A local numerical reference is useful only when a justified flow image enters its domain.In autonomous source refresh, that obligation involves the state of the apparatus, not merely itsselected output registers. We study a supplied 44-coordinate source model and give a conditionalcomposition theorem that retains the original state and time, the interaction contrast, accumulatedpulse area, mechanical histories, and passive transport records. Exact normal-mode reconstructiondistinguishes interval wrapping from physical uncertainty. For an inactive router with positivemomentum, we derive the attained position extrema and a restricted forward marginal-hull equal-ity: retaining correlations does not remove the genuine spread of this particular projection. Aseparate guarded receiving domain admits the wider router record while preserving the originaltime and other 43 coordinate requirements. Source-local factorization then transfers the referenceobservables without deleting a physical coordinate or replacing an evolved state by a stored cen-ter. Smooth continuation, exact negative free transport, and a source-matched integral left-entrylayer are composed under prospective whole-path guards. A positive moment expansion suppliesdirected bounds for the isolated reference, with its interior momentum trace distinguished from thearchived cutoff convention. Fresh bounded calculations reproduce 18 accepted and three rejectedcandidate comparisons at each of two scale labels under their separately declared input hypotheses;these are not service probabilities. The result closes a local conditional implication, not physicalinitialization, complete two-use validation, global floating-program accuracy, or a common-originderivation.
A local numerical reference is useful only when a justified flow image enters its domain.In autonomous source refresh, that obligation involves the state of the apparatus, not merely itsselected output registers. We study a supplied 44-coordinate source model and give a conditionalcomposition theorem that retains the original state and time, the interaction contrast, accumulatedpulse area, mechanical histories, and passive transport records. Exact normal-mode reconstructiondistinguishes interval wrapping from physical uncertainty. For an inactive router with positivemomentum, we derive the attained position extrema and a restricted forward marginal-hull equal-ity: retaining correlations does not remove the genuine spread of this particular projection. Aseparate guarded receiving domain admits the wider router record while preserving the originaltime and other 43 coordinate requirements. Source-local factorization then transfers the referenceobservables without deleting a physical coordinate or replacing an evolved state by a stored cen-ter. Smooth continuation, exact negative free transport, and a source-matched integral left-entrylayer are composed under prospective whole-path guards. A positive moment expansion suppliesdirected bounds for the isolated reference, with its interior momentum trace distinguished from thearchived cutoff convention. Fresh bounded calculations reproduce 18 accepted and three rejectedcandidate comparisons at each of two scale labels under their separately declared input hypotheses;these are not service probabilities. The result closes a local conditional implication, not physicalinitialization, complete two-use validation, global floating-program accuracy, or a common-originderivation.
Motivated by Complementarity-First's distinction between a relational operation and its physical implementation, we study one-use source refresh in a supplied autonomous classical Hamiltonian model. A moving courier must reach a stationary source, exchange arbitrary internal contents with it, and expose the fresh input and retained old output at one fixed readout time. A positive completed-square kinetic coupling makes the completed exchange depend on spatial area rather than entering speed. A separate position-dependent recovery interaction transfers energy from a main mechanical carrier to a courier that may be nearly stopped. For four real four-coordinate internal registers and two mechanical pairs, the combined construction admits the fixed product domain ||(x,y,b,c)|| ≤ 1, 18.6193 ≤ α ≤ 18.6248, with mechanical preparation independent of the unknown internal data. Exact-rational certificates for a reduced three-coordinate release law, together with recovery estimates, give source completion before time 47 and an interaction-free readout at 33π/2. Certification here concerns reduced-system arithmetic, not a formal-kernel proof or validated integration of all twenty coordinates. Fresh loading is exact; the source/old-output pair has error at most ε_pre ||(x,y,b)||, with ε_pre < 0.007338, against its declared ideal comparison. This is not a whole-processor accuracy bound. The full endpoint retains two additional main-port passages, the remaining internal registers, and both mechanical records. Its mechanical energy allocation excludes immediate reapplication of the same ready-state contract, while a source-only witness excludes an exact payload-only linear reduction. The result supplies a bounded one-use implementation conditional on the stated architecture. Physical rearming, all-input/all-word linear control, and primitive selection of the carrier and interactions remain outside the theorem.
An autonomous refresh mechanism is not completed by specifying a local exchange map:the same apparatus must reach the exchange, leave it, finish its remaining motion, and deliver itsstate at the prescribed readout time. We construct this connected late route for one specifiedpreparation in the exact-real interpretation of a frozen classical numerical source. The input isa previously certified twenty-coordinate state after entry into coupled occupancy; its validity andretained preparation history are explicit premises. Source-valid continuation establishes occupancyclearing, physical recovery and courier reversal, complete courier passage, a smooth main-wallexcursion, and the remaining main processing. The resulting full state is enclosed at the originalexact dyadic readout time, without an independently chosen switch, momentum reset, or deletion ofinternal phase history. The courier’s completed-square coupling distinguishes canonical momentumfrom physical velocity. At its frozen right cutoff, canonical momentum is continuous while velocityand the assigned source value have different one-sided traces. The completed spatial exchangediffers strictly from the ideal exact-period swap; its small algebraic port residual is distinct fromthe much wider full-state enclosure. Event-dependent times are composed into a common final clockby retaining the actual remaining duration. The result is a conditional fixed-time completion for thesupplied source and preparation, not a time-zero reconstruction, an all-input ideal-model theorem,physical auxiliary attachment, or a second service. Returned local Lean evidence supports selectedcomposition, branch-map, accounting and conditional endpoint results; the cumulative trajectorytheorem is not fully formalized.
No-signalling constraints on isolated outcome probabilities do not determine whether an apparatus preserves the joint law of its accessible records under repeated readout, network composition, or reuse. We study these extensions through a conditional-record criterion and four explicit models. For a wrapped-Cauchy retrocausal source, protection under two conditionally independent continuous readouts forces an angle-independent response; three readouts suffice without continuity. A finite angular-bandwidth assumption gives a quantitative leakage bound, whereas continuity alone gives no uniform stability. At a specified two-source Bell junction, the required correction matrix has full nonnegative rank on every even uniform angular grid. A Gaussian controller can cancel branch-mass dependence while leaving a distinguishable joint record, for which we derive exact noise and law-change tradeoffs. Finally, a fixed singlet with shared classical noise reproduces the variable-width isolated outcome family, but retained and refreshed controllers generate different temporal statistics. We prove a finite-resource obstruction and construct positive finite-controller approximations. Numerical studies comprising 8.3 million independent simulated histories support the model-specific predictions. Together, the results identify conditions that must accompany marginal no-signalling when extending an apparatus to richer records and interventions.
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.
Can an autonomous quantum apparatus produce approximately Born-distributed outcomes and faithfully preserve its own earlier measurement label without assuming complete initial quantum equilibrium? This paper presents two finite autonomous Schrödinger–Bohmian constructions with explicit error bounds under restricted nonequilibrium initial laws. Each construction addresses three separate requirements: outcome calibration, copying the actual earlier configuration label, and retaining that label throughout a specified holding interval. The first model combines periodic focusing, a finite recoiling reference rotor, and a delayed compensated oscillator. Its record-distribution error is at most 0.004434840087, while failure to copy and retain its actual earlier label is at most 0.002327254798. The second model begins with a small radial Gaussian, includes autonomous preparation and activation, and incorporates the record writer from the original entrance. The joint law of its earlier label and complete symbolic holding record differs from an ideal constant Born-labelled record by less than 0.006086770113 in total variation. Its actual-label copy-and-hold failure probability is below 0.000552421956. The proofs combine bounded-variation calibration, exact recoil reduction, conditional-rank current estimates, finite-clock derivative bounds, and absolute probability-current estimates over the entire holding interval. The specialized derivations and coefficient calculations are included in the manuscript’s appendices. The two models retain distinct Hamiltonians and statistical assumptions. Neither establishes universal relaxation from arbitrary initial laws, supplies physical sources for every prescribed interaction, or demonstrates laboratory realization. External specialist verification remains open.
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.