A persistent internal state can retain preparation information without improving prediction of achosen output. We construct a finite-time comparison, under matched dynamics, between theconditional risk of a policy that reads the record once and the worst-preparation risk of everypredictor restricted to one common output law. Separately controlled reference and policy-riskbudgets turn elementary metric separation into a sufficient certificate of predictive advantage.For a stochastic Schlögl network, a coupling stopped at record-band exit combines write failure,escape and the accumulated variation of reporter rates. An analytic boundary-crossing boundtransfers these budgets from a finite-capacity chain to the unbounded count process. The selectedchemical point gives certified upper and lower risks of 0.04423 and 0.13845. An engineeredbound-pair model gives corresponding trace-distance bounds of 0.06683 and 0.28242. For thesame stored channel distinction, a fixed position measurement retains a positive certificate, whilemomentum measurement has exactly zero separation. These results distinguish persistence ofa record, access through a reading policy and predictive effect on a selected observable. Forthe charge-field programme, which separates inherited identity from active internal degreesof freedom, the comparison supplies a calculable test of finite-time record-based constraint.Chemical reporter feedback is neglected; quantum preparation and detection are specified modelinputs. The capacity bound controls state-space truncation separately from numerical integrationerror.
A persistent internal state can retain preparation information without improving prediction of achosen output. We construct a finite-time comparison, under matched dynamics, between theconditional risk of a policy that reads the record once and the worst-preparation risk of everypredictor restricted to one common output law. Separately controlled reference and policy-riskbudgets turn elementary metric separation into a sufficient certificate of predictive advantage.For a stochastic Schlögl network, a coupling stopped at record-band exit combines write failure,escape and the accumulated variation of reporter rates. An analytic boundary-crossing boundtransfers these budgets from a finite-capacity chain to the unbounded count process. The selectedchemical point gives certified upper and lower risks of 0.04423 and 0.13845. An engineeredbound-pair model gives corresponding trace-distance bounds of 0.06683 and 0.28242. For thesame stored channel distinction, a fixed position measurement retains a positive certificate, whilemomentum measurement has exactly zero separation. These results distinguish persistence ofa record, access through a reading policy and predictive effect on a selected observable. Forthe charge-field programme, which separates inherited identity from active internal degreesof freedom, the comparison supplies a calculable test of finite-time record-based constraint.Chemical reporter feedback is neglected; quantum preparation and detection are specified modelinputs. The capacity bound controls state-space truncation separately from numerical integrationerror.
Quantum records may be well encoded upstream yet fail to support downstream inference when routing, decoder resources, memory, or comparison channels are restricted. We analyze finite routed quantum-record networks using task-specific Bayes risk. Three exact results provide the constructive core. First, a single Bell-coded pair supplied as the same complete input to GLOBAL and finite-round LOCC-late decoders yields the risk gap qμ/2, where q is pair-delivery probability and μ retained coherence; memory policy determines whether this access is preserved. Second, fixing local binary experiments with error rates 0.20 and 0.30, we determine the entire compatible joint-risk interval [0.15, 0.20] and an attaining decoder with minimax risk 0.20 that does not know the coupling. Third, a binary sufficient-statistic delivery law is attained by a three-record majority compression with nonzero raw risk, yielding a tight physical reconstruction defect. A generated-outcome model gives an exact receiver-availability boundary, while four reduced-description witnesses and continuity bounds delineate when source, state/resource, local-coupling, or clock/memory information is insufficient. The results distinguish record formation, receiver use, memory retention, and task-adequate comparison within standard finite quantum processes. Author manuscript, preprint version 1.0. This manuscript is not a peer-reviewed journal publication. A separate technical Supplement provides complete model definitions, proofs, channel specifications and numerical details. Reproducibility materials are available separately as Software release 1.3 on OSF: https://doi.org/10.17605/OSF.IO/5TDRB. The preprint and reproducibility record are separate, linked scholarly outputs.
This paper develops task-relative reader certification within Shadow Theory’s source–readout programme. It asks how many source copies are needed to validate a fixed quantum reader, and when access to its retained measurement records reduces that cost compared with observing only its final correct-or-incorrect decisions. The reader is fixed before validation and must return a sign after each complete experiment. A trusted evaluator knows the preparation label for scoring, while the reader does not. The analysis distinguishes terminal-error observations, records with trusted ancillary information, and readable measurement trajectories governed by a shared physical response model. For terminal-error validation, the paper establishes exact finite-budget tests and sequential likelihood tests, together with nearly matching information-theoretic lower bounds on expected source consumption. An unrestricted-record result shows why additional archived data alone need not improve certification: an advantage requires specified information or response constraints. A separate trusted-seed construction gives an exact information-projection test. The central quantitative comparison concerns a fixed fourteen-copy reader in a trusted shared-visibility family. To certify a maximum label error of one percent, a trajectory-based validator achieves a false-certificate probability of at most one in 1,048,576 and a nominal acceptance probability of at least 95 percent, using no more than 720,160 source copies. Under the same preparation and physical-response promises, every validator restricted to terminal errors and meeting the same acceptance requirement needs more than 1,992,371 source copies in expectation. At the exact nominal device, an uncapped trajectory-based test accepts almost surely with expected consumption below 254,746 copies. A separate robustness analysis extends likelihood-test validity to a quantified class of conditional response deviations. Finite-cap guarantees and expected stopping costs remain explicitly distinguished. The results demonstrate a certified resource advantage from access to retained classical measurement trajectories, rather than merely an improvement over one chosen terminal-error test. Supporting materials provide the fixed reader, exact certificates and reproducibility procedures. All guarantees depend on the stated preparation, response, information-access and deployment-stability assumptions. Certification concerns the nominated reader’s decision error; it does not establish complete quantum-instrument equivalence or identify an underlying source. The analysis uses standard quantum probabilities within Shadow Theory’s research programme.
We construct a source-matched reference calibration for distinguishing the coherent law h = 0 from the specified added generator h = J/25 in a finite quantum-record model. A complete-carrier transfer bound connects an accessible reference contrast to a delayed target contrast while retaining the original drift and actual law. On reference coverage and measured admission, restricting both hypotheses by the same observed reference band yields a directed target-set separation of at least 0.0071230619 whenever both compatible classes are nonempty. Rival worlds may have different admitted nuisance parameters. The constants are uniform over independently known finite lengths m >= 65, with tag 65, access prefix 1:32 and target deadline 25.02/J. Finite reference preparation and work/source reset satisfy the declared response budgets. Reset acts on the actual joint input and consumed supply, with the retained reference R exactly inert. It does not assign a universal reduced channel to correlated supplies. Under almost-sure conditional resource and supply bounds, the comparison extends to history-dependent acquisition. A prespecified rule, including abstention below the calibration threshold, has each-law unconditional wrong-declaration probability at most 2 exp(-6) + exp(-85/16) < 0.01. This is neither a posterior probability nor an error bound conditional on admission. The statistical step uses standard bounded-martingale concentration. Global target states and specified auxiliary supplies remain resources. Reset requires additional globally conditioned controls and has very large finite costs. The result is a conditional theorem for the two declared laws; it does not establish practical implementation or a unique diagnosis of fundamental collapse. A supporting Lean 4 companion contains 85 theorem declarations and 30 supporting definitions across Calibration, Operations and Risk. It provides checked scalar, sufficient-bound, resource-arithmetic and finite-history risk consequences under explicit premises. The companion has an executed Linux preflight and accepted returned macOS compilation, named-type and transitive-axiom evidence. The quantum constructions and conditional moment, filtration, tower and tail derivations remain written proofs or external premises; this is not an end-to-end formalization of the physical theorem chain. The original checked companion targeted author v0.4 and is carried forward unchanged, with its scope and execution chronology preserved.
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.
A quantum record can have locally accessible populations while its distinguishing phase remains correlated with a distant source. We study this access boundary in a specified finite recording apparatus with a returning, vacuum/one-excitation source. Source-position weights give finite-time cutoff bounds without removing any physical hopping or return coupling. For an initially far-tagged record and its incoherent mixture, every reader in a declared position-preserving local-control class obeys a phase-visibility envelope and a necessary access-time inequality. A separate finite local probe shows why that state-discrimination limit is not an unrestricted limit on distinguishing two dynamical laws: fresh local coherence can reveal a stipulated added dephaser without recovering the old record phase. We construct a bounded, degree-three reader accessing only source positions 1–32. With the initial tag fixed at position 65, exact finite-source arrival certificates and an analytical tail comparison yield recovery uniformly over every finite total source length at least 65. Accessible control faults are charged against the arrived signed-phase norm, while source-moving drift is charged against the full norm. At the fixed deadline 25.02/J, the matched phase-dependent difference between coherent and archive-gated dynamics exceeds 0.005, including the stated simultaneous control faults and four additional absolute response allowances of 0.008 each. Complementary results cover a vacuum-ready source and a stronger endpoint-prepared finite example. The conclusions are conditional on the supplied model, preparations, common realized controls and calibration bounds. They are not a switchable-fundamental-law experiment, a nuisance-independent classifier, an optimal-speed theorem or an end-to-end formal verification. The supporting Lean companion contains 59 declarations in eight modules with accepted returned Mac compilation, type and axiom evidence; physical analytical interfaces remain explicit hypotheses or outside that subset. The deposit files include the main manuscript, technical supplement and full research/reproduction package.
Continuous quantum error correction depends on which parts of a measurement record remain available to the controller at recovery. We study this temporal access using continuous strategic codes, formulated as open continuous process tensors contracted with causal instruments and conditional recovery. For monitored stabilizer codes, a common-optimal-correction criterion identifies when records can be merged without loss and yields a factorization for separated CSS inference. Terminal decision labels and causal persistent memory carry different costs: five persistent classical states reach the complete-record optimum in a finite-strength counting model, whereas its two-window diffusive counterpart has nine terminal decision labels and requires unbounded memory for exact causal recovery. Uniform lower and constructive upper bounds establish an optimal fidelity deficit of order m−2 for m persistent states in the diffusive model. Counting and diffusive stability bounds translate open-process approximation into guarantees for deployed-controller performance and retained access advantages, quantified by explicit quantizers and model-mismatch curves. The same process--instrument framework connects storage with persistent environments and irreversible noise to spacetime interrogation and record-selected paths for a logical T gate. These results specify which record distinctions recovery needs, the memory cost of approximating them, and the logical protection retained under model error.