bond dimension, continuum process extensions, broader non-Markovian compression, and extensions beyond the finite-dimensional declared collision architecture. Claim-boundary preservation No theorem herein is asserted beyond its declared domain. Markov-order minimality, infinite-memory claims, continuum recovery geometry, and unrestricted process reconstruction remain open. Conclusion update Phase E1 terminology rule. In current scientific assertions, “canonical” means invariant only up to the equivalence explicitly proved by the cited theorem, and “generally” or “necessarily” is read only under the theorem’s stated quantified domain and hypotheses. Historical uses remain genealogy and do not broaden current claim status. The ETP framework now contains synchronized finite-dimensional theorems, conditional reconstructions, model executions, counterexamples, and open programs spanning symmetry selection, recovery geometry, observer accessibility, collision-generated records, non-Markovian diagnostics, and intervention-relative process-tensor extensions. The governing claim status is supplied by the current ledgers and correction notices; no summary upgrades an open or conditional program. These results do not complete RSCP, universal recovery-content/recovery-geometry exhaustion, or basement-level physical provenance. VERSION 50 RECONCILIATION EXECUTION RECORD R3B historical status. This paragraph records the Version 50 synchronization view and is superseded by the Phase D1 current-status corrections where FIT or PTE completeness is concerned. Finite-dimensional process-tensor construction, enlarged-carrier Markovity, memory-emptying benchmarks, source-only causal-break analysis, and finite-memory compression remain model- and intervention-relative results. Thermal, correlated-input, continuum, minimal-bond-dimension, and minimal reduced-source Markov-order questions remain open. SYM synchronization. Bounded finite-dimensional SYM-1 through SYM-4 are treated as completed within declared scope. SYM-5 remains the active observer-accessibility execution stage and SYM-6 remains a conditional escalation frontier. Observer-reconstruction synchronization. S1A reconstruction and S1B.1A record-orbit no-selection are treated as completed. Recoverability completeness and universal RSCP closure remain open and are not promoted to theorem status. Phase D1 FIT synchronization. FIT-E and FIT-F retain conditional architecture and invariant-packaging status in their declared categories. FIT-G retains class-relative conditional compression. None supplies universal recovery-semantic completeness, universal minimality, or general lossless compression. Phase D1 process-tensor claim boundary. The manuscript does not claim minimal MPO bond dimension, finite universal source-only Markov order, infinite Markov order, instrument-independent Markov order, or continuum process-tensor closure. Every finite-memory statement is indexed by the declared process, intervention family, carrier, approximation convention, and model scope. R4 preservation audit summary. Existing theorem statements, scope declarations, conditional claims, falsification results, and open-problem boundaries were preserved. This reconciliation record supplements rather than replaces earlier theorem packages. VERSION HISTORY CLARIFICATION AND RELEASE IDENTIFICATION Historical release designation at the Version 51 stage: Version 51 Matrix Integration Addendum Edition. Historical references throughout this manuscript such as Version 18, Version 20, Version 34, Version 49, and Version 50 identify developmental milestones and synchronization passes. They are retained as part of the internal research chronology and should not be interpreted as conflicting current document versions. Version-line interpretation rule: references embedded in sections, theorem histories, reference lists, synchronization records, and research milestones denote the version at which a result first entered the manuscript. The governing current release is Version 55 as identified by the author-designated filename and current front matter. Earlier release sections are preserved as historical genealogy and do not independently define the current release. Cleanup result: historical version markers preserved for traceability; current release identity clarified; no theorem statements, claim boundaries, synchronization records, or reference chronology removed. Historical-status notice. The following table records the program status at the Version 52 stage. It is preserved for genealogy and does not by itself govern the current Version 55 claim status. Phase D1 current-status notice. The Version 52 table is retained for genealogy. Its “Complete” labels do not govern Version 55. Current statuses are: FIT-E conditional architecture with source/proof linkage required; FIT-F conditional invariant packaging with universal completeness open; FIT-G conditional class-relative compression with the broader frontier open; PTE-A finite-dimensional construction in a declared intervention scope; and PTE-B an executed persistent-memory benchmark with minimality and continuum questions open. Phase D1 cross-program dependency rule. FIT/PTE results may support model-specific recovery and memory claims, but they do not close the Recovery Semantic Closure Principle, universal content-geometry exhaustion, universal uniqueness or basement minimality of the recovery representation, or the basement-level physical provenance problem. VERSION 52 TABLE-AWARE RECONCILIATION PASS Table-aware reconciliation completed for identified SYM roadmap inconsistency and matrix integration tracking. Historical-status notice. This section records the Version 53 release-candidate state and is superseded only where a later Version 55 theorem, audit, or restricted status determination addresses the same claim under the same scope. VERSION 53 FINAL RELEASE CANDIDATE AUDIT Audit result: dependency-map process-tensor layer inserted; Table XVI.1 process-tensor frontier classification updated; open-and-memory-bearing clocks frontier updated; prior SYM-5 reconciliation retained. VERSION 54 DUAL-OUTPUT COLLISION RECOVERY INTEGRATION This release integrates the complete dual-output Kraus-product commutant execution into H22.6A and synchronizes the abstract, SYM-5 channel orientation, recovery-adequacy separation, ETP synthesis, required-work program, claim-status ledgers, dependency frontier, and restored repeated-interaction execution. No previously established theorem is weakened. The new result is bounded to the declared finite-dimensional two-fragment model and distinguishes exact recovery through the reduced final system from exact recovery through the complementary retained environment. Later BNP correction phases preserved the underlying research questions and valid scoped results but reclassified or narrowed C17.3D, C17.3E, C17.3F, C17.3G, FIT/PTE, approximate-recovery, and nonfinite claims where the stated theorem breadth exceeded the available proof. The earlier release statement is retained as historical evidence and does not govern the corrected Version 55 claim status. Preservation audit. The edit was performed on the supplied DOCX binary. Existing text, tables, theorem statements, references, and historical version markers were retained; additions were localized to the relevant recovery, collision, synthesis, status, and release sections. PHASE E9 CONTROLLED FORMATTING REPAIR AND VISUAL CERTIFICATION Scope. Layout and style normalization only. The Phase E7 registry was restored as a titled set of body-text entries; the Version 55 narrative and Abstract headings were separated; the stale Version 36 footer was normalized to the governing Version 55 release identity. No theorem, proof, formula, benchmark, identifier, claim status, or scientific wording was changed by this phase. Certification disposition. PASS after full rerender and page-image review. This checkpoint records formatting certification only and does not enlarge any mathematical or physical claim. PHASE E10 ACCESSIBILITY AUDIT AND DISPOSITION Automated structural audit. The high-return OOXML checks covered heading-level continuity, image alternative text, table-header semantics, and hyperlink labeling. The initial audit returned no high-severity findings and identified missing header-row semantics on manuscript tables. Every table was manually confirmed to begin with a genuine lexical header row, and each first row was marked as a repeating table header. Document title, subject, author, keywords, comments, and en-US language metadata were normalized. Truth-first disposition. CONDITIONAL PASS. The remediated DOCX passes the available automated OOXML accessibility checks with no reported high-, medium-, or low-severity findings. Full WCAG conformance is not claimed. Advanced mathematical notation remains predominantly linear text rather than MathML or Office Math with independently verified assistive rendering, and no human screen-reader usability test was executed. These limitations are recorded rather than hidden. No theorem, proof, formula, benchmark, identifier, claim status, or scientific wording was changed by Phase E10. PHASE E11 FINAL INTEGRATED SUBMISSION CERTIFICATION AND AUTHORITATIVE FREEZE Integrated certification. The Version 55 manuscript has completed the E2-E8 authority, typing, proof-import, quantifier, cross-ledger, definition-to-use, residual-alias, and theorem-local map-signature program; Phase E9 visual remediation; and Phase E10 structural accessibility audit and safe remediation. The governing scientific claim ceilings, historical-genealogy rules, residual-type escalation rule, and accessibility limitations remain in force. Frozen authority. This file is the historical frozen Version 55 release candidate, superseded as current authority by later checksum-distinct Stage 9 successors. Its identity is determined by its exact filename and SHA-256 digest in the accompanying Phase E11 release manifest. Any later byte-level change creates a new binary and must receive a new filename, checksum, ledger entry, and applicable regression checks. Truth boundary. The freeze certifies integration, binary provenance, structural accessibility within the recorded conditional scope, and render integrity. It does not convert conditional, restricted, model-relative, open, historical-only, or undetermined scientific results into proved results. It does not claim full WCAG conformance or independently verified screen-reader rendering of advanced mathematical notation. Disposition. PASS FOR AUTHORITATIVE FREEZE, subject to the explicit accessibility limitations above. No theorem, proof, formula, benchmark, identifier, or scientific claim is altered by Phase E11. PHASE E12 POST-E11 MANUSCRIPT-CONTROL RECONCILIATION Scope. Release-control and ledger synchronization only. The two cumulative E2-E8 workflow rows are superseded by cumulative E2-E11 manuscript-control rows, and the recurring release footer is synchronized to the Version 55.1 successor identity. Truth boundary. This reconciliation does not alter any theorem, proof, formula, benchmark, identifier, scientific status, open problem, accessibility limitation, or claim ceiling. The Phase E11 frozen binary remains preserved as the prior authoritative release artifact. Disposition. PASS FOR POST-E11 CONTROL RECONCILIATION. This Version 55.1 binary is a successor control release and requires its own checksum, manifest entry, and applicable regression records. VERSION 55.1 RC36 INTEGRATED AUTHORITATIVE SYNCHRONIZATION PASS Authority freeze and controlled RC-20 through RC-36 manuscript integration Authority Freeze Authoritative manuscript source: ETP_Local_Relational_Time_Version_55.1_Post_E11_PB01_AGG02_AGG03_RC16_Authoritative_Synchronized 03.docx Source binary SHA-256: 30d386281960060d9d1822a5ce46c5f0f142cfd229f199c45aaaa3d99b822819 Freeze rule. The named RC16 binary is preserved as the immediate source authority. This RC36-integrated binary is a successor artifact and does not overwrite or retroactively alter the RC16 source. Controlling integration ledger: RC_MASTER_INTEGRATION_LEDGER.md; pre-integration SHA-256: b882bd4e983677db8e34dc5e1041d669286c42ce64828a692cad6ab886cccf6d Coverage statement. The controlling ledger contains detailed RC-20 through RC-36 entries. The preexisting manuscript already contains the synchronized RC11-RC16 theorem and model material. No standalone RC17-RC19 ledger entries were available in the controlling ledger, so this pass makes no invented RC17-RC19 scientific additions. RC-20 Through RC-36 Integrated Status Integrated Clock and Claim Disposition RC36 synchronization does not convert numerical, conditional, restricted, model-relative, or undetermined findings into proved universal theorems. The original RC30 fixed chart remains rejected across the tested perturbation box. The shorter RC35 chart on T in [1.4, 1.6185] remains a numerical corner-grid candidate with approximately ten-percent symmetric speed deviation and a very small margin. RC36 found no interior counterexample in the executed differential-evolution and Sobol searches, but the continuous parameter box remains uncertified because the searches are not exhaustive interval proofs. The full operating trajectory remains Class E. Intrinsic S1 candidate-record derivation and S2 operational-boundary derivation remain open. No RC result supplies basement-level physical actuality, a unique ontology, or an observer-independent global time. All RC findings remain subordinate to BNP-C1 through BNP-C7, the proof-import firewall, selected-output discipline, complete-carrier accounting, typed nullity, and the current claim-status ledgers. RC36 Synchronization Claim-Status Addendum Successor Authority and Regression Gate Successor authority. This RC36-integrated file is the current manuscript binary for subsequent controlled edits, subject to its exact filename and checksum. The RC16 source remains frozen and preserved as predecessor authority. Regression requirements. Any subsequent binary edit requires a new filename, SHA-256 digest, ledger entry, structural accessibility audit, and complete render review. Scientific changes additionally require theorem, formula, notation, cross-reference, dependency, claim-status, and provenance review appropriate to the affected scope. No-change declaration. Except for this controlled synchronization appendix and successor-authority metadata, the source manuscript text, theorem statements, formulas, tables, references, and earlier historical records are preserved. SI-5 CONTROLLED BINARY INTEGRATION Publication-Grade RC-29 Through RC-36 Integration Local Relational Time from a Globally Constrained Substrate | Version 55.1 CONTROL STATUS: This section is merged into the supplied RC36 authoritative binary as a checksum-distinct successor artifact. The supplied source remains preserved unchanged. SOURCE AUTHORITY: ETP_Local_Relational_Time_Version_55.1_RC36_Integrated_Authoritative_Synchronized-001.docx CLAIM CONTROL: Numerical propositions, counterexamples, and witnesses remain distinct from analytic theorems. Full-domain Class E remains authoritative. Abstract Integration The post-RC-29 timing program introduces a reverse-paired local-detailed-balance event-memory carrier and evaluates detector-accessible elapsed-time Fisher information, entropy production, calibration, robustness, and numerical certification. The carrier admits finite local calibration windows but remains a full-domain Class-E event timer. Uniformization, subdivision, perturbation-corner analysis, common-chart identification, and parameter-interior searches provide progressively stronger numerical evidence for local timing behavior while preserving explicit boundaries between numerical witnesses, counterexamples, and theorem-level results. No global affine duration theorem, parameter-independent certification theorem, or emergent spacetime geometry result is obtained. Results: Local Fisher Calibration of a Detector-Accessible Event Timer Reverse-paired local detailed balance and event-timer classification The enlarged event-memory carrier is X=(K,D₁D₂), where K∈{0,1,2} and D₁D₂∈{00,01,10,11}, giving a twelve-state stochastic carrier. Let Q(λ) denote the parameterized generator and p(T;λ)=p(0)e^{Q(λ)T} its elapsed-time distribution. Δs_env(x→y) = ln[w_xy / w_yx] Σ̇(T) = Σ_{x0.10. Counterexample D.1. The unchanged RC-30 detector chart is not valid throughout the tested parameter neighborhood. This falsifies universal fixed-chart robustness, not the existence of every local chart. Uniformization and numerical enclosure controls e^{QT}=e^{−ΛT} Σ_{m=0}^∞ [(ΛT)^m/m!] P^m, P=I+Q/Λ For each fixed generator, stochastic uniformization provides a positivity-preserving series with an explicit omitted Poisson-tail bound. The tail target is 10⁻¹⁵; the largest nominal omitted mass is 9.999648909974597×10⁻¹⁶, and the maximum nominal discrepancy from direct matrix exponentiation is 1.2212453270876722×10⁻¹⁵. The calculation reproduces the same 8-of-16 fixed-chart result and the same worst-corner deviation. Parameter subdivision into 1, 16, and 256 cells contracts the maximum detector-probability interval width at T=2.2 from 0.143556830773 to 0.074216908903 to 0.037729896582. All detector-probability lower bounds remain positive in all 256 cells at the two chart endpoints and midpoint. These are numerical enclosure controls, not proof-grade MPFI/MPFR certification. Shorter common-chart witness and parameter-interior search Across all sixteen tested corners and 5,201 time samples on [1.4,2.7], a shorter shared interval is found: I_common=[1.4,1.6185]. The optimized corner envelope is v_min=0.299348295485 and v_max=0.365850089223, with α*=0.332599192354 and δ*=0.099972873156. The tolerance margin is only 2.7126843744×10⁻⁵. The minimum detector probability over the corner-time grid is 6.59993187464×10⁻⁴. Proposition D.3. A parameter-independent numerical common chart exists across the sixteen tested parameter corners and sampled times. It is not a certified continuous-parameter theorem. Differential-evolution candidates over chart time and four parameters remain in [0.299348304267,0.365850059982], inside the optimized corner envelope. A separate Sobol design uses 2,048 parameter points and nine chart times, totaling 18,432 evaluations, with sampled range [0.299562182015,0.365718855111]. Proposition D.4. No executed interior search produced a detector Fisher speed outside the optimized corner envelope. Absence of a discovered counterexample does not prove that global extrema occur at corners. Publication Figures Figure SI5-1. Autonomous battery thermodynamics. Battery/excitation transfer and entropy-production diagnostics are retained as model-relative complete-carrier evidence; they do not convert the detector token into a globally adequate clock. Figure SI5-2. Retained-memory clock diagnostics. Cursor, detector-memory, and full-carrier Fisher-speed behavior illustrate local timing information and full-domain saturation; all clock classifications remain tied to the declared carrier and readout. Figure SI5-3. S1 multi-fragment scaling. Exact and approximate record-quality trends remain model and task relative; the figure does not close intrinsic S1 record-channel derivation or S2 boundary selection. Publication Tables Methods Appendix: Numerical Verification, Robustness, and Enclosure Methodology Perturbation design and calibration functional Fixed-chart robustness holds the nominal interval and calibration unchanged. Recalibrated robustness allows each parameter corner to choose its own chart. These answer distinct questions and are reported separately. The ten-percent threshold is a declared numerical convention rather than a theorem of adequacy. Uniformization stopping and solver comparison The Poisson series is truncated when the omitted mass is below 10⁻¹⁵. Direct matrix exponentiation is retained as an independent point-arithmetic comparator. Agreement controls solver consistency for the fixed tested generators but does not provide directed-rounding certification. Subdivision and denominator checks Because detector Fisher information contains terms ṗ_d²/p_d, every reported enclosure checks positivity of detector-probability lower bounds. Parameter subdivision reduces dependency inflation by propagating smaller boxes separately. Continuous time between audited points remains unenclosed in RC-34. Common-chart search and interior sampling The RC-35 search evaluates 5,201 uniform time samples. The longest contiguous interval meeting the ten-percent criterion simultaneously for every tested corner is selected. RC-36 then compares continuous-time corner optimization, differential-evolution candidates over the parameter interior, and a scrambled Sobol design. These are extensive numerical searches, not exhaustive interval proofs. Discussion Integration The RC-29 through RC-36 program demonstrates that elapsed-time distinguishability can arise within an explicitly modeled detector-memory carrier while remaining insufficient for globally calibrated duration. The full-domain Class-E result separates finite timing information from stronger clock adequacy. The RC-30 local charts show that global failure does not entail complete timing failure. RC-31 and RC-32 show that calibration cannot be promoted beyond its tested scope without robustness evidence. RC-33 and RC-34 strengthen numerical control but retain the distinction between truncation control and formal interval certification. RC-35 and RC-36 provide a shorter common numerical witness and extensive interior search, while preserving the open status of continuous-box certification and corner-extremum reduction. Conclusion Integration The detector-accessible timing branch supports finite local calibration and event ordering but does not establish global affine duration, parameter-independent certification, or spacetime geometry. The scientifically controlling result is therefore a hierarchy: coarse local-detailed-balance event timing, local numerical Class-C charts, an explicit fixed-chart counterexample, truncation-controlled uniformization, sampled subdivision and denominator controls, a shorter common corner-grid witness, and nonexhaustive interior-search support. Every stronger global claim remains open. Required Work Microscopic reservoir derivation of event, dark-write, and reset affinities. Preserve the RC-29 independent reproduction, accumulation-convention audit, adaptive-quadrature settings, and superseded coarse-grid benchmarks in the publication provenance package. Proof-grade directed-rounding or arbitrary-precision interval evaluation. Continuous-hyperrectangle common-chart certification or counterexample. Analytic or verified boundary-reduction theorem for Fisher-speed extrema. Restricted observer-access and detector-POVM reruns. Regeneration and inspection of every publication figure from source code. Full manuscript-wide notation, bibliography, cross-reference, accessibility, and render audits. Data Availability and Provenance The supporting evidence consists of phase audits, machine-readable outputs, verification and artifact-generation programs, diagnostic figures, sanitation and research ledgers, and checksum manifests preserved in the RC17-36 recovery archive. Publication authority requires source-code regeneration and reconciliation against the controlling result registry. RC-29 has been independently reconstructed and audited. The original standalone RC-29 package remains unrecovered, while the reproduction package and RC-29C2 accumulation-convention audit control the corrected integrated values and preserve the superseded coarse-grid benchmarks for provenance. SI-5 Successor Control Note This binary is a controlled, checksum-distinct successor of the supplied RC36 manuscript. The predecessor binary remains unchanged. The SI-5 material is integrated as a localized postscript section so the predecessor theorem architecture, historical records, and formatting remain preserved. Any later scientific edit must receive a new filename, SHA-256 digest, ledger entry, and applicable theorem, notation, cross-reference, accessibility, and render regression checks. Publication Packaging Certification Certification status: CONDITIONAL PASS for referee circulation and journal-format preparation. This status is not submission acceptance and does not close the listed scientific or reproducibility gates. RC-29C2 CHECKSUM-DISTINCT MANUSCRIPT CORRECTION Correction scope. This successor replaces only the two RC-29 integrated numerical values and synchronizes associated table, provenance, required-work, and packaging-control statements. The pointwise Fisher diagnostics, correct-prefix probabilities, nonnegative entropy-production rate, stationary Fisher-information collapse, and full-domain Class-E classification are unchanged. Superseded benchmarks. The historical values 16.102858284219 and 1.779360987325 are retained as coarse-grid provenance benchmarks consistent with approximately 600 uniform trapezoidal subintervals on [0.01,30]. Controlling values. The publication-controlling values are 16.086782080046472 nats for accumulated entropy production and 1.744817513152730 for Fisher length, obtained by adaptive quadrature using v(T)=1/2 sqrt(F_T(T)). Authority rule. This file is a checksum-distinct successor of the supplied Publication-Packaged Successor-03-1 binary. The predecessor remains unchanged and must be retained with its checksum and the RC-29C2 audit artifacts. S1/S2 STAGE 7 CHECKSUM-DISTINCT MANUSCRIPT INTEGRATION Source authority. ETP_Local_Relational_Time_Version_55.1_RC36_Publication_Corrected_Successor 04.docx Source binary SHA-256. 8e493010a57f71d54f2151ecc3d644fc4ccc17791b1a4b8b9f44731e2a2a81cc Integration scope. Cumulative S1/S2 Closure Stages 1 through 7, including exact countermodels, refinement-equivalence classification, the compatibility-hypergraph countermodel, the six-component candidate invariant, recovery-map benchmark boundary, status synchronization, and Stage 8 required work. Truth boundary. This successor does not convert conditional, restricted, model-relative, numerical, open, or undetermined claims into universal theorems. The predecessor binary remains unchanged. Authority rule. Any later byte-level change requires a new filename, SHA-256 digest, ledger entry, and applicable scientific, structural, accessibility, and render regression checks. M8 S1/S2 STAGE 8R CHECKSUM-DISTINCT MANUSCRIPT SYNCHRONIZATION Source authority. ETP_Local_Relational_Time_Version_55.1_S1S2_Stage7_Integrated_Successor(05).docx Integration scope. Cumulative Stage 8 response-law, instrument, multitime-process, causal-screening, divergence, carrier, comb, memory, and recovery findings through Stage 8R. Preservation rule. The supplied source binary remains unchanged. This successor preserves the complete source package and performs localized editable-DOCX changes with a new filename, checksum, manifest, accessibility audit, and full render review. Truth boundary. No conditional, model-executed, finite-benchmark, numerical, open, or undetermined result is promoted to a universal theorem. ETP remains the active research name; Theory of Relational Recoverability remains a viable candidate only. M8-R1 REFEREE REMEDIATION AND CHECKSUM-DISTINCT BINARY CORRECTION Source authority. ETP_Local_Relational_Time_Version_55.1_S1S2_Stage8R_M8_Integrated_Successor 06.docx Correction scope. Referee remediation of the Stage 8R M8 integration: abstract synchronization; canonical master-ledger authority; local theorem-status typing; Stage 8R methods signature; current claim-status rows; Required Work synchronization; primary process-tensor and recovery references; and checksum-distinct release control. Truth boundary. This correction does not promote any conditional, finite-benchmark, numerical, model-relative, open, or undetermined statement. Nearest-causal projection remains unexecuted; only exact causal feasible-point status is retained. Intrinsic S1 and S2 remain open. Preservation rule. The supplied Successor-06 binary remains unchanged. This document is Successor-07 and requires its own SHA-256 digest, manifest, accessibility record, and full render review. M8-R1 disposition. PASS FOR REFEREE-REMEDIATED SUCCESSOR RELEASE, subject to the explicitly retained external artifact-release, nonfinite, minimality, and intrinsic S1/S2 gates. M9 STAGE 8S-8Z CHECKSUM-DISTINCT MANUSCRIPT SYNCHRONIZATION Source authority. ETP_Local_Relational_Time_Version_55.1_S1S2_Stage8R_M8R1_Referee_Remediated_Successor (07).docx Integration scope. Stages 8S through 8Z, including the process-operational recovery bridge, complete-carrier target alignment, collision record gates, event-net provenance, autonomous activation, controller readability, calibration regress, physical-anchor admissibility, quotient semantics, Stage 8 completion, and the Stage 9 S1/S2 frontier. Truth boundary. No conditional, model-executed, finite-benchmark, numerical, open, or undetermined claim is promoted. ETP remains the active research name. TRR remains a viable candidate only. Intrinsic S1 and S2 remain open. Preservation rule. The supplied Successor-07 binary remains unchanged. This Successor-08 binary requires its own SHA-256 digest, manifest, accessibility audit, structural reports, render review, and ledger entry. M9 STAGE 9B.1-9B.11 CHECKSUM-DISTINCT MANUSCRIPT CONTROL NOTE Source authority. ETP_Local_Relational_Time_Version_55.1_M9_Stage8S-8Z_Synchronized_Successor-08 - Copy.docx Source binary SHA-256. 76a411bf2f92f8cd95a17a277480cc068069609154c0dcb90ddee73081e117a6 Integration scope. Interim narrative, theorem-boundary, claim-status, thermodynamic-direction, evidence, and next-frontier synchronization through Stage 9B.11. Preservation rule. The supplied Successor-08 copy remains unchanged. This Successor-09 binary is a new artifact. Earlier theorem statements, formulas, figures, tables, references, and historical genealogy remain preserved except for the visible release-title synchronization and the localized Stage 9B integration described above. Truth boundary. No conditional, model-executed, finite-benchmark, numerical, restricted, open, historical, or undetermined result is promoted. ETP remains the active research name; TRR remains a viable candidate only. Intrinsic S1 and S2, physical-anchor selection, thermodynamic orientation, primitive-resource provenance, and nonfinite extension remain open. Regression gate. Any subsequent binary edit requires a new filename, SHA-256 digest, ledger entry, structural accessibility audit, full render review, and scientific regression checks appropriate to the affected theorem, formula, notation, cross-reference, dependency, claim-status, and provenance scope. M9 STAGE 9C.4 FINAL STAGE 9 RELEASE CONTROL NOTE Release meaning. Final audit release, not intrinsic S1/S2 closure and not laboratory confirmation. Naming. ETP remains active; TRR remains candidate only. Regression rule. Any later byte-level change requires a new filename, checksum, ledger entry, and appropriate regression audit. BNP-TIII5R CONTROLLED BINARY INTEGRATION Release authority. This checksum-distinct successor integrates BNP-TIII5R without modifying the predecessor binary. The release preserves truth-first claim ceilings, the three-target firewall, complete-carrier discipline, selected-output discipline, and the distinction between informational order and metric duration. Order-layer closure disposition The record-induced informational-order layer is provisionally closed within the exact declared scopes. Eligible reachability supports a preorder, and mutual reachability supports quotient order, only under the applicable channel, composition, carrier, equivalence, and witness hypotheses. Approximate order remains construction dependent; path-sensitive recovery categories remain controlling when endpoint reachability suppresses route, resource, disturbance, fragment, or loop information. K01-K10 mandatory clock-entry contract Metric-duration nonbootstrapping No order coordinate, update index, collision count, path cost, statistical distance, Fisher length, persistence-decay parameter, entropy change, or notation t is promoted to elapsed duration without an independently justified calibration. Fisher information quantifies precision for a declared parameter-estimation task. It does not determine the physical interpretation of that parameter. Successor disposition The existing LRT theorem architecture remains preserved within its stated scopes. The BNP-TIII5R additions narrow interpretation and govern the next metric-time audit. Intrinsic S1 and S2 remain open. Metric duration and fundamental time remain unestablished. The controlling next gate is BNP-TIII6. BNP-TIII9R CONTROLLED BINARY RELEASE Metric-Time Closure, Manuscript Synchronization, Authority Promotion, and Target III Locality Entry Release authority This checksum-distinct successor integrates BNP-TIII6 through BNP-TIII9 from the exact attached BNP-TIII5R predecessor binary. The predecessor remains unchanged and frozen. The release preserves the truth-first constitution, three-target firewall, complete-carrier discipline, selected-output discipline, typed nullity, proof-import restrictions, and the separation of informational order, operational duration, synchronization, global-time representation, locality, geometry, and fundamental ontology. Metric-time layer disposition The metric-time layer is provisionally closed and reopenable within its declared operational scopes. A clock is a complete K01-K10 physical-estimation-calibration package. Accessible Fisher information is a task-relative sensitivity and precision quantity, not duration by identity. Local metric duration is conditionally constituted only on identifiable, independently calibrated, stable clock domains with declared uncertainty and recurrence treatment. Cross-clock synchronization requires experiment-wide comparison kernels or channels together with composition, inverse, overlap, noise, drift, access, gauge, and loop-consistency audits. Duration atlas, gauge, and global-time boundary A path-independent parallel duration atlas is licensed only on a compatible connected component with identity holonomy under the declared domains and tolerances. Identity holonomy supports existence, not uniqueness: a positive-affine global scale and origin freedom remains unless additional physical or conventional data fix it. Disconnected components retain independent freedoms. Partial overlaps, noisy kernels, and protocol choices can produce extension, statistical, or conventional nonuniqueness. Nonidentity holonomy is stronger because it obstructs one path-independent global parallel calibration. Controlling claim audit Target III locality entry certificate Locality research is authorized only from explicitly admitted physical events, records, carriers, propagation or influence relations, support or net structures, operational norms, complete carriers and complements, and the closed metric-time claim package. The locality program must not assume spatial coordinates, background topology, graph adjacency as physical locality by definition, tensor-factor adjacency as neighborhood, finite propagation as causal locality without interpretation, or clock holonomy as geometric curvature without an independent bridge. Three-target firewall and reopening Target I remains provisionally closed as unresolved and reopenable. Target II remains provisionally closed as partitioned, unresolved for universal dynamical generation, and reopenable. No qualifying reopening trigger was established by BNP-TIII6 through BNP-TIII9. Target III may use admitted event and record inputs but may not retroactively derive their occurrence, ontology, or provenance. Release status This file is the controlling Layer 2 temporal-reconstruction successor for BNP-TIII9R. It supersedes Successor-13 only within the Local Relational Time manuscript family. The metric-time block is provisionally closed within the declared scopes; universal, unique, fundamental, cosmological, basement-derived, and geometric time remain unestablished. Controlling next gate BNP-TIII10: Locality-Candidate Constitution, Influence-versus-Adjacency Separation, Operational Neighborhood Reconstruction, and Topology-Smuggling Audit. ETP-CPR Batch 2 Release Control Successor role Controlling Layer 2 temporal authority under the admitted-actuality conditional reconstruction constitution. Paper-specific starting package and scope The admitted package includes model-specific physical events or distinctions, complete carriers, interactions, output channels, records, instruments, accessible experiments, physical equivalences, calibrations, references, and resources. The manuscript may derive only scoped temporal structures from those inputs. Fisher information remains a task-relative metrological quantity, not duration by identity. Clock success does not establish ultimate provenance. Scientific preservation All preceding theorems, proofs, countermodels, calculations, figures, tables, citations, appendices, and historical records retain their prior scope unless this successor explicitly narrows a statement. This synchronization changes the governing constitution and dependency disclosure. It does not silently enlarge a theorem, erase a negative result, convert an admitted input into a derived output, or use downstream success to justify ultimate provenance. Controlling next project gate ETP-CPR3: Supporting-Paper Constitutional Synchronization, Cross-Corpus Claim Regression, and Referee-Facing Consistency Audit. OIRP Conditional-Closure Synchronization for Local Relational Time Program disposition. The core Observer-Invariant Reconstruction Program (OIRP) is conditionally closed as a restricted truth-first constitution for conditional reconstruction, not as a substrate theory. Physical actuality remains admitted rather than derived; ultimate provenance remains unresolved and reopenable; downstream structures may not retroactively justify their inputs; no universal ontological core, physical-instantiation bridge, definite-event selector, or positive empirical ETP claim is authorized. Layer 2 effect. LRT remains a conditional temporal-reconstruction authority under admitted physical actuality. OIRP closure does not alter existing theorem statements or promote clock adequacy, record order, memory, comparison, synchronization, or thermodynamic direction into fundamental time, definite-event generation, or ultimate provenance. Integrated OIRP findings. Memory, learning, agency, observer, and object claims appearing in LRT must be read through their completed OIRP constitutions. Retention is not primitive storage; adaptation is not primitive optimization; control is not intention; access is not a primitive subject; and persistence or boundary stability is not primitive thinghood. Composite certification. Local clock adequacy, record recovery, observer access, comparison consistency, and constituent process certification do not by themselves certify a complete temporal composite. Exact and approximate, finite and nonfinite, local and global, Markovian and non-Markovian, and selected-view and complete-carrier statuses remain distinct. Inherited K1-K9 Kernel K1. Admitted actuality. Physical actuality, events, and carriers are admitted for reconstruction and are not derived from downstream success. K2. Typed target and scope. Every claim states its target, domain, task, regime, and exact or approximate status. K3. Complete carrier and boundary. Retained and discarded systems, environments, branches, controls, references, resources, boundaries, correlations, and relevant continuations are declared. K4. Witness and access. Measurements, testers, readouts, access limits, and witness-completeness status are explicit. K5. Equivalence before identity. Operational, representational, gauge, and physical equivalence are typed before nullity, identity, objectivity, or invariance is promoted. K6. Transformations, composition, and history. Operations, ordering, interfaces, feedback, composition laws, and relevant histories are declared; constituent certification does not certify the composite. K7. Criteria and uncertainty. Metrics, tolerances, calibration, uncertainty, error propagation, and resource bounds are explicit. K8. Provenance, rivals, falsifiers, and ceilings. Inputs, serious alternatives, falsifiers, evidence class, reopening triggers, claim ceilings, and nonretroactivity are preserved. K9. Plural and null admissibility. Empty, null, quotient-valued, plural, nonunique, Pareto, restricted, and underdetermined outcomes remain legitimate. Authorized LRT Successor Work Nonfinite and device-independent recovery and witness completeness. Approximate recovery under declared error, resource, and composition rules. Complete sequential-process and constituent-to-composite certification. Physical implementation and empirical comparison among rival temporal reconstructions. OIRP-S2.MS1 Corpus Synchronization Addendum Status: SUCCESSOR CANDIDATE. Scientific cutoff: verified record through HHLWBUILD2F Level 5. HHLW Level 6 excluded because no completed raw result was available. Authority and scope This checksum-distinct manuscript successor is synchronized through OIRP-S2.MS1 using the verified OIRP-S2 record available on September 18, 2026. The core OIRP program remains conditionally closed only as a truth-first constitution for conditional reconstruction. OIRP-S2 remains open. Physical actuality is admitted rather than derived, ultimate provenance remains unresolved and reopenable, and no downstream reconstruction may retroactively justify its admitted physical inputs. OIRP-S2 composition update The current successor program separates constituent certification from composite certification. Channel, instrument, process, comb, superchannel, adaptive-network, controller-inclusive, and memory-bearing claims require their own complete carriers, interfaces, histories, witnesses, equivalence relations, metrics, tolerances, and composition laws. Equality or recoverability under restricted access does not establish complete physical equivalence, and a finite-dimensional computation does not establish physical realization or a nonfinite theorem. Current CP4.1 numerical and theorem status CP4.1 has established a restricted finite-dimensional CPTP telescoping theorem for compatible channel sequences, genuine diamond-norm semidefinite-program executions on declared finite-dimensional cases, noncommuting-channel tests, dimension-changing examples, and an ancilla-advantage benchmark. These results do not establish a universal physical composition law, arbitrary process composition, unrestricted adaptive equivalence, nonfinite closure, or OIRP-S2 closure. HHLWBUILD2F status firewall The patched repository hierarchy record is L1 = 0.9268200938227155, L2 = 0.9998804689772093, L3 = 1.0000370777829688, L4 = 1.00015286168964, and L5 = 1.0000000197695689. Values slightly above one are treated only as solver-tolerance-compatible with probability one, not as literal probabilities. Level 6 was not available for this synchronization. Formal patch correctness, formal convergence, exact probability one, equality with the strict-classical optimum, equality with the Definition-23 factor-affine hierarchy, CP4.1 closure, and CP4.2 entry are not established. External relationship and prohibited inference The external constrained-separability literature formulates limited-memory and classically adaptive channel discrimination using separable variables with additional affine constraints and discusses PPT and constrained symmetric-extension outer approximations. That framework supports keeping the repository patched hierarchy distinct from the full factor-affine constrained hierarchy unless an explicit equivalence and implementation audit is completed. No external source is used here to replace proof, raw execution, or authority-grade numerical certification. Evidence and release classification This addendum is a manuscript-synchronization action. It changes cross-program status language and interfaces but does not alter previously proved theorem statements, raw calculations, figures, datasets, or experimental status except where this addendum explicitly narrows a claim. No physical-device experiment, definite-event selector, substrate ontology, universal witness completeness, or ultimate-provenance result is claimed. Layer-2 import boundary For temporal reconstruction, constituent channel recovery, one-step distinguishability, or selected-output equality does not establish multitime process recovery, clock adequacy, comparison consistency, or a temporal arrow. Controller memory, outcome registers, retained environments, feedback histories, reset domains, and future interventions must be included or explicitly excluded. Approximate errors require a declared transport and accumulation rule. Source note Internal authority: OIRP_LEDGER.md and the applicable OIRP/OIRP-S1/OIRP-S2 reproducibility and correction packages in the Recovery corpus. External relationship checked against T.-A. Ohst, S. Zhang, H. C. Nguyen, M. Plavala, and M. T. Quintino, “Characterising memory in quantum channel discrimination via constrained separability problems,” arXiv:2411.08110. Publication status must be verified independently before journal submission. OIRP-S2 Restricted Closures and the Local Relational Time Interface The CP4.1-CP4.3 results constrain how channels, instruments, controllers, memories, histories, postselection, tomography, operational quotients, and composite certificates may enter temporal reconstruction. They do not change the existing metric-time theorem status or derive time from ultimate provenance. OIRP-S2 remains open. These closures do not establish physical realization, laboratory confirmation, empirical confirmation, indefinite-causal-order closure, a universal process-matrix composition law, a universal operational-memory formula, definite-event generation, substrate ontology, or ultimate physical provenance. Physical actuality, events, carriers, interactions, records, instruments, calibrations, and empirical inputs remain admitted rather than derived. Ultimate provenance remains unresolved and reopenable. No downstream result may retroactively justify an admitted input. Restricted closure status • CP4.1 is closed at a restricted channel-level and HHLW claim ceiling. The exact strict-classical optimum, complete arbitrary-history factor-affine hierarchy, physical realization, and empirical confirmation remain excluded. • CP4.2 is closed at a restricted instrument, comb, superchannel, adaptive-memory, postselection, tomography, identifiability, implementation, and empirical-preregistration ceiling. • CP4.3 is closed at a restricted operational-quotient, extension, tensor-descent, operational-memory, fixed-context composite-certification, and general-process no-go ceiling. Composite-certification firewall Certified constituents do not automatically certify a composite. Composite use requires matching typed interfaces, complete carriers, causal order, controllers, interleavings, memory and environment access, branch and history treatment, tester class, normalization, success floors, equivalence or gauge, uncertainty model, and implementation class. The restricted fixed-context hybrid theorem does not imply a universal composition rule for arbitrary process matrices or indefinite causal order. Direct operational tensors must be defined on declared operator-system domains or proved extension independent. Operational-memory firewall Operational memory is defined only relative to a fixed realization class by minimizing over all admitted realizations of one operational quotient. Selected ambient-extension Choi rank, selected Stinespring multiplicity, and nonminimal dilation size are not automatically operational-memory invariants. Authority and reopening Detailed theorem, numerical, execution, dataset, figure, and empirical authority remains package specific. This manuscript successor does not replace the certified research packages or the canonical OIRP program ledger. The affected branch reopens if a new carrier, environment, branch, controller, tester, causal architecture, tensor structure, gauge, identifiability result, correction, retraction, counterexample, hardware result, or empirical dataset changes the restricted claim ceiling.