# Source extract: FA-QD-PRXQ-01.docx

Causal Nonseparability and Environmental Objectivity

Independence, Control-Assisted Records, and Quantum-Switch Benchmarks

Scott Muniz

Dr. Ware Technology Services | scott@drware.com

Research Article | August 2026

Popular Summary

Quantum Darwinism explains how information copied into many parts of an environment can become independently available to multiple observers. A different line of research studies what happens when a quantum system controls the order in which two operations occur. This paper shows that the two ideas do not imply one another. A process can create records that multiple observers can independently read without quantum-controlled ordering, and it can have quantum-controlled ordering without creating such records. When both are present, the control can contain extra information, but only when it is measured together with an environmental record. Ignoring the control removes that contribution. The result provides a practical accounting rule: quantum control can change what is learned when the control is measured, but it does not automatically create independently readable records or establish how operations are causally ordered.

Abstract

Environmental objectivity and indefinite causal order answer different operational questions. This article establishes a rigorous independence framework for their interface. Within a composite class containing fixed-order record circuits, the unaltered quantum switch, and deterministic record-augmented switch processes, environmental objectivity is neither necessary nor sufficient for causal nonseparability. The independence theorem is the organizing result. Supporting it, we derive a nonnegative access decomposition for a classical pointer label Λ, order control T, and fragment F; branchwise criteria for exact pointer preservation; and probability-weighted postselection bounds controlled by I(Λ:T|F). In a declared binary record model, coherent order leaves equal-prior Helstrom discrimination exactly neutral because the coherent conditional-difference operator adds only two null directions while preserving the nonzero singular values. This coexists with a resource-matched joint-access Holevo advantage of 0.083977 bit and many-fragment diagnostics showing no coherent fragment-only redundancy advantage in the tested class. The significance of the positive Holevo difference is not its absolute size but its coexistence with fragment-only and equal-prior decision neutrality, which isolates control-assisted joint information from environmental objectivity in the same declared model. The result establishes an operational distinction within the declared access and comparator architecture: quantum-controlled ordering can supply a bounded, task-dependent, access-dependent information resource, but it is not a universal mechanism for environmental objectivity. Validation is computational; no physical-device measurements are claimed.

Claim-interface status. This operational distinction is not an intrinsic S2 boundary derivation. Environmental objectivity does not select an intrinsic observer partition, and process-causal certification does not derive geometric causal order or Lorentzian structure. Validation remains computational.

Truth-First Basement and Reconstruction Interface

Authority and role. This Version 13 successor is synchronized with the Core Synthesis Draft-07, the Basement Provenance and Instantiation Protocol Version 03, Local Relational Time Version 56, and the truth-first Local Relational Geometry architecture. The paper remains a bounded operational study of environmental objectivity and causal nonseparability. It does not select a basement, derive physical instantiation, derive definite events from a merely formal possibility structure, or reconstruct geometry.

• Environmental objectivity and causal nonseparability are Layer 1 and later operational properties of already instantiated preparations, controls, laboratories, interactions, records, fragments, and instruments.

• Relational potentiality remains a candidate basement, not an established premise of this paper.

• Formal process matrices, channels, state vectors, causal witnesses, Holevo quantities, and record structures do not establish physical instantiation.

• Objectivity, record availability, causal nonseparability, recoverability, operational boundaries, causal order, and Lorentzian geometry remain distinct claims.

• Computational validation establishes only the declared model calculations and does not constitute laboratory evidence or an instantiation bridge.

• A positive or null result may not be promoted beyond the declared comparator class, access structure, witness family, carrier, and numerical or analytic scope.

Three-Target Firewall

Target I concerns physical instantiation of a basement. Target II concerns definite-event generation from an instantiated basement. Target III concerns reconstruction and certification of operational structures among instantiated events. This paper addresses a bounded Target III interface. Its independence theorem and information-access results do not establish Target I or Target II.

Inherited Physical-Input Certificate

Revised Claim Ceiling

The independence theorem remains a theorem within the declared composite record-process class. The access decomposition, pointer-preservation results, postselection bounds, Helstrom-neutrality result, matched-family Holevo comparison, many-fragment diagnostics, and witness-pullback proposition retain their stated scopes. None establishes an intrinsic observer, intrinsic operational boundary, physical instantiation, a unique causal ontology, geometric causal order, topology, metric, Lorentzian signature, or spacetime light cone.

I. Introduction

Quantum Darwinism describes operational objectivity through redundant environmental records of a system pointer [1-8]. Process-matrix and higher-order-operation frameworks instead classify transformations and processes that cannot be reduced to a definite or probabilistic causal order [9-13,19]. Objectivity asks which information is available in multiple disjoint fragments. Causal nonseparability asks whether the higher-order process connecting local laboratories lies outside the causally separable cone.

This work reframes the interface as an independence problem. The central result is that environmental objectivity and causal nonseparability are independent operational resources: neither implies the other, and evidence for one should not be used as evidence for the other. The central contribution is an exceptional connection: advantages from quantum-controlled operation order are not objectivity advantages unless the access, comparator, and independent-certification conditions support that inference. The connection is consequential because it supplies a common reporting standard for two communities that can use the same quantum-control evidence to answer different operational questions.

Objectivity claims are operational and access-relative. In the broader ETP framework, observer access is represented through operationally justified observable domains and accessible records rather than through primitive assumptions about observer structure.

Methodologically, this work adopts the broader principle that operational resources should be certified independently unless a theorem establishes their equivalence. Evidence for causal structure, record accessibility, redundancy, or information gain is therefore interpreted only within the certification route appropriate to that resource.

The independence result changes how controlled-order experiments and simulations should be interpreted. Evidence of causal nonseparability cannot, by itself, establish environmental objectivity, while redundant environmental records cannot certify causal nonseparability. Any claimed connection must therefore identify three elements separately: the accessible subsystem, the causally separable comparator, and an independent certification route for causal structure. This separates genuine objectivity claims from control-assisted information-access effects.

The supporting results identify where control-assisted information can reside, when pointer records survive controlled ordering, and why postselection cannot create unrestricted record gain. The independence theorem is the conceptual advance; the exact Helstrom-neutrality theorem is its sharpest technical instance in the declared binary model.

Scope is deliberately bounded. We do not claim an advantage over the full causally separable process cone, a universal enhancement of objectivity, a global distance-to-spectrum-broadcast theorem, or physical-device validation. The reported numerical advantage is a resource-matched separation within the declared family, not an optimization over the full causally separable cone; the requirements for that broader optimization are stated in Sec. VII.

Table I. Theorem roadmap and logical architecture.

II. Operational setting and notation

A. Global assumptions

All Hilbert spaces are finite dimensional. Logarithms are base two. The trace norm is ||X||₁ = Tr√(X†X). Unless otherwise stated, the binary decision problem uses equal priors. Classical registers are represented by orthogonal basis states, and I(A:B|C)=S(AC)+S(BC)-S(C)-S(ABC).

Complete-carrier declaration. The task-relative carrier includes the pointer label, order-control system T, system register, all retained environmental fragments and record subsystems, local-operation interfaces, classical instrument outcome Y when measured, references used to define branch basis and access, discarded carriers, and any correlations or future continuations relevant to the claimed task. A conclusion after discarding T or other carriers is a selected-view statement, not complete-process equivalence.

B. Pointer labels, fragments, and objectivity

Let Λ be a classical pointer label with prior qλ and conditional states ρ_TF|λ.

ρ_ΛTF = Σλ qλ |λ⟩⟨λ| ⊗ ρ_TF|λ.   (1)

For a fragment F, I(Λ:F) equals the Holevo information of the conditional ensemble. Operational redundancy at deficit δ requires many disjoint fragments to carry at least (1-δ)H(Λ). Ordinary redundancy is not identified with strong Quantum Darwinism or spectrum-broadcast structure without their additional conditions [20].

Witness-family discipline. Equality or failure to separate under the finite diagnostics used here does not establish complete physical process equivalence unless the admitted witness family is proved complete for that distinction. Otherwise the appropriate status is RESTRICTED-NULL or UNDERDETERMINED.

C. Causal separability and composite process class

W_sep = q W^(A≺B≺C) + (1-q) W^(B≺A≺C).   (2)

A causal witness is nonnegative on every causally separable process and negative on at least one causally nonseparable process [10-12]. Record quality and causal certification are evaluated separately.

Record formation, exact recoverability, and environmental objectivity are treated as distinct operational properties. The presence of one does not by itself certify the others, and each requires its own verification criteria.

Definition 1. Composite record-process class. Let C_rec contain fixed-order processes with deterministic local record channels, the unaltered quantum switch without appended records, and record-augmented switch processes. The class is closed under adjoining deterministic local record channels and deterministic local discard of those records.

III. Independence and control-access decomposition

Theorem 1. Logical independence

Within C_rec, environmental objectivity is neither necessary nor sufficient for causal nonseparability. A fixed-order process with compatible repeated records is objective but causally separable. An unaltered causally nonseparable switch, or one with blank or incompatible record channels, is nonobjective but causally nonseparable. A switch with common-pointer-preserving record channels can satisfy both properties. A fixed-order nonrecording process satisfies neither.

Proof sketch. The objective, causally separable sector is populated by a fixed-order process whose compatible local record channels copy the same pointer label into disjoint environmental subsystems. The nonobjective, causally separable sector is obtained from a fixed-order process with blank, incompatible, or absent record channels. The objective, causally nonseparable sector is populated by adjoining common-pointer-preserving deterministic record channels to a causally nonseparable switch; closure of C_rec under those channels keeps the construction in the declared class, while deterministic discard recovers the underlying switch. Finally, the unaltered switch, or a switch with blank or incompatible records, populates the nonobjective, causally nonseparable sector. Thus all four sectors are nonempty within C_rec, proving that neither property is necessary or sufficient for the other. □

Figure 1. Logical independence within the composite record-process class C_rec. Explicit constructions populate all four combinations of environmental objectivity and causal nonseparability. The construction establishes that neither property implies the other within the declared class.

Theorem 2. Nonnegative access decomposition

I_coh - I_F = Δ_coh + Δ_label,   Δ_coh ≥ 0,   Δ_label ≥ 0.   (3)

Here Δ_coh=I(Λ:TF)_ρ-I(Λ:TF)_D_T(ρ) and Δ_label=I(Λ:TF)_D_T(ρ)-I(Λ:F)_ρ. Data processing under D_T proves Δ_coh≥0. Dephasing T leaves the ΛF marginal unchanged; the chain rule gives Δ_label=I(Λ:T|F)_D_T(ρ)≥0. The split is tied to the declared branch basis and access structure. □

IV. Controlled-order record dynamics

U_ord = |0⟩⟨0|_T ⊗ U_B U_A + |1⟩⟨1|_T ⊗ U_A U_B.   (5)

Definition 2. Exact pointer preservation. Pointer projectors {Qλ} act on H_S and are lifted to larger spaces by identities on all nonpointer subsystems. An operator V preserves them exactly when [V,Qλ]=0 for every λ after this extension. This is equivalent to block diagonality in the pointer-sector decomposition.

Theorem 3. Common-pointer sufficiency

If [U_A,Qλ]=[U_B,Qλ]=0 for every pointer projector, then both definite orders, U_ord, and every nonzero postselected operator Kφ=⟨φ|U_ord|τ⟩ preserve Qλ. Products and linear combinations in the commutant remain in the commutant.

Theorem 4A. Operator-level orthogonal-branch necessity

Exact operator-level preservation by U_ord holds if and only if both order products preserve the pointer projector, because

[U_ord,I_T⊗Qλ] = |0⟩⟨0|⊗[U_B U_A,Qλ] + |1⟩⟨1|⊗[U_A U_B,Qλ].   (6)

The two terms occupy orthogonal control blocks, so cross-order coherence cannot cancel operator-level leakage. □

Corollary 4B. State-supported leakage criterion

For a fixed input control state with branch weights p₀ and p₁, define Lλ(V)=||(I-Qλ)VQλ||₂². Tracing out T yields Lλ(U_ord)=p₀Lλ(U_BU_A)+p₁Lλ(U_AU_B). Zero unconditional leakage is equivalent to zero leakage in every occupied branch. If one weight vanishes, the criterion applies only to the occupied branch. □

V. Postselection information budget

Let an instrument {M_y} on T store the accessible outcome in a classical register Y. Define χ_y=I(Λ:F)_ρ|y and χ=I(Λ:F)_ρ.

Theorem 5. Exact average identity

Σ_y p_y χ_y - χ = I(Λ:Y|F) - I(Λ:Y).   (7)

The result follows from the two chain-rule expansions of I(Λ:FY). □

Theorem 6. Average and selected-outcome bounds

Σ_y p_y(χ_y-χ) ≤ I(Λ:T|F),   p_y(χ_y-χ) ≤ I(Λ:T|F)+(1-p_y)χ.   (8)-(9)

Conditional data processing gives the average bound. Bounding every complementary deviation below by -χ gives the selected-outcome bound. A favorable outcome must therefore be reported with its probability and complementary outcomes. □

VI. Exact equal-prior decision neutrality

For equal priors, let Δ_AB, Δ_BA, and Δ_coh be the fixed-order and coherent conditional-difference operators. In the declared physical bases, the archived symbolic certificate constructs the operators and verifies unitary reduction. With s=sin(α_B/2)cosθ,

V_AB Δ_AB V_AB†=diag(s,-s),  V_BA Δ_BA V_BA†=diag(s,-s),

V_coh Δ_coh V_coh†=diag(s,-s,0,0).

p_fix(λ)=λ²-sin²(α_B/2)cos²θ,   p_coh(λ)=λ²[λ²-sin²(α_B/2)cos²θ].   (10)-(11)

||Δ_coh||₁=||Δ_AB||₁=||Δ_BA||₁=2|sin(α_B/2)cosθ|,   P_H=1/2+|sin(α_B/2)cosθ|/2.   (12)-(13)

The coherent operator adds two null directions and preserves the nonzero spectrum. This is an exact, task-specific equality, not a generic statement that coherent order is useless.

VII. Numerical and operational results

A. Matched-family joint-access capability

For fixed dimensions, initial states, one use of each operation, and comparison with the two fixed orders, the model reaches a joint control-fragment Holevo advantage of 0.0839771759 bit near θ=0.700108 and full record strength. Discarding T removes the cross-order contribution and produces no fragment-only advantage.

Operational interpretation. The increase is 17.59% over the matched fixed-order value 0.477538 bit, raising the joint-access Holevo quantity to 0.561515 bit. Its importance is the coexistence pattern: joint control-plus-fragment information increases while fragment-only redundancy and equal-prior Helstrom discrimination remain neutral in the same declared model.

For a binary label, Fano's inequality relates an achieved classical mutual information to the minimum decoding error of a specified measurement and decoder. The present comparison concerns Holevo quantities, which upper-bound accessible information, so the increase cannot by itself be converted into a guaranteed error reduction. Such a conversion would require an explicit measurement or coding construction.

Table II. Comparator hierarchy.

A full-cone comparison would require optimization over all causally separable processes compatible with the declared laboratory dimensions, input-output interfaces, allowed ancillary systems, number of channel uses, and accessible output systems. That optimization is distinct from the matched-family question, which holds the physical resources and record-generating operations fixed while changing their ordering structure.

Table III. Access-dependent numerical results.

Relation to communication-power results. Zhao et al. study one-shot classical-message transmission and asymptotic comparison with shared entanglement [22]. Reference [22] is a recent preprint used to distinguish communication-power tasks from the environmental-record access task studied here; none of this manuscript's central results depends on that preprint. A numerical comparison would require re-expressing both studies within a common encoding ensemble, accessible-system specification, task metric, and comparator class; absent that common optimization problem, the numerical values are not commensurate.

B. Many-fragment objectivity

Complete subset enumeration through eight records and mismatch-dependent surfaces for N=2,4,6 show no coherent fragment-only redundancy advantage over the better fixed order under mean-fragment or worst-fragment 0.9-bit thresholds. The complete-family control-assisted bonus remains bounded, so the bonus per record vanishes asymptotically.

C. Local spectrum-broadcast diagnostics

At the benchmark mismatch, coherent environment-only singleton records remain nonorthogonal and at finite optimized trace distance from every ideal orthogonal qubit record pair. These finite diagnostics are not a global minimum distance to the full SBS set.

D. Noise and phase structure

Control dephasing removes the coherence-sensitive contribution. Partial strengths and tested noise models leave positive joint-access regions but do not generate a fragment-only redundancy advantage. Robustness is regime specific.

VIII. Supporting record-augmented causal certification

Let W_sw be a causally nonseparable base switch process and W_ext append local record outputs. Let R be the deterministic local recovery/discard map satisfying R(W_ext)=W_sw. Deterministic local processing maps the extended causally separable cone into the base causally separable cone. This result is a supporting certification check, not the main conceptual advance.

Proposition 1. Ancilla stability and witness pullback

S_ext=R*(S_sw),   Tr[R*(S)W]=Tr[S R(W)].   (14)-(15)

Tr[S_ext W_ext^sep]≥0,   Tr[S_ext W_ext]=Tr[S_sw W_sw]<0.   (16)-(17)

Therefore S_ext is a causal witness for W_ext, and W_ext remains causally nonseparable. □

IX. Four-qubit validation route

The minimal logical register is T,S,E₁,E₂ with tensor ordering T⊗S⊗E₁⊗E₂. Fixed AB, fixed BA, coherent-order, dephased-control, and postselected-control arms test neutrality, biased-prior activation, coherence loss, and joint-versus-fragment access.

U_A(α_A)=|0⟩⟨0|_S⊗I_E₁+|1⟩⟨1|_S⊗R_y(α_A)_E₁,   (18)

U_B(θ,α_B)=R_y(-θ)_S[|0⟩⟨0|_S⊗I_E₂+|1⟩⟨1|_S⊗R_y(α_B)_E₂]R_y(θ)_S.   (19)

Qiskit statevector validation reports maximum error 3.33×10^-16; independent NumPy gate-level validation reports 2.22×10^-16; reduced normal form matches at stored double precision; finite-shot ideal sampling has maximum absolute error 0.010897 across 99 settings with 8,192 shots per label.

X. Scope and limitations

Draft-07 and Layer 0-3 interface. This paper supplies operational independence results, access-relative distinctions, and finite counterexamples relevant to the ETP corpus. It is downstream of physical instantiation and complete event-structure assumptions. It does not derive an intrinsic observer, record channel, operational boundary, metric, topology, Lorentzian signature, geometric causal order, or physical instantiation. Intrinsic S1 and S2 remain open. The controlling corpus authorities are Core Synthesis Draft-07, Basement Provenance and Instantiation Protocol Version 03, Local Relational Time Version 56, and the truth-first Local Relational Geometry v6 successor.

The Holevo advantage is resource matched, not optimized over the full causally separable cone.

Causal nonseparability does not imply environmental objectivity, and objectivity is not a causal witness.

The inherited-witness result requires deterministic local recovery.

SBS diagnostics are finite local and correlation diagnostics, not a global distance theorem.

Validation is computational and contains no physical quantum-processor measurements.

The access decomposition is relative to the declared branch basis and access structure.

Helstrom neutrality is exact only for the declared equal-prior binary model.

OIRP Conditional-Closure Interface

Program status. The core Observer-Invariant Reconstruction Program (OIRP) is conditionally closed as a restricted truth-first constitution for conditional reconstruction. This closure does not establish a substrate theory, physical-instantiation bridge, definite-event selector, universal ontological core, or positive empirical ETP result. Physical actuality remains admitted rather than derived, ultimate provenance remains unresolved and reopenable, and no downstream result in this manuscript may retroactively justify the admitted systems, events, carriers, controls, records, probabilities, access structures, or equivalences on which it depends.

Manuscript effect. The theorem statements, numerical results, figures, and venue-specific claims of this manuscript retain their existing scopes. OIRP closure adds a governing claim discipline; it does not upgrade any analytic or computational result into physical implementation, laboratory evidence, intrinsic observer or object structure, intrinsic boundary or topology, Lorentzian geometry, or ultimate provenance.

Inherited OIRP Constitutional 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 remains indexed to 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 must be declared.

K4. Witness and access: measurements, testers, readouts, access limits, and witness-completeness status must be explicit.

K5. Equivalence before identity: operational, representational, gauge, and physical equivalence must be typed before promoting nullity, identity, objectivity, or invariance.

K6. Transformations, composition, and history: operations, ordering, interfaces, feedback, composition laws, and relevant histories must be declared; certification of constituents does not certify the composite.

K7. Criteria and uncertainty: metrics, tolerances, calibration, uncertainty, error propagation, and resource bounds must be explicit.

K8. Provenance, rivals, falsifiers, and claim ceilings: inputs, alternatives, falsifiers, evidence class, reopening triggers, and nonretroactivity must be preserved.

K9. Plural and null admissibility: empty, null, quotient-valued, plural, nonunique, Pareto, restricted, and underdetermined outcomes remain legitimate scientific results.

Reopening and Successor-Research Control

This manuscript should be reopened or promoted only by qualifying evidence, including a new physical bridge, a noncircular event selector, a witness-completeness theorem or counterexample, a newly relevant carrier or branch, a composition theorem or failure, a revised physical-equivalence result, an approximation failure or stronger error bound, measured rival discrimination, a theorem contradiction, or an artifact-integrity failure. Otherwise future work proceeds through restricted successor programs for recoverability completeness, composition and boundary closure, physical realization and empirical discrimination, or separate provenance and definite-event research.

Publication statement. OIRP provides the governing methodology for this manuscript’s future revisions: truth-first, carrier-complete, witness-explicit, equivalence-controlled, composition-aware, provenance-preserving, falsifiable, and nonretroactive.

XI. Conclusion

Environmental objectivity and causal nonseparability are independent operational structures. That independence is the central result. When record-generating operations are coherently ordered, pointer compatibility determines deterministic preservation, while the control provides a bounded joint-access information resource. It can increase Holevo information for T+F without improving equal-prior binary discrimination or fragment-only redundancy. Postselection is probability constrained, record replication does not make the bonus extensive, and record outputs preserve an independently certified causal witness under deterministic local recovery.

More generally, operational resources require independent certification. Environmental objectivity certifies the existence of redundantly accessible records, while causal witnesses certify properties of causal structure. Neither certification route substitutes for the other, even when both concern the same underlying physical process.

The independence theorem therefore constrains later geometric reconstruction without supplying it. Causal nonseparability is a process property under a declared laboratory structure, not a derivation of a Lorentzian causal order; environmental objectivity is an access-and-record property, not intrinsic observer or boundary selection.

The communication-power comparison shows why coherent order must be assessed task by task. An advantage in message transmission does not imply an advantage in environmental-record discrimination, redundancy, or spectrum-broadcast structure.

Natural extensions include multipartite process structures, optimization over broader causally separable comparator classes, and experimentally accessible tests that independently evaluate record redundancy and causal structure.

Data and code availability

A compact reviewer-facing package accompanies the working submission and contains the claim-to-artifact map, disclosure and scope statements, metadata, validation summary, checksum manifest, and verification script. The complete deterministic scripts, prediction tables, exact matrix outputs, finite-shot simulated counts, interval calculations, local-simulator outputs, theorem certificates, reports, and release artifacts are preserved in the author-controlled archive. The materials contain computational results and no measurements from a physical quantum processor. No public GitHub URL or verified Zenodo record is claimed in this version; released identifiers should be inserted before journal submission.

Author contributions, declarations, and AI assistance

S.M. is the sole author, conceived and directed the research, and takes responsibility for the manuscript and its claims. The author is affiliated with Dr. Ware Technology Services and declares no competing interests. Microsoft 365 Copilot (M365 Copilot based on the GPT-5 reasoning model) assisted with language editing, code review, bibliographic organization, consistency analysis, and draft preparation under the author's direction. The author independently rederived symbolic identities, regenerated numerical results locally from the archived scripts, checked references against DOI or arXiv metadata, and verified every scientific claim. No AI system is an author.

Appendix A. Physical-basis bridge for the neutrality theorem

The gate-level construction exports Δ_AB, Δ_BA, and Δ_coh directly in the declared physical bases before any reduction. For each grid point the certificate checks Hermiticity, zero trace, and unitary similarity to the normal forms. Writing the eigenvector matrices as V_AB, V_BA, and V_coh gives

Δ_AB=V_AB† diag(s,-s)V_AB,  Δ_BA=V_BA† diag(s,-s)V_BA,

Δ_coh=V_coh† diag(s,-s,0,0)V_coh.

Thus the characteristic-polynomial factorization follows from matrices first constructed in the physical E₂ and T⊗E₂ bases, rather than from assuming diagonal representatives.

Appendix B. Referee reproducibility checklist

References

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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.

Controlled-order interface update

Environmental objectivity and causal nonseparability remain independently certified operational properties. Fixed-order, controlled-order, adaptive, feedback, and unrestricted higher-order strategy classes must not be conflated. Resource-matched computational comparisons do not establish optimization over the full causally separable cone or laboratory evidence.

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 CP4.1-CP4.3 Interface Addendum

This causal-nonseparability and environmental-objectivity manuscript retains its independence and control-assisted-record claims at their existing scope. CP4.3 adds a no-go firewall against universal composition of arbitrary general processes and does not promote the manuscript to indefinite-causal-order closure.

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.