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.
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.
We distinguish complete unitary preservation, subsystem quantum recovery, destructive observation, and operational discard in a reference-preserving channel formulation. A finite acyclic typed fragment represents allocation, specified unitary transformations, destructive computational readout, and classical forgetting. Quantum recovery returns a quantum register; terminal readout returns a classical record and is a different task. For a complete isometry V from C to R tensor B, we give an explicit global recovery channel and prove that exact recovery from R is equivalent to constancy of the complete complementary channel to B. A process retaining only classical outcomes cannot reconstruct an arbitrary d-dimensional quantum code with half-diamond error below 1 − 1/d. Unobserved readout of B, followed by discarding its outcomes, is exactly partial trace over B and leaves the radiation channel unchanged. Its coherent dilation retains the information omitted from the classical interface. Independently, a finite-band quartic source–scalar-reservoir Hamiltonian is constructed and approximated by finite-dimensional proxies with explicit complete-channel allowances. Separate qubit dictionaries preserve the accessible–inaccessible partition. For a commuting four-Majorana control, the exact radiation recovery error is (3 − √(1 − c_R(t)²))/4, the residual recovery error is (1 − c_R(t))/2, and the complete residual leakage is the maximum of 1/2 and (1 + 2c_R(t))/4. Thus, the complete output is reversible although neither output alone need recover the diary. Access-preserving endpoint continuity and fixed-proxy recurrence are proved separately from syntactic termination. The gravitational implication remains conditional on a same-model unitary dilation, outgoing access construction, and constant endpoint residual. No positive noncommuting recovery window, physical field-to-memory receiver, quantum-gravity unitarity theorem, or gravitational evaporation endpoint is established.
We construct a supported local spacetime adaptation of the observation-boundary contract of VII-E and the finite hybrid-channel contract of VII-G. The two contracts are proved independently within one laboratory on I-B’s prescribed conical Lorentzian product. For the observation boundary, we prove the effect-pullback criterion, its reference-stable form, the exact sharp-measurement coherence criterion, and a qubit separation of trace distance 1/2 between ideal residual targets with identical records. For hybrid evolution, we prove a complete branch representation, history-sensitive observation closure, spectator consistency, and the exact recovery–constant-complement criterion. An informative classical record consequently excludes recovery of every unknown quantum input, even when the declared residual is also available. The additional contribution is an end-to-end finite observation package that keeps these distinctions visible in supplied spacetime. We provide a compactly supported 3+1-dimensional lift of a four-face transverse adjacency exchange, an exact guarded reversible encoding, and a separately constructed qubit instrument selecting strictly ordered slices without changing the conical metric. A two-step specialization has two histories with the same terminal slice but different conditional residuals. After coarse aggregation, the record remains unchanged while two complete residual targets have the exact half-diamond gap 4(1 − 1/√2)/25. An explicit residual audit enlarges the required input observation space from the diagonal algebra to the full qubit algebra. A destructive finite clock, complete-channel approximation bounds, and a causal route for both payload and classical control complete the local Grade S construction. The package may serve as a precise testbed for distinguishing record-level claims from instrument-level claims in geometric feedback models. The channel and recovery principles themselves are established finite-dimensional facts; the claimed contribution is their supported assembly and the explicit certificates and separations. No metric emergence, gravitational dynamics, field instrument, radiative coefficient, or quantum Ward identity is constructed.
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.
This preprint studies a cycle-reflection operation on a supplied six-qubit realization of paired trigram configurations. A four-matched-pulse identity and a minimum within its stated serial grammar are separated from an ordinary-exchange endpoint construction. The raw full-source reflection and its full domain-safe alternative are different tasks: the latter is excluded by a collective-symmetry obstruction in the admitted source-only exchange model. Strict preservation of the addressable sector throughout the path is also excluded for the full task, while finite-action lower bounds quantify peak no-address population and time-integrated exposure. The paper presents an explicit low-excursion balanced word and a conditional joint clock–amplitude error certificate. Root settings and trajectory inputs remain tied to their supplied computer-assisted certificates. The second internal review reproduced all ten advertised numerical routes and twenty-four outputs using the original programs; this is same-program reproducibility, not independent implementation or external peer review. A separate Lean supplement covers the finite raw/safe operator algebra and the obstruction for arbitrary finite words of the declared algebraic factors. It does not formalize the numerical roots, matrix exponentials, continuous-time dynamics or resource theorems. No global pulse optimum, measured hardware feasibility, general noise tolerance, or automatic connection to the companion recovery code 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.
Abstract Decoherent-histories methods identify time-extended alternatives whose diagonal weights may support probabilistic interpretation, while environmental-record approaches ask which alternatives a specified observer can recover. We formulate their intersection as a finite branch-record packing problem generated from one cross-branch construction. A codebook is admissible only if its declared coarse-graining obeys a bounded probability-sum-rule error and its accessible states can be decoded under a specified observer policy with finite error. Uniform reweighting defines a geometric packing number; a communication interpretation additionally requires a preparation instrument. Retrospective resolution is stricter: the base history family is exhaustive, selected histories must cover sufficient physical diagonal weight and be individually recoverable, and the same declared quantum-to-classical measurement channel used for decoding must preserve approximate consistency after histories are refined by its registered outcomes. This measurement-refined condition quantitatively controls the difference between the actual outcome distribution and the normalized diagonal classical-quantum prediction. For uniform branch-record packing and its communication realizations, general converses follow from accessible dimension, fixed outcome count, Holevo-Fano information, and hypothesis testing. Exact models quantify record compression, access, memory, measurement policy, temporal dependence, and shared latent errors. In a jointly realized four-bit benchmark, scalar consistency and recovery admit six histories, whereas direct measurement-refined consistency under the declared local Helstrom instrument admits four at the same tolerances. Finite-sample procedures distinguish held-out recovery achievability, structure-conditional consistency certification, structural converses, independently constructed parameter intervals, and abstention.