We introduce a contrast-based recoverability diagnostic for relativistic and mediated-entanglement experiments. The diagnostic compares a homogeneous reference spectrum with spectra obtained after controlled parameter variation, species change, hidden-mode access, finite-record feedback, or null intervention. In the minimal two-qubit setting, proper-time and transport effects are locally recoverable dressings. For $n$-constituent configuration sectors, one-round $k$-block recovery gives a scale-resolved spectrum. A diagonal phase is exactly recoverable at block size $k$ precisely when it factorizes over a partition whose blocks have size at most $k$. Homogeneous occupation shells provide the symmetry null; heterogeneous sectors test the factorization of the observed phase. The four-constituent, four-level case is the smallest compact example with nontrivial block hierarchy, shell leakage, and constituent ambiguity. A staged fermion--fermion (FF)/boson--fermion (BF) comparison separates ordinary parameter nonidentity from species and statistics constraints. Extensions to unused levels, redundant modules, and larger connected clusters remove specific diagnostic blind spots. A finite classical record is bounded by a record-overlap contraction theorem, while a coherent hidden mode is not. The framework organizes which observed phases, visibility losses, and mediated-entanglement witnesses are locally recoverable, block-recoverable, hidden-mode recoverable, or record-limited.
We introduce a contrast-based recoverability diagnostic for relativistic and mediated-entanglement experiments. The diagnostic compares a homogeneous reference spectrum with spectra obtained after controlled parameter variation, species change, hidden-mode access, finite-record feedback, or null intervention. In the minimal two-qubit setting, proper-time and transport effects are locally recoverable dressings. For n-constituent configuration sectors, one-round k-block recovery gives a scale-resolved spectrum. A diagonal phase is exactly recoverable at block size k precisely when it factorizes over a partition whose blocks have size at most k. Homogeneous occupation shells provide the symmetry null; heterogeneous sectors test the factorization of the observed phase. The four-constituent, four-level case is the smallest compact example with nontrivial block hierarchy, shell leakage, and constituent ambiguity. A staged fermion--fermion (FF)/boson--fermion (BF) comparison separates ordinary parameter nonidentity from species and statistics constraints. Extensions to unused levels, redundant modules, and larger connected clusters remove specific diagnostic blind spots. A finite classical record is bounded by a record-overlap contraction theorem, while a coherent hidden mode is not. The framework organizes which observed phases, visibility losses, and mediated-entanglement witnesses are locally recoverable, block-recoverable, hidden-mode recoverable, or record-limited.
This work establishes the Black-Hole Information Sector Connection of Return Theory (RT) in finite-dimensional quantum hosts. It applies RT’s typed observational roles—observer, host stage, validation reference, representation reference, access, record, and reconstruction—to black-hole information problems while retaining the supplied quantum dynamics. The central result is a three-level conditional classifier that locates an apparent conflict at its first failed level: host-state feasibility, same-run record and access compatibility, or equivalence between reconstructed images. In the Hayden–Preskill host, an old black hole is maximally entangled with early radiation, a one-qubit diary is scrambled by a fixed Haar unitary, and recovery is evaluated through two access channels on the same dynamics: emitted radiation R alone and the combined system ER. Classical diary content is assessed by exact binary minimax error, while coherent recovery is assessed separately by Petz entanglement fidelity. Across 36 fixed unitaries with total coherent register sizes of 8, 10, and 12 qubits, ER reaches both declared recovery thresholds after two emitted qubits in every run, whereas R reaches them only after complete emission. In the selected 12-qubit realization at k = 2, adding access to E changes the minimax error from 0.472087 to 0.031872 and the Petz fidelity from 0.252652 to 0.948306, with the unitary dynamics held fixed. The AMPS audit is evaluated under two explicit factorization declarations. With independent tensor factors E, b, and b̃, simultaneous exact Bell targets for Eb and bb̃ define an empty marginal-feasibility set. The conflict is therefore classified at the host-state level. When b̃ is instead declared as a logical algebra encoded in E by a [[4,1,2]] quantum error-correcting code, one consistent joint state exists and the issue moves to access, records, and reconstruction. Within the encoded realization, the disjoint logical-Z representatives Z₁Z₃ and Z₂Z₄ generate exact, order-independent records against a stable external validation reference and satisfy the declared RT operational-objectivity conditions on the binary logical-codeword ensemble. These objective classical records reduce logical coherence, while persistent coherent extraction transfers full logical fidelity to the first output and leaves only Bell fidelity 1/4 for a later E-only output. The calculation therefore distinguishes redundant objective records of commuting classical content from reconstruction of the complete logical quantum state. The finite connection further includes five modules: a sequential evaporation host that instantiates the RT Time role and distinguishes monotone admitted-record histories from observer accesses that lose records; an exact independent-factor Bell-fidelity frontier with symmetric optimum 3/4; a graded early-radiation-entanglement sweep that moves the ER and R recovery thresholds in opposite directions; a three-qutrit code whose two-share recoverability coexists with record redundancy one; and typing witnesses showing how verdicts change when minimax admission, joint realization, records, occupied order, or certified validation are replaced by inequivalent proxies. The resulting sector connection shows how factorization, observer-supported access, reference stability, record admission, and reconstruction jointly determine the consistent observational reading of a supplied black-hole information model. It provides formal statements, executable finite-dimensional witnesses, independent numerical verification, machine-readable results, complete source code, and cryptographic checksums.
We derive an explicit test of whether a reduced quantum-memory model overstates recoverability. For channels with orthogonal logical outputs in distinguishable sectors, the minimum recovery error follows directly from differences between environmental record amplitudes, without numerical recovery optimization. Applying this test to two-excitation oscillator–qubit and four-qubit encodings reveals that matching single-loss responses need not preserve finite-time recoverability. Under collective loss, the four-qubit encoding is perfectly recoverable, whereas oscillator emission records reveal logical information. The resulting unavoidable error initially grows as the cube of storage time. The test quantifies omitted-level effects and benchmarks reduced models and recovery algorithms.
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
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.
We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four σ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-σ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply PrL(fail)≤C∣ΩL∣e−cL below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.
This record contains aexpanded research-program manuscript and reproducible companion package for a finite-dimensional framework connecting recoverability frustration, entropy-loss calculus, observer screening, path consistency, higher-order memory diagnostics, and quantum statistical comparison in quantum channel networks. The work separates microscopic recoverability from observer-visible distinguishability. The formal core includes a four-level ladder between visible defects, microscopic recovery defects, entropic proxies, and entropy-loss actions; structural tests for post-processing monotonicity, hidden convex channel mixing, tensor-product factorization, continuity, path consistency, and observer monotonicity; and a probabilistic observer layer based on Blackwell garbling, forward Le Cam deficiency, visible Kullback-Leibler rates, and visible Fisher information. The accompanying code regenerates figures, validation summaries, and theorem-level regression tests.