Fragment-size averages of the pointer Holevo information summarize how quantum environmental records are accessed without specifying their microscopic locations. We give a focused N=4 non-identifiability result: assuming the previously archived interval-certified root for a four-qubit two-branch witness, a second, two-carrier pure-branch ensemble exists with exactly the same complete size-averaged Holevo curve (A1, A2, A3, A4). The key additional step is a rational-interval verification of g(c) < 4A1 < 2g(sqrt(c)), where g(t) = h2((1+t)/2) and c is the overlap of the original branches. Continuity then provides the matching two-carrier ensemble, and the support-projection identity forces agreement at the remaining fragment sizes. Independent 16-dimensional state-vector calculations reproduce the matching averaged curves to floating-point precision; these diagnostics are not used as a substitute for the existence proof. The result concerns one certified curve in an equiprobable two-branch pure-state model, not equality of every individual fragment's information, an open family of curves, or arbitrary environment sizes. The original Krawczyk certificate is a documented imported premise and was not rerun in the present extension audit. Unrefereed preprint. Scientific claims are conditional on the archived original OBS01 certified-root premise. Supporting files provide numerical and interval checks, recorded base certificates and their hashes. Base Krawczyk/Arb computations were not rerun in this extension. This is not a claim of global priority or experimental realization. Manuscript, original notes and data are CC BY 4.0; original Python code is MIT; embedded archived materials retain their applicable component notices and rights. AI-assisted work is disclosed in the manuscript.
Fragment-size averages of the pointer Holevo information summarize how quantum environmental records are accessed without specifying their microscopic locations. We give a focused N=4 non-identifiability result: assuming the previously archived interval-certified root for a four-qubit two-branch witness, a second, two-carrier pure-branch ensemble exists with exactly the same complete size-averaged Holevo curve (A1, A2, A3, A4). The key additional step is a rational-interval verification of g(c) < 4A1 < 2g(sqrt(c)), where g(t) = h2((1+t)/2) and c is the overlap of the original branches. Continuity then provides the matching two-carrier ensemble, and the support-projection identity forces agreement at the remaining fragment sizes. Independent 16-dimensional state-vector calculations reproduce the matching averaged curves to floating-point precision; these diagnostics are not used as a substitute for the existence proof. The result concerns one certified curve in an equiprobable two-branch pure-state model, not equality of every individual fragment's information, an open family of curves, or arbitrary environment sizes. The original Krawczyk certificate is a documented imported premise and was not rerun in the present extension audit. Unrefereed preprint. Scientific claims are conditional on the archived original OBS01 certified-root premise. Supporting files provide numerical and interval checks, recorded base certificates and their hashes. Base Krawczyk/Arb computations were not rerun in this extension. This is not a claim of global priority or experimental realization. Manuscript, original notes and data are CC BY 4.0; original Python code is MIT; embedded archived materials retain their applicable component notices and rights. AI-assisted work is disclosed in the manuscript.
Correction in version 1.1 (2026-10-18) The Decimal/Fraction interval engine distributed with version 1.0 used context-sensitive Decimal unary negation and absolute value on interval endpoints. These operations can round inward, so the recorded P02G certificate in v1.0 should not be treated as a rigorously outward-rounded enclosure. Version 1.1 repairs those operations using exact copy_negate() and copy_abs(), hardens the second interval implementation's row-bound selection, and regenerates the interval reports, checksums, and reproducibility manifest. The certified rectangle, theorem scope, and all 34 mathematical environments in the manuscript remain unchanged. Repaired P02G and the separately implemented mpmath.iv P02J both pass the original theorem gates. The result remains conditional computer-assisted mathematical evidence, not formal machine verification or external peer review. Previous version (v1.0): https://doi.org/10.5281/zenodo.23438614 This unrefereed research preprint establishes a computer-assisted, quantitatively certified non-identifiability result for fragment-size-averaged Holevo information in an N=4 environment with two equiprobable pure pointer-conditioned branches. For every point in an explicitly specified closed rational rectangle of coordinate area 4 × 10^-14, two realizations of the complete averaged Holevo information curve exist: one with minimum pure-common-rest carrier size two and another with carrier size four. The proof combines a uniform parameterized contraction mapping, directed interval arithmetic with a second interval implementation, and an intermediate-value construction. Reproducibility materials include the certification programs, frozen numerical certificates, SHA-256 checksums, and a standalone 200-state numerical comparison against direct 16-dimensional partial traces. Both interval implementations share the underlying analytic Holevo reduction and do not constitute independent symbolic proofs. The conclusions are restricted to N=4, equiprobable pure branches, fragment-size-averaged Holevo information, and the specified carrier definition. They do not establish non-identifiability for arbitrary N, mixed states, unequal priors, or individual labeled-fragment information. This is an unrefereed preprint, not a formally verified or peer-reviewed mathematical theorem. Comprehensive novelty priority is not claimed. Licensing: The manuscript and original documentation/data are provided under CC BY 4.0 where applicable. Original Python research code is provided under the MIT License. The accompanying reproducibility archive specifies licenses on a per-file basis. AI assistance disclosure: OpenAI ChatGPT assisted with mathematical analysis, Python implementation, numerical and interval verification, manuscript preparation, and quality assurance. Claude contributed critical assessment and adversarial review assistance. Neither AI system is a coauthor or an external scientific referee. The human author, Yoichi Tsujisawa, assumes responsibility for the research and its publication.
Fragment-size-averaged Holevo-information curves omit the identities of environmental fragments. We construct a restricted four-qubit example in which two equal-prior orthogonal pure-branch ensembles have exactly the same fragment-size-averaged Holevo curve (0, 1/6, 1/2, 1) and all four labeled singleton Holevo values equal to zero, while their minimum pure-common-rest carrier sizes differ (2 versus 4). Subject to the correctness of a reduced entropy formula and validated directed interval arithmetic, this coincidence persists for every choice of six raw amplitude coordinates in an explicit product box, with a unique two-coordinate correction inside a common root box. The solution family is open in the six free coordinates, not in the full coefficient or observational space. Analytic exact-rational sign bounds show that the two ensembles are separated by the pattern of labeled two-site fragment information (one positive pair versus three). Combining quantum Pinsker's inequality and an exact-rational simplex bound yields a permutation-invariant second-moment gap larger than 0.6128 bit², conditional on the existence certificate. The release includes the manuscript source, exact reports, four frozen phase source packages, original figure data, checksums, code, and AI-assistance disclosure. This is an unreviewed preprint, not a theorem about general carrier estimation, operational quantum data hiding, local indistinguishability, or finite-shot measurements. Independent priority assessment and comprehensive external peer review remain outstanding.
Environmental fragments provide indirect access to pointer information, but fragment-averaged accessibility need not determine where microscopic branch dependence resides. For an environment of N subsystems, we quantify fragment pointer information by the Holevo quantity, which upper-bounds accessible classical information, and average it over fragments of equal size. Assuming two pure pointer-conditioned environmental branches share an identical, branch-independent rest outside a carrier, we establish an exact support-projection identity and a hypergeometric formula for the complete averaged accessibility curve. A carrier of size at most two is sufficient for a finite-N order-2 constraint; for N = 4, this yields A2 = A4/6 + 4A1/3 and A3 = A4/2 + A1. The converse fails in general: an interval-certified, computer-assisted N = 4 state-space construction satisfies these relations while having minimal carrier size four. We distinguish this distributed state-space witness from realizations in the tested five-qubit controlled-gate circuits. Using the pre-operation conditional branch state and the known gate, a prospectively frozen mechanism catalog correctly classified 417/417 protocol-defined, outcome-conditioned scored events at theta = pi/2; 28 post-R0 events among 445 eligible instances were handled separately. Paired sampled-parameter tests classified 320/320 scored events at each of theta = pi/3 and 2pi/3; these paired evaluations are not independent circuit replications. A separate retrospective numerical census classified 70/70 near-zero-residual R0 events as low-carrier within the sampled circuit population. Finally, an exact product-local preservation lemma supplies a sufficient condition for invariance of fragment accessibility, not a complete characterization. Together, these results separate exact operational constraints, nonunique microscopic realizations, and circuit-specific dynamical selection.
This record contains the manuscript and complete verification artifact for the Bell inequality J62_4422 of Pál and Vértesi.It certifies the exact rational bracket750075557472719639 / 10^18 ≤ β_Q ≤ 750091086083765047288227625 / 10^27,and exact attainment of 3/4 by the stated two-qubit strategy. The lower endpoint is an exact rational strategy. The upper endpoint is certified by a sum-of-squares chain for the retained 365-word relaxation level.The Krawczyk certificate localizes the underlying KKT zero and establishes a strict local maximum, including negativity of the border form in all 256 boundary strata; it does not establish global optimality of that localized point. Accordingly, this work does not claim the exact quantum value, qubit-restricted optimality, minimum optimizer dimension, or global optimality of the localized KKT point.The archive includes SHA-256 manifests, standard-library replay programs, independently written certificate checks, mutation controls, and pinned full Krawczyk replays. The acceptance programs are fail-closed under optimized Python. The manuscript and certificates are licensed CC BY 4.0; verification code is licensed MIT.Version 1.1 is a hygiene-only update: it replaces one machine-local default path in a helper with the portable repository root ".", updates the corresponding checker and certificate source bindings, and leaves the mathematical calculation, inputs, claims, and replay outputs unchanged.
Abstract Decoherence suppresses interference, but an observer’s ability to reconstruct a classical pointer value depends on additional resources: distinguishable conditional environmental states, an allowed measurement, sufficient accessible fragments, and sufficiently independent records. We provide an explicit minimal quantum-to-operational bridge that places these requirements in one chain. A system qubit is correlated with conditional environmental qubit states, permitting exact expressions for residual coherence, quantum mutual information, pointer Holevo information, accessible information, and Helstrom discrimination error. Full density-matrix calculations validate the formulas across 280 endpoint and interior cases. An accessible-sector/hidden-bath construction shows that coherence can fall below 2.3 × 10⁻⁵ while the observer’s record information and error remain fixed at 0.0803 bits and 0.4009. Measurement policies shift recovery thresholds: for record overlap 0.8 and error tolerance 0.05, collective or optimized adaptive-local discrimination requires four fragments, a fixed canonical-basis measurement requires six, and repeated one-fragment Helstrom measurements followed by classical decoding require seven. More importantly, records matched at the one-fragment level can have radically different collective behavior. For a binary pointer, the common-mode classical channel has an exact accessible-information ceiling of 1 − h₂(0.1) = 0.5310 bits, while the noncommuting hidden-memory ensemble approaches the same Holevo-information ceiling; matched independent records approach one bit and zero error. Disjoint observer counts and unanimous agreement likewise depend on shared error structure. Finally, declared local measurements induce classical record channels that are subjected to held-out model selection, predictive checks, abstention, and simultaneous finite-sample threshold certification. The resulting framework supplies the validated quantum-to-operational record model used in the companion analysis of operational capacity bounds for distinguishable decoherent histories.
Calibrated support observations can constrain a complete quantum instrument without identifying its action by unrestricted tomography. We study what those observations guarantee when the noise law is not supplied. On a fixed four-dimensional incidence-record experiment, we define the set of all complete two-outcome completely positive instruments compatible with the exact raw rejection and outcome probabilities, retaining the classical flags. Exact ideal data identify the target, but an explicit family of complete instruments reproduces the nonzero-data category laws of a low-error contamination device while admitting coherent error of square-root order. A target-ray mass certificate and a common complete-dilation comparison give a universal upper, while an exactly feasible rational instrument supplies a stronger lower witness. At the diagnostic parameter r = 1/300000000, the unhalved target-relative worst-case radius lies between 0.000258065115532250 and 0.000951566242566. The result separates avoidable inequality conservatism from information absent from the measured data. Fixed-functional and fixed-map optimizations have separately certified scopes; they do not solve the global radius. The conclusions are conditional on the supplied carrier, calibration and common-device model. They do not establish an apparatus, device-independent certification, a finite-sample power guarantee or complete formal verification. Selected supporting declarations have one recorded successful Lean 4.19.0 macOS execution and declaration-specific source, type and axiom review; this is not an end-to-end formalization.