# Source extract: FA-QD-PS-01.docx Postselection Tradeoff Theorem A universal information bound for conditional environmental records Third research memorandum for the Draft 3 decision gate This block continues the irreducible-coupling investigation and converts the numerical postselection effect into a general finite-dimensional theorem. Draft 2 remains unchanged. Executive result A selected control outcome can have better fragment information than the unconditional record, but the probability-weighted improvement is bounded by the total control information available beyond the fragment, plus an explicit rarity term. The average postselection gain obeys a sharper identity and cannot exceed the control conditional mutual information. Thus coherent postselection redistributes information; it does not create unlimited objective information from nothing. Truth-First Instantiation and Information-Accounting Interface Authority and role. This successor is synchronized with Core Synthesis Draft-07, Basement Provenance and Instantiation Protocol Version 03, Local Relational Time Version 56, and the truth-first Local Relational Geometry architecture. This manuscript remains an interpretation-neutral Target III information theorem for finite-dimensional classical-quantum states and instruments on an already admitted control. It does not select a basement, derive physical instantiation, or derive definite events from a merely formal structure. • Relational potentiality remains a candidate basement, not an established premise or conclusion of this manuscript. • Physical systems, controls, channels, Hamiltonians, fragments, pointer alternatives, instruments, access structures, outcome registers, probabilities, and records are supplied physical or model inputs whose provenance must be declared. • Formal states, process matrices, commutators, entropies, mutual informations, Holevo quantities, causal witnesses, postselection identities, and numerical simulations do not establish physical instantiation. • Objectivity, common-record preservation, redundancy, recoverability, accessibility, causal nonseparability, operational boundaries, topology, and geometry remain distinct claims. • Target I concerns basement instantiation. Target II concerns definite-event generation. This paper addresses a bounded Target III reconstruction or certification question and does not establish Targets I or II. • Every conclusion remains indexed to its candidate class, complete carrier, witness or instrument family, selected output, access structure, comparator, tolerance, and analytic or computational status. Inherited Input Certificate and Claim Ceiling 1. Setting Let Lambda be a classical pointer label, F an environmental fragment, and T a quantum control. Let an arbitrary quantum instrument be applied to T, with classical outcome Y. The resulting classical-quantum state on Lambda, F, and Y defines p_y, the outcome probability, and chi_y=I(Lambda:F) conditioned on Y=y. Let chi=I(Lambda:F) before the control measurement. The theorem applies to finite-dimensional cq states and any instrument on T. It does not require the controlled-order model, causal nonseparability, or a specific pointer Hamiltonian. 2. Exact average-gain identity Theorem 1. Average postselection identity sum_y p_y chi_y - chi = I(Lambda:Y|F) - I(Lambda:Y). (1) Proof. The chain rule gives I(Lambda:FY)=I(Lambda:Y)+I(Lambda:F|Y) and also I(Lambda:FY)=I(Lambda:F)+I(Lambda:Y|F). For a classical outcome register, I(Lambda:F|Y)=sum_y p_y chi_y. Equating the two expansions proves (1). QED. Equation (1) shows that average conditional fragment information can increase only when the measurement outcome becomes more informative about Lambda after F is known than it is marginally. The subtraction of I(Lambda:Y) prevents an outcome label that directly reveals Lambda from being misinterpreted as an increase in the fragment record itself. 3. Universal average bound Theorem 2. Control-information budget sum_y p_y(chi_y-chi) <= I(Lambda:T|F). (2) Proof. Equation (1) gives the average gain as I(Lambda:Y|F)-I(Lambda:Y), which is at most I(Lambda:Y|F). Because Y is generated from T by a local channel, conditional data processing gives I(Lambda:Y|F)<=I(Lambda:T|F). QED. The right side is exactly the total information supplied by joint access to T and F beyond access to F alone: I(Lambda:T|F)=I(Lambda:TF)-I(Lambda:F)=Delta_label+Delta_coh. (3) Therefore the classical-label and coherence-sensitive bonuses derived in the first execution block form an information budget for the average conditional improvement. 4. Outcome-resolved tradeoff Theorem 3. Selected-outcome bound For every outcome y with probability p_y, p_y[chi_y-chi] <= I(Lambda:T|F) + (1-p_y)chi. (4) Proof. Let G be the average gain in (2). Because every conditional mutual information chi_z is nonnegative, chi_z-chi >= -chi. Hence the contribution from all outcomes z not equal to y is at least -(1-p_y)chi. Therefore p_y(chi_y-chi)<=G+(1-p_y)chi. Applying (2) proves (4). QED. Equation (4) formalizes the probability tradeoff. A rare selected outcome has a larger rarity allowance, while a high-probability improvement must be paid for primarily by the control-information budget. The inequality is universal but not always tight. 5. Immediate corollaries If I(Lambda:T|F)=0, the average postselection gain is nonpositive. A control carrying no conditional pointer information cannot improve environmental records on average. If chi=H(Lambda), no outcome can improve the fragment beyond the complete-record limit. If p_y=1, the selected-outcome gain is at most I(Lambda:T|F). Large unweighted conditional gains at very small p_y do not establish deterministic enhancement. Decomposing I(Lambda:T|F) into Delta_label and Delta_coh distinguishes classical branch information from coherence-sensitive information. 6. Relation to branch distinguishability and leakage The universal theorem does not require separate branch-distinguishability or pointer-leakage variables. Those quantities enter through model-dependent bounds on I(Lambda:T|F), chi, and the attainable postselected ensemble. A universal formula solely in terms of one branch distance and one leakage number would require precise metric definitions and additional assumptions; none is asserted here. For the irreducibly coupled model, pointer leakage and branch incompatibility determine the output cq state, which in turn determines I(Lambda:T|F). Thus the information budget is the invariant operational quantity, while branch distances and leakage can be used for sharper model-specific corollaries. 7. Numerical adversarial test The theorem was tested on both environmental fragments throughout the 41 by 41 irreducible-coupling grid. The dataset contains 3,362 fragment-parameter rows. For each row and both coherent-control outcomes, the calculation evaluated p_y, chi_y, chi, the weighted gain, the bound in (4), the average identity in (1), and the decomposition in (3). The maximum apparent violation of the selected-outcome bound was 3.65 times 10^-15 bits, consistent with floating-point rounding. The maximum average-identity error was approximately 1.23 times 10^-32 bits. The largest probability-weighted selected gain on the grid was 0.1607331472 bits. The largest unweighted selected gain was 0.1933890481 bits. The largest available total control-information budget was 0.5 bit. Figure 1. Probability-weighted selected gains remain below the universal theorem bound. Figure 2. Conditional information gain versus coherent-control outcome probability. 8. Strongest tested case The largest probability-weighted gain occurred for fragment E1 near theta=0.903208 radians and alpha=pi. The positive-control outcome had probability 0.845819, conditional gain 0.190032 bit, and weighted gain 0.160733 bit. The complementary outcome had negative gain. The total control-information budget at that point was 0.344059 bit, and the universal selected-outcome bound was 0.402617 bit. This case demonstrates why the average identity matters: the favorable outcome is balanced by a less informative complementary outcome. Postselection redistributes record quality across outcomes rather than uniformly improving every branch. 9. Physical interpretation The theorem separates three claims that are often conflated. First, a control measurement may reveal a classical label directly. Second, conditioning on the outcome may change the quality of the fragment ensemble. Third, coherent access may make additional information available before dephasing. Equation (1) removes the directly revealed outcome information, while equations (2) and (4) cap average and selected improvements by the control-information budget. An experiment should report the complete outcome distribution, every conditional fragment diagnostic, the unconditional fragment information, and I(Lambda:T|F). Reporting only the best postselected outcome can overstate the operational gain. 10. Draft 3 significance Together, the three execution blocks now support a coherent theorem sequence: General separation: objectivity and causal nonseparability are independent in a broad composite class. Access decomposition: fragment, classical-control, and coherence-sensitive information form a nonnegative hierarchy. Irreducible coupling: common branchwise pointer compatibility controls unconditional preservation. Postselection tradeoff: conditional fragment improvements are constrained by probability and the total control-information budget. The postselection theorem is suitable as a headline mathematical result because it turns the observed conditional enhancement into a universal operational law. Its novelty should be claimed as the controlled-objectivity application and decomposition, not as a new proof of data processing or strong subadditivity. 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. 11. Remaining highest-value target The next downstream target is a formally certified process-matrix example using the same admitted record-generating laboratories. This remains a Target III technical extension and is not the controlling route to basement instantiation. The upstream priority is the truth-first Layer 0 program. 12. Reproducibility and limitations Included: deterministic source code. Included: 3,362-row tradeoff dataset. Included: top weighted-gain cases. Included: two figure files and SHA-256 checksums. Not included: process-matrix causal-witness calculation. Not included: physical hardware data. Not included: ETP interpretation; the result remains interpretation neutral. 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. Postselection-composition boundary The exact average identity and selected-outcome information bounds remain unchanged. Postselected improvement is selected-output and outcome-register relative; it does not establish an unconditional composite resource, complete-process equivalence, physical realization, or objective event selection. 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 Postselection and Composite-Certification Addendum This postselection tradeoff theorem retains its stated finite-dimensional information-accounting scope. CP4.2 and CP4.3 require named success events, explicit denominators, positive success floors where normalized stability is claimed, branch-specific accounting, and fixed composite contexts. 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.