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 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 derivationally self-contained sector connection instantiates Return Theory (RT): Foundations 1.0.0 (doi:10.5281/zenodo.23286579) in finite-dimensional quantum-record systems. It studies a friend–Wigner laboratory, general two-branch systems, and an exactly solvable branching environment. Quantum mechanics remains the unchanged host law. The RT connection explicitly separates candidate generation, record admission, validation reference, representation reference, observer access, image fibres, inter-observer agreement, and operational objectivity. The release establishes four mathematical results: complementarity for one access, complementarity for two disjoint systems, agreement bounds for jointly realised admitted records, and the equal-tolerance threshold for disjoint content and phase observers. Nine host-law consequences include the friend’s exact content record, Wigner’s exact phase record and content non-identifiability, persistence of internal agreement with loss relative to an outside record, the exact cost of a partial inter-observer copy, agreement without a record, fragment admission and redundancy, phase exclusion from fragments, the classical mutual-information plateau, and an overlapping-observer code construction. The discriminating RT-connection prediction class is explicitly empty in version 1.0.0: every observable distribution in the declared systems is fixed by quantum theory together with the stated preparations, channels, and measurements. Four prospective protocols, V1–V4, are therefore classified as tests of host-law consequences and calibrations rather than as independent RT physical predictions. The reproducibility package contains the release PDF and LaTeX source, exact state-vector code, fixed-seed validation calculations, JSON results, generated tables and figures, machine-readable B1–B10 declarations, the three-way result ledger, checksums, and release-audit scripts. No external data are required. This release continues the earlier public Quantum Records record (doi:10.5281/zenodo.23282709), which is cited only for lineage and timestamping and is not used as a premise.
This document connects Return Theory (RT) to quantum mechanics through two standard settings: Wigner's friend and quantum Darwinism. Each setting is read with RT's declared roles: record, reference, access and reconstruction. The document does not choose an interpretation of quantum mechanics and does not claim to solve the measurement problem. It states, in operational terms, what each observer can reconstruct, which reference validates that reconstruction, and when two observers can agree. Main results: • Content and phase are different records with different references. Wigner's coherence access gives no information about the friend's content (error 1/2). It records the relative phase exactly. • Record–interference complementarity (one access): (1−2ε₁)² + (1−2ε₂)² ≤ 1. Equality holds for projective measurements in the plane of the two tasks. • Two-system bound: if one system holds an ε₁-record of the content, any access to another system reads the phase with error ε₂, where 1−2ε₂ ≤ 2|αβ|·√(1−(1−2ε₁)²). • Agreement between observers follows from records when both are realised jointly in one run. Agreement alone is not a record (Bell-pair example). • Operational objectivity is defined by four conditions. In a branching environment, fragment errors are ε_f = ½(1−√(1−c^{2f})), and redundancy R_ε ≥ 2 is required. Mutual information shows a classical plateau at the pointer entropy. • Complementary tolerances for disjoint observers have a threshold of ½(1−1/√2) ≈ 0.146. A [[4,2,2]] code example shows task-separate (not simultaneous) objectivity of content and phase. All numbers in the text are generated from code (Python, exact state vectors). The package reproduces the JSON results and the generated TeX byte-for-byte. Figures may differ slightly across Matplotlib versions. Prior work (Wigner, Frauchiger–Renner, Brukner, Zurek, Englert, and others) is cited for comparison. Interactive language-model tools were used in preparing the text and code. The author is responsible for all content.
This sector-connection document instantiates the admitted-record order of Return Theory (RT): Foundations 1.0.0 (doi:10.5281/zenodo.23286579) in reversible classical and quantum memory systems. It uses no result from another RT sector connection as a premise. Canonical sector DOI: 10.5281/zenodo.23291979. The connection is developed in five declared settings: one classical observer on a Kac ring, several observers with a witness channel, quantum CNOT records, open memories supplied by a readiness reservoir, and noisy approximate records with redundancy and refresh. The package contains full proofs, executable models, fixed-seed numerical checks, machine-readable B1–B10 boundary declarations, and a three-way result ledger. The ledger separates ten mathematical results from twelve consequences of the declared host laws and explicitly reports that the discriminating-prediction class is empty in version 1.0.0. The mathematical results cover entropy bridges for controlled and general write channels, finite record capacity, exact and noisy occupancy registration, readiness as distinguishability, approximate-admission bounds, entropy headroom and preparation cost, redundancy, and noise-limited orientation. The host-law results show, among other things, two record directions away from one low-entropy instant, the absence of an order for a random memory, the failure of purity or low entropy to guarantee readiness, the boundary dependence of forgetting, and the warm-reservoir threshold. Four prospective validation protocols test: (V1) readiness rather than purity; (V2) conditional agreement q, exact validity q², and the threshold q ≥ 1 − ε; (V3) loss of memory-only inclusion under forgetting while inclusion persists on the memory-plus-sink boundary; and (V4) the false-conflict law η²ⁿ for a noisy two-snapshot witness. These protocols test the declared host laws, independence assumptions, and calibrations; they are not claimed as physical predictions supplied independently by the RT bridge. This release is a Foundation-aligned restatement and extension of the earlier time-direction sector record (doi:10.5281/zenodo.23277154). That record is cited only for public lineage and timestamping, not as a premise. The executable models and numerical results are retained byte for byte. The reproducibility package includes the release PDF and LaTeX source, Python code, JSON outputs, figures, generated tables, environment and integrity manifests, release-audit scripts, and source checksums. No external data are required.
We derive a causal law of observer-indexed entropy retrieval: retrieved reference information grows toward the supply available to a finite observer. A recorded admission-and-readout instrument produces orthogonal operational sectors from arbitrary coherent radiation; exact diary fidelity and decoupling give the flagged family, while branch errors bound its approximate realization. An exact quantum transfer channel yields the remaining-gap law and its continuous Lindblad evolution. The channel has an explicit unitary collision realization: admitted reference information moves from radiation to memory, while the capture record remains diary-blind. Its memory marginal is the Choi state of an erasure channel, giving an intrinsic positive-coherent-information boundary at retrieved fraction r = 1/2. A strip-holomorphic modular speed limit bounds how rapidly the channel can activate, with the hyperbolic tangent selected uniquely at extremality. The integrated retrieval dose is invariant under a change of clock, making the Kerr lapse cancellation one geometric instance of a general result. At fixed supply and kinetic rate under an affine clock, the transfer and activation laws compose into a closed proper-time retrieval trajectory on the equality branch, identify that trajectory as the sharp fastest-retrieval envelope, and fix the earliest admissible retrieval horizon. In the continuum formulation, Araki relative entropy defines retrieval on restricted observable algebras, and a split Type-I factor provides the regulated density-operator representation. The experimental program follows the same causal sequence through retrospective analysis and staged intervention. Tier 0A determines whether archived analog-horizon data retain the shot-level, timing, protocol, and processing information needed to ask the retrieval question. Tier 0B then tests whether qualifying data contain observer-, protocol-, readout-, or analysis-operator-indexed correlation structure that standard pooling or filtering suppresses. A positive Tier 0B result identifies retrieval-relevant structure and motivates Tier 1; it does not by itself confirm the complete law. The prospective Tier 1 BEC program targets the held-out g²(t₁,t₂) surface, bandwidth scaling, and protocol dependence. Independent activation readout tests the modular bound and tanh rigidity. Reference-tagged supply and memory test cumulative dose, composite-hazard recovery, degraded-channel ordering, and two-memory allocation. The v2 verification artifact provides synthetic benchmarks and numerical consistency checks at bond dimensions D = 4 and D = 8 for the transfer law, inverse recovery, finite-resolution robustness, and matched adversarial alternatives. Standard RT/HRT geometry and a separate accessibility functional place the retrieval dynamics within holographic language. The retrieval law closes the observer-local path between global encoding and retained access. Information conservation, radiation encoding, causal collection, finite-observer retrieval, formal reconstruction, and comprehension are distinct physical operations. The black-hole information problem contains a proper-time access dynamics that global entropy accounting alone does not determine. Physical complexity enters through encoding, supply, clock, activation, transfer, retention, and observation. Those layers locate each source of retrieval failure instead of idealizing it away. The theory orders the encoding, supply, and retrieval clocks; distinguishes late supply from early supply through their overlap with future transfer dose; and gives an exact finite-lifetime retrieval criterion. Here rᵢ(τ) is the normalized reference information retained by observer i, eᵢ(τ) is the causally available supply, kᵢ(τ) is the composite access hazard, and τ_char is the characteristic activation scale.
This record deposits the frozen scientific protocol for a standalone Return Theory (RT) Sector Connection in constrained quantum cosmology. The declared hosts are a spatially flat FLRW universe sourced by a free massless scalar field, quantized in matched Wheeler–DeWitt and solvable loop quantum cosmology (sLQC) forms. The protocol keeps four structures distinct: the host constraint and physical inner product; the assignment of probabilities through consistent or decoherent histories; explicit memory dilations producing accessible and validated records; and the representation reference and reconstruction map used to form an observer image. It thereby provides a typed implementation of observer, relational clock, record order, reference, access, validation, and reconstruction without modifying either host dynamics. The frozen design addresses three central non-definitional questions: whether a time-symmetric sLQC geometry combined with different physical memory-readiness conditions produces different admitted observer images; whether an observer can construct a globally usable volume reference across the resolved bounce from its own admitted branch records; and how a declared relational monitoring channel changes the sLQC bounce-history probability relative to the unmonitored host, including null, ordinary, and Zeno-like outcomes. Results are classified at four distinct levels: host/constraint, history/decoherence, record/access, and image/reference. The package contains the complete human-readable protocol, machine-readable scientific freeze card, frozen state families, instruments, clock windows, volume bins, thresholds, Wheeler–DeWitt/sLQC matching rules, numerical tolerances, risk and source ledgers, engineering-pilot contract and results, deterministic consistency audits, validators, and the external execution gate. In the Chimera implementation, sLQC remains the host: the exact sLQC difference operator is used on the inner lattice, while the large-volume Wheeler–DeWitt tail serves as a bounded-error numerical approximation. This record reports no authoritative scientific outcome. Its purpose is to establish a public and independently verifiable timestamp for all scientific choices before authoritative execution. Authoritative frozen archive: RT_Constrained_Quantum_Cosmology_Sector_Frozen_0.2.1.zip SHA-256: 250f8ee99ac7188568e363616f8f9649583eb70157f19840e6af727992c839d0 Authoritative execution is permitted only after publication of this timestamped record and creation of a later atomic execution-start record.
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.