AbstractReplicated environmental records are often summarized by mutual information or by the number of fragments required to recover a target variable. Both summaries can be misleading when the observed joint alphabet is sparse, errors are dependent, or a selected channel model is used without predictive checking. We analyze a classical replicated-record framework motivated by quantum Darwinism. If every observed full record tuple is unique, plug-in mutual information equals empirical history entropy even at a zero-capacity endpoint. Miller–Madow correction retains the structural failure, whereas permutation subtraction removes spurious information at p=0.5 but can also remove genuine information as informative tuples become sample identifiers. Exact repeated-channel calculations provide a controlled baseline: at whole-history error tolerance 0.10, the IID threshold is 7 fragments for p=0.20 and M=3, 2,065 for p=0.48 and M=3, and 5,905 for p=0.48 and M=100. We distinguish population recovery from finite-sample certification and collective recovery from disjoint operational redundancy. In the exact IID control with K=48, six disjoint seven-fragment groups are population-sufficient, with six fragments unused. End-to-end access uncertainty was evaluated with 240 independently regenerated matrix-and-subset replications per family. The candidate library contains six explicitly defined families, including an interleaved four-group latent model distinct from consecutive blocks. Predictive cutoffs were calibrated from 220 of 300 independent null pipelines per family and evaluated on the remaining 80; each pipeline used 20,000 predictive-reference rows and 1,999 correlated Gaussian max-statistic draws. There were 19 rejections among 480 held-out null pipelines; family-specific counts ranged from 2/80 to 5/80. In 60 repetitions of each of six unseen in-family configurations, structural selection was correct in every run and 3.89% were rejected and abstained. Among 231 reachable, non-abstained runs, mean absolute threshold error was 0.31 fragments (median 0; 90th percentile 1; maximum 3). Predictive checks rejected 70% of 400 runs from ten out-of-family alternatives. Of 120 false negatives, 108 produced the exact reference threshold, seven differed by one fragment, and five by two; none reversed reachability. Near p=0.5, mapped confidence sets retained nominal coverage but remained wide. Recovery claims therefore require a controlled estimator, explicit access policy, dependence-aware scoring, separate population and certified quantities, calibrated abstention, and propagated threshold uncertainty. Keywords: replicated records; sparse alphabets; mutual information; correlated channels; recovery thresholds; redundancy certification; predictive checks; quantum Darwinism
AbstractReplicated environmental records are often summarized by mutual information or by the number of fragments required to recover a target variable. Both summaries can be misleading when the observed joint alphabet is sparse, errors are dependent, or a selected channel model is used without predictive checking. We analyze a classical replicated-record framework motivated by quantum Darwinism. If every observed full record tuple is unique, plug-in mutual information equals empirical history entropy even at a zero-capacity endpoint. Miller–Madow correction retains the structural failure, whereas permutation subtraction removes spurious information at p=0.5 but can also remove genuine information as informative tuples become sample identifiers. Exact repeated-channel calculations provide a controlled baseline: at whole-history error tolerance 0.10, the IID threshold is 7 fragments for p=0.20 and M=3, 2,065 for p=0.48 and M=3, and 5,905 for p=0.48 and M=100. We distinguish population recovery from finite-sample certification and collective recovery from disjoint operational redundancy. In the exact IID control with K=48, six disjoint seven-fragment groups are population-sufficient, with six fragments unused. End-to-end access uncertainty was evaluated with 240 independently regenerated matrix-and-subset replications per family. The candidate library contains six explicitly defined families, including an interleaved four-group latent model distinct from consecutive blocks. Predictive cutoffs were calibrated from 220 of 300 independent null pipelines per family and evaluated on the remaining 80; each pipeline used 20,000 predictive-reference rows and 1,999 correlated Gaussian max-statistic draws. There were 19 rejections among 480 held-out null pipelines; family-specific counts ranged from 2/80 to 5/80. In 60 repetitions of each of six unseen in-family configurations, structural selection was correct in every run and 3.89% were rejected and abstained. Among 231 reachable, non-abstained runs, mean absolute threshold error was 0.31 fragments (median 0; 90th percentile 1; maximum 3). Predictive checks rejected 70% of 400 runs from ten out-of-family alternatives. Of 120 false negatives, 108 produced the exact reference threshold, seven differed by one fragment, and five by two; none reversed reachability. Near p=0.5, mapped confidence sets retained nominal coverage but remained wide. Recovery claims therefore require a controlled estimator, explicit access policy, dependence-aware scoring, separate population and certified quantities, calibrated abstention, and propagated threshold uncertainty. Keywords: replicated records; sparse alphabets; mutual information; correlated channels; recovery thresholds; redundancy certification; predictive checks; quantum Darwinism
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.
What does a rising redundancy curve actually prove? This deposit accompanies “Raw-Fragment Benchmarking for Redundancy Tests in Quantum Darwinism and Spectrum Broadcast Structure,” a paper built around a practical but often hidden problem in tests of quantum objectivity. Aggregate mutual-information curves can show that information grows with fragment size, but they do not reveal whether independent fragments really carry accessible, mutually consistent records of the same pointer variable. The missing evidence lives one level lower: in the raw, shot-indexed fragment records. The paper turns that gap into a reproducible benchmark. Starting from a fixed synthetic binary pointer-copy model with 100,000 shots and eight heterogeneous fragments, it asks what a raw-fragment evidence package should expose before redundancy, consensus, or Spectrum Broadcast Structure-style objectivity is inferred. The analysis enumerates all fragment subsets, separates mean-over-subsets and all-subsets threshold crossings, measures fragment disagreement, reproduces analytic binary-channel threshold maps, applies coupling-breaking negative controls, and stress-tests the result against hidden common-mode correlated errors. The resulting picture is sharper than an aggregate curve. In the baseline model, the 0.9 information-fraction threshold is crossed at subset size m=3 under the mean criterion, but only at m=4 under the stricter all-subsets criterion. Negative controls remain below threshold, while a hidden common-mode flip exposes the main fragility axis: many agreeing fragments can still fail as evidence for independent records if they share the same unobserved error source. The attached PDF gives the manuscript-level argument and results. The arXiv source package supports manuscript reproduction, and the ancillary benchmark archive contains the synthetic raw-shot data, subset diagnostics, disagreement tables, threshold maps, negative-control outputs, common-mode sensitivity maps, figures, scripts, manifests, and checksums. The scope is deliberately bounded. This is a synthetic/model-level benchmark and reproducibility package, not an experimental claim that Quantum Darwinism or Spectrum Broadcast Structure objectivity has been observed. Its purpose is to make explicit what future experimental raw-fragment evidence should provide before a plotted redundancy signature is treated as a claim of objective, independently accessible records.
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.
Quantum Darwinism explains effective classical objectivity through the selective proliferation of information about preferred quantum states into multiple environmental fragments. Its mature formulations operate within quantum theory, using a system-environment decomposition, decoherence or einselection, and information-theoretic or broadcast-structure criteria for redundant accessibility.This paper asks an upstream question from Registration Theory (RT): what makes a physical distinction a record before that record is described as information available to an observer? Canonical RT answers that retained registration is persistent organization constraining later evolution, that a distinction becomes a physical record only when it remains consequential, and that persistent physical organization is prior to information extracted from it.The paper assembles a theorem-backed finite-model chain now available in the RT corpus. Canonical ACRF dynamics supplies write-retain-read consequential history. The exact finite-amplitude ACRF-4 quotient identifies a nine-dimensional host-host-boundary state. The certified consequence-minimality theorem (DOI 10.6084/m9.figshare.34063464) shows that the retained relational boundary contributes irreducible future-consequence directions. The spectrum-preserving ACRF-8 redundant-record theorem (DOI 10.6084/m9.figshare.34063500) then proves an explicit local witness in which one quotient-valid source distinction is written into two disjoint retained fragments that are locally recoverable, counterfactually separable, and independently readable, giving RRT(q)≥2 under the stated criteria.The resulting distinction from Quantum Darwinism is precise but bounded. RT recordhood is consequence-based, while Darwinian objectivity is an accessibility-and-redundancy criterion formulated within quantum theory. The ACRF witness demonstrates redundant retained record formation without assuming Hilbert space, density operators, mutual information, decoherence, einselection, Born probabilities, or an observer-defined environment. Therefore, multiple persistent records of one distinction are not definitionally identical to Quantum-Darwinian redundancy.The paper does not claim that Quantum Darwinism is false or that RT has derived quantum mechanics, decoherence, Born probabilities, a mutual-information redundancy plateau, or macroscopic objectivity. Instead, it proposes a disciplined hierarchy: consequential distinction → retention → persistent record → branching → redundant retained records → redundant accessibility → environmental objectivity. The final two steps remain open in RT. A future bridge must dynamically select and scale multi-host redundancy and map consequence-bearing RT records into quantum environmental fragments that satisfy decoherence and strong objectivity conditions.
This paper is the seventh in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of objects independently existing within spacetime. Two prior results set up its question. The third paper (Uchida, 2026c) characterized public finality—recoverability of a committed fact by many mutually independent accessors, non-disturbingly and repeatably—as a broadcast redundancy: an effectively classical pointer value proliferated across many disjoint environmental fragments, approximately in the form of spectrum broadcast structure. The fourth paper (Uchida, 2026d) read the evaporating black hole as the regime in which a committed interior record crosses, at the Page time, from hidden to recoverable encoding, reconstructible from the radiation through entanglement-wedge reconstruction; and the parent framework (Uchida, 2026a, §10) had flagged holographic quantum error correction as a place its language of protected, redundantly encoded, locally recoverable records might find content. The present paper makes that content precise as a structure theory of recoverable records, kept separate from any claim of new physics or any suggestion that the framework derives the underlying results. The central claim is conditional and structural. The framework does not claim that every recoverable piece of information is a code, that the universe is literally a quantum error-correcting code, or that AdS/CFT describes our universe. It claims that when information qualifies as an erasure-recoverable record because an encoding preserves a logical degree of freedom—or a logical operator algebra—against a specified family of physical erasures, then that recoverability is formalized as an exact or approximate quantum error-correcting structure, in which each correctable erasure leaves an authorized complementary region supporting a recovery map. That formalization is itself an established mathematical classification; the paper's interpretive contribution is to read the series' recoverable record through it, and to set it against a second, distinct architecture. Broadcast redundancy makes the same effectively classical value legible from many disjoint fragments; erasure-correcting redundancy protects noncommuting logical quantum information by keeping each correctable erased region ignorant of—approximately private with respect to—the protected logical content, while suitable remaining regions recover it. The formal engine of this distinction is the duality between correctability and privacy (the information–disturbance tradeoff), not the no-cloning theorem; no-cloning, with the access structure of quantum secret sharing, survives as the reason independently usable recovery regions for arbitrary unknown logical quantum information must overlap. Three commitments keep the reading honest. First, it separates three senses of "record"—protected logical quantum information, a committed classical record, and a publicly final record—and insists that recovery or decoding restores access to logical quantum information but does not, by itself, commit a classical value; commitment additionally requires measurement, pointer imprinting, decoherence, and stabilization (Uchida, 2026f), and public finality a still later broadcast. Second, holographic quantum error correction is offered as the sharpest worked model of the distinction, with exactly solvable tensor-network toy codes (exact within the model) kept distinct from realistic semiclassical holographic reconstruction beyond those limits (generally treated as approximate), and both from the further question of state-dependence. Third, code-recoverability does not by itself imply public finality: protected quantum information can be recoverable from authorized regions while remaining non-public. The framework relabels code structure as the architecture of recoverability and derives none of it, making no independent empirical prediction. Its disciplined thesis is that codes protect recovery; broadcasts create publicity—different architectures that a single physical history may nonetheless realize in sequence.
Quantum Darwinism explains classical objectivity through the redundant spread of information across fragments of the environment. This note studies the opposite regime: an environment whose record is distributed through entangled correlations rather than independent fragments. The model considers a mediator carrying a which-branch displacement record while being purified by an unobserved environment. The central question is: how much of the original quantum information can an observer recover when only part of the environment is accessible? The paper proves that any fragment excluding the record-bearing mode contains exactly zero which-branch information. No individual fragment carries a readable record, and redundancy disappears completely. A closed-form partial-information curve is derived. The amount of accessible evidence depends only on the captured fraction of the environment purification and is governed by a single rational function. The curve connects two limits: the minimal information available to an external observer and the full information recovered when the complete environment is captured. A precise threshold is obtained: strong environmental entanglement requires capturing almost the entire environment before objectivity emerges. The results show that decoherence and objectivity are not equivalent. The same loss of coherence can either create a public, redundantly observable fact or leave information hidden entirely in nonlocal correlations. The structure of the environment determines whether quantum information becomes an objective classical fact or remains inaccessible. All results are derived in exactly solvable Gaussian models, verified analytically and numerically, and accompanied by a reproducibility package.