We study classical-communication simulation of the joint source and probe statistics of a qubit-controlled momentum kick. Competitors retain quantum local matter, arbitrary causal classical memory and two-way communication, but no shared entanglement or quantum communication. For fixed Gaussian probe preparation and specified source preparations and readouts, an explicit local instrument approximates the complete continuous momentum statistics arbitrarily well while preserving branch-control distributions exactly. An independently bounded full output position second moment gives a strictly positive total-variation lower bound against the entire competitor class; the explicit simulator attains the same asymptotic order. At fixed nonzero kick separation, the minimum RMS spatial resource for error d scales as Theta(d^(-1/2)); at fixed relative size greater than one, the minimum error scales as Theta(eta^2) in the weak-kick parameter eta. Fixed finite bins admit exact simulation at sufficiently large finite resource. These are statements about specified readout statistics, not full-channel approximation, a Newtonian realization of the simulator, or demonstrated experimental feasibility. The manuscript discusses conditional routes to independent resource certification. This record contains the preprint, editable LaTeX source and an offline reproducibility archive. The archive reproduces Table 1 and runs 22 computational tests; three project-history tests requiring undistributed materials are explicitly excluded. The preprint has not undergone external peer review. Licensing: the manuscript, documentation and numerical records are licensed under CC BY 4.0; the Python software and tests are licensed under MIT. The archive specifies these separate scopes.
We study classical-communication simulation of the joint source and probe statistics of a qubit-controlled momentum kick. Competitors retain quantum local matter, arbitrary causal classical memory and two-way communication, but no shared entanglement or quantum communication. For fixed Gaussian probe preparation and specified source preparations and readouts, an explicit local instrument approximates the complete continuous momentum statistics arbitrarily well while preserving branch-control distributions exactly. An independently bounded full output position second moment gives a strictly positive total-variation lower bound against the entire competitor class; the explicit simulator attains the same asymptotic order. At fixed nonzero kick separation, the minimum RMS spatial resource for error d scales as Theta(d^(-1/2)); at fixed relative size greater than one, the minimum error scales as Theta(eta^2) in the weak-kick parameter eta. Fixed finite bins admit exact simulation at sufficiently large finite resource. These are statements about specified readout statistics, not full-channel approximation, a Newtonian realization of the simulator, or demonstrated experimental feasibility. The manuscript discusses conditional routes to independent resource certification. This record contains the preprint, editable LaTeX source and an offline reproducibility archive. The archive reproduces Table 1 and runs 22 computational tests; three project-history tests requiring undistributed materials are explicitly excluded. The preprint has not undergone external peer review. Licensing: the manuscript, documentation and numerical records are licensed under CC BY 4.0; the Python software and tests are licensed under MIT. The archive specifies these separate scopes.
A source-inclusive coherence observable is useful only when its measurement resources are admitted. This preprint constructs a bounded local reader for the conserved exchange observable of a specified finite quantum-record apparatus. A degree-three graph uses one independently prepared probe, twenty relays with arbitrary joint states, and finite-duration two-register extraction and full-SWAP interactions. The original apparatus Hamiltonian and an independently added archive-gated process remain active throughout their respective implementations. A prefix-dependent control-frame comparison, symmetry-resolved effect estimate, and explicit gain, phase and finite-slew waveform bounds yield a sufficient response separation greater than 0.005139889 for the stated fixed, fully accessible sixteen-position source and remaining response-error budgets. This is a conditional mathematical result, not measured hardware calibration, a partial-source-access theorem, or a unique diagnosis or derivation of fundamental collapse. The package contains the main manuscript, technical supplement, editable LaTeX, direct numerical reproduction sources, and accepted returned Mac evidence for 56 paper-specific supporting Lean declarations. That subset verifies selected algebra, scalar inequalities and the actual-constant numerical expression; it does not formalize the full physical/continuous-time theorem end-to-end. Separate-session AI review, numerical reproduction and returned kernel evidence are distinguished. Author-owned manuscripts, documentation and mathematical data are CC BY 4.0; newly authored numerical and Lean software is MIT. Predecessor and third-party notices remain unchanged; the reproduction package contains the licence map. No external funding; no competing interests declared. Public preprint v1.0; edition date 18 October 2026; DOI 10.5281/zenodo.23442579.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.
This preprint studies readout-dependent prediction in a specified classical-quantum continuous-time model with an autonomous classical source and a driven qubit memory. A dynamic model (D) is compared with a scalar-contact model (MATCH) reconstructed to match stationary contact occupancy and both directional switching fluxes. The predictor observes only the qubit memory and the declared classical record j; the target is a future binary signal at an independent exponential horizon. Two distinct results are established by exact interval certificates. At B04, (s,u,v,Omega)=(-1/16,0,0,1), D has lower Brier loss under fixed-Z readout by at least 1/5,000,000, but higher loss under each model's optimal quantum readout by at least 1/2,000,000. This is a commuting conditional-state example. Separately, at nine D-only points with u=1/8, v=0, s in {-1/16,0,1/16}, and Omega in {1/16,1/4,1}, both j sectors have squared normalized commutator norm greater than 10^-8, and the same state's gain over fixed-Z readout is greater than 10^-3. These shared nine-point bounds were extracted after certification. The standard Sylvester readout construction is explicitly attributed. The results are fixed-point statements; they do not establish a continuous parameter region, an unrestricted classical-resource advantage, a causal role of noncommutativity in the B04 reversal, or equal thermodynamic costs. Exact evidence and editable source accompany the preprint in a separate Dataset item. OpenAI ChatGPT/Codex and Anthropic Claude systems assisted the work. Their roles and the limitations of the archived checks are disclosed. AI checks are not human or institutional independent validation. This preprint is not a peer-reviewed journal article. Companion evidence and editable source: https://doi.org/10.6084/m9.figshare.33756607
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 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.
Hamiltonian recognition asks which candidate Hamiltonian generated an unknown quantum evolution when the evolution angle is not observed. This preprint solves a binary two-query qubit setting in which the two unknown angles range from independently resampled to fully shared. A task-specific symmetrization theorem reduces the full sequential optimization exactly to a three-dimensional correlation elliptope. A nine-dimensional legal subset of the most general tester dual is then shown to reduce, by semidefinite-program duality, to the same elliptope problem, yielding P_GEN(beta,eta) = P_SEQ(beta,eta) throughout the stated parameter domain. The resulting PAR-versus-GEN phase diagram contains an exact interior equality curve and an exact high-angle reentrant regime in which the globally optimal causal-processing advantage vanishes at both zero and full temporal correlation but is positive at every intermediate strength. Arbitrarily weak temporal correlation can activate a strict advantage. In this archival v1.1 release, the weak-correlation statement is conservatively formulated as explicit constructive lower bounds of orders eta^2 below 45 degrees, eta^(4/3) at 45 degrees, and eta above 45 degrees. No matching upper-bound derivation is claimed for exact asymptotic equivalence of those exponents. The record includes the main preprint, Supplemental Material, and a reproducibility archive containing proof notes, symbolic verification scripts, numerical cross-check data, revised manuscript source, figures, and integrity hashes. Status: public preprint; not independently peer-reviewed.