We present a bounded analysis of the finite, nonsingular, occurrence-ON quantum sequential-growth presentation at dimension d=2 and source stages n<=4. The preprint records five ledger-bound results: a frozen strong/strong commutativity baseline; an exact rational fixed-vector/general-covariance and reachable-state/martingale witness of noncommutativity; a source-native Eq. (120) lemma; commutativity on a proper reconstruction slice; and conditional statewise-to-operator recovery lemmas. Its purpose is to make the distinction between statewise and operator semantics auditable, rather than to classify all finite semantics. In particular, it does not claim the full 955 or 721 profiles, a complete finite occurrence-ON classification, or a reachable-visible witness. The bounded U2 auxiliary-ideal investigation is retained as a reproducibility and resource-boundary record, not as a unit-ideal theorem.
We present a bounded analysis of the finite, nonsingular, occurrence-ON quantum sequential-growth presentation at dimension d=2 and source stages n<=4. The preprint records five ledger-bound results: a frozen strong/strong commutativity baseline; an exact rational fixed-vector/general-covariance and reachable-state/martingale witness of noncommutativity; a source-native Eq. (120) lemma; commutativity on a proper reconstruction slice; and conditional statewise-to-operator recovery lemmas. Its purpose is to make the distinction between statewise and operator semantics auditable, rather than to classify all finite semantics. In particular, it does not claim the full 955 or 721 profiles, a complete finite occurrence-ON classification, or a reachable-visible witness. The bounded U2 auxiliary-ideal investigation is retained as a reproducibility and resource-boundary record, not as a unit-ideal theorem.
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
STATUS (18 October 2026). The physics programme that motivated this work was concluded in October 2026 and its claim of a unified derivation withdrawn. The results in this paper do not depend on that programme and stand as stated. Can an exact account of recorded events also determine what happens next? We examine this question in a finite quantum model with 48 preparations and an apparatus with ten distinguishable outcomes. Two reversible evolution rules reproduce exactly the same first-use specification but predict different results when the apparatus is used again. A separate example fixes one evolution rule: two possible histories then have identical counts of reads, commits and service events, yet different probabilities for the next outcome. Counts alone are therefore insufficient for prediction in this model. What remains visible depends on the observation. At eight preparations, even the complete individual-read distributions agree between the two rules, while correlations between the reads distinguish them. We classify two explicitly specified families of observations and determine exact ranges for the simpler family, showing both how distinctions are lost and how they can survive. The results separate event accounting, the choice of evolution, retained memory and the conversion of records into measured responses. They are conditional finite-model results, not a derivation of a physical dynamics or a measured coupling. The contribution is a set of exact counterexamples and classifications, rather than new principles of quantum theory. Exact computations and explicit proofs identify the additional response, weighting and sampling choices that a predictive physical interpretation must supply. Version 0.2 incorporates corrections following an author-requested preprint review. The deposit supplies the PDF, editable LaTeX with three TikZ figures, bibliography, exact fixture data for 96 two-read processes over 48 preparations, and a portable Python/SymPy verifier. The revised verifier passes 2,926 exact assertions. These are conditional finite-model results; no physical law, bridge, weighting or sampling measure is selected. The unchanged historical audit sources are available in a separately linked companion archive. Frozen audit sources and INT99 result dossier: https://doi.org/10.5281/zenodo.22541080.
This theoretical and computational preprint integrates standard QND measurement and posterior-martingale results with finite-confidence decisions and physically retained observer records. Under a declared quantum instrument and its standard outcome probabilities, the posterior is a bounded martingale on an increasing observation filtration. A precise sufficient identification condition yields asymptotic terminal weights; a binary Gaussian benchmark gives exact mean stopping times, finite-threshold latch probabilities, and a controlled population-transfer bias bound. Version 2.3 is the first public release. It distinguishes accumulated observations from lossy current memory; assumed Born/instrument probabilities from derived consistency results; expected entropy from pathwise ambiguity; continuous confidence endpoints from discrete overshoot and unresolved runs; and Kramers first escape from endpoint readout error. It also clarifies the conditional D18/D19 persistence-selector interface and the separate D1092 finite-resource ledger, with direct attribution to established QND convergence and filtering literature. The companion archive includes the editable manuscript, PDF, equation sources, a fixed validation protocol, executable independent checks, numerical results, a revision ledger, and unchanged historical inputs. Historical simulation summaries are explicitly distinguished from new executions; some older executable sources were unavailable. The new audit includes exact finite-kernel and stopped-posterior checks, Gaussian likelihood-mixture calculations, five neutral and four population-transfer boundary-value cases, and retention/accounting controls. All examples are analytic or synthetic. No laboratory measurements, independent Born-rule derivation, measurement-problem solution, universal physical selector, or new empirical anchor are claimed.
A Kernel-Role Determinacy Theorem for Bell-Witness Fixation, No-Signaling, and Tensor-Level Compatibility This manuscript develops a fixed-domain AASC analysis of Bell-nonlocal quantum entanglement as a theorem of reusable witness determinacy. It does not introduce hidden variables, modify quantum mechanics, derive Hilbert-space formalism from below, or propose a new signaling mechanism. Instead, it asks a prior structural question: once a Bell-type operational family is used as the same non-degenerate witness across settings, lawful transformations, retained records, comparison, and later classification, what admissible same-domain standing form is forced? The governing claim is that AASC is not an optional interpretive overlay on physics. Once an operational Bell family is asserted as a determinate reusable witness, admissibility, standing, reference, and irreversibility are already active as minimal necessity conditions of same-witness objecthood. Target instantiation therefore carries governance occupancy: an alleged determinate Bell witness with no admissible governance realization is not an empty alternative, but a refusal or retyping of the target. The manuscript defines finite operational Bell objects with explicit setting carriers, outcome families, channel/interface data, standing-token valuations, continuation and report/reuse structure, comparison profiles, and act-time status. It then defines object-indexed query sorts, dependent response families, exactly recoverable operational simulations, lawful Bell groupoids, six governance-effect functors, and target-active extensional governance equivalence. The result is a typed interface in which arbitrary native formalisms can be compared by their target-bearing governance effects rather than by their internal syntax. The central Bell-specific theorem states that every same-target candidate-admissible Bell-governance construction is forced component by component into the certified boundary-compatible standing form on the complete target-active practice. Candidate admissibility does not assume the canonical form at the outset. Instead, target occupancy, lawful transport, standing-flow exclusion, compatibility certification, coordinate-deviation exhaustion, and componentwise closure force exact target-bearing realization and governance equivalence to the canonical representative CP∗C^{*}_{P}CP∗. The target-induced governance quotient therefore has exactly one element. The Bell carrier is treated as a compatible empirical model on a finite measurement cover. Bell-local completion is identified with the existence of a global section, and Bell nonlocality is the obstruction to such a section. The manuscript separates operational marginal invariance, absence of controllable communication, and exclusion of same-domain standing transport. No-signaling is therefore preserved without retyping Bell nonlocality as superluminal communication, metric transport, distant repair, or hidden standing transfer. The finite-dimensional quantum specialization uses the standard bipartite tensor-product and local-POVM carrier. Every separable realization in that carrier is Bell-local; therefore, if the fixed Bell-nonlocal empirical model has such a finite-dimensional quantum realization, every realizing state in that carrier is entangled. Post-quantum Bell-nonlocal no-signaling models remain within the general witness theorem but outside the quantum-entanglement corollary. The accompanying Lean 4 development, release v0.8.0, directly verifies the finite Bell-side witness carrier: denominator-32 vector/Born weights, structural no-signaling, CHSH numerator 808080, all sixteen deterministic local extrema, finite denominator-32 Bell-local-envelope exclusion, and conditional endpoint routing. Its direct witness certificate is vectorBorn_flagship_spine, and its deepest released endpoint is reasonableDeepestFiniteEntanglement_spine. The manuscript-level typed governance interface, empirical lawful groupoid, compatibility equivalence, coordinate-deviation exhaustion, completed role collapse, candidate-to-canonical derivation, and finite-dimensional quantum specialization are not attributed to the released Lean layer. This paper is the Bell-witness member of the paired AASC quantum determinacy arc. The record-side manuscript, Admissible Record Construction in Physically Non-Selective Quantum Regimes, fixes when local physical record occurrences stand as the same evidential record objects. The present manuscript fixes when the Bell operational family sampled by such records stands as the same reusable witness. The paired endpoint is therefore: record-role fixation+Bell-witness compatibility.\text{record-role fixation} + \text{Bell-witness compatibility}.record-role fixation+Bell-witness compatibility. The manuscript is classificatory and eliminative rather than mechanistic. It does not claim that AASC generates the Bell probabilities or supplies a hidden physical influence. Its claim is that reusable Bell-witness determinacy strongly constrains how Bell-nonlocal correlations may be coherently classified on a fixed domain: not as independently complete local states later connected by a mechanism, not as Bell-local hidden-variable value assignments, not as superluminal signaling, and not as same-act repair, but as boundary-level compatibility carried by one standing-preserving Bell witness.
Paper 6 of 11 — A General Theory of Answerability The eleven-paper TUS OS mathematical research programme Abstract This paper develops a general theory of how the exact resource cost of representing an independently fixed native system depends on the operational access contract imposed on that system. Rather than assigning a single intrinsic “dimension” to a process or operational object, the paper treats realisation as a fully typed problem determined jointly by the native system, the access contract, the admitted realisation category and the tolerated error. The framework defines contract-indexed achievable resource regions, declared-memory closure, lawful contract restriction and resource monotonicity. Strengthening a contract can require a realisation to preserve or recover distinctions that a weaker behavioural contract is permitted to compress, while the independently fixed native system itself remains unchanged. Lawful contract restriction induces reverse inclusion of achievable regions, but strict separation requires additional lower and upper certificates. For stochastic processes, the paper introduces causal-state transparency as a recovery obligation imposed on an exact autonomous predictor at a declared inspection boundary. All persistent quantum, classical, controller, history and source-side information available at that boundary is included in the resource account. A direct-sum classical–quantum identity-capacity theorem proves that uniform worst-case recovery probability τ across N intrinsic causal classes requires identity capacity at least ⌈τN⌉. The theorem is then applied to two external stochastic-process families. Periodically modulated decay processes exhibit an unbounded finite separation: exact autonomous prediction has minimum identity capacity 2, exact causal-state transparency has minimum capacity N, and every fixed positive transparency threshold forces a lower bound growing as ⌈τN⌉. For the dual-Poisson process, an exact qubit predictor exists under the weak prediction contract, while every positive uniform worst-case transparency threshold makes the finite-capacity achievable region empty. This is a contract-induced loss of finite feasibility on the same native process, not a claim that the process itself ceases to exist or that no larger infinite-capacity realisation could exist. The paper then returns to the TUS OS discovery instance from which the general problem arose. The same independently fixed 64-state canonical target admits a complete static quantum-operational Atlas of exact minimum Hilbert dimension 4, while recovery-complete LANDING selected-line instrument access, GUIDANCE selected-line readout and GUIDANCE selected-line-and-endpoint joint-instrument access each have exact minimum Hilbert dimension 64 within the same declared representation category. The native target does not change; the operational access obligation does. Together, the TUS OS, periodically modulated decay and dual-Poisson results establish three distinct exact contract-relative regimes: strict finite separation: 4 → 64; unbounded finite separation: 2 → N; finite-capacity breakdown: 2 → ∅finite for every τ > 0. These are three instances of one general structural law, not three identical recovery mechanisms. TUS OS uses admitted experiment families that collectively separate its canonical carriers, whereas the stochastic-process results use direct causal-state decoding at a declared inspection boundary. The results establish that finite-capacity behavioural sufficiency does not imply positive uniform worst-case recoverability of native predictive identity, and that exact representation cost is not a context-free property of the native system alone. It depends on the complete typed obligation imposed on an admitted realisation. The paper does not claim physical quantum execution, quantum advantage, a universal scalar resource, equivalence of TUS OS recovery with causal-state transparency, or finite-capacity breakdown for prior-weighted dual-Poisson transparency. Supporting General Theory Programme This paper is Paper 6 in the TUS OS quantum-operational and Answerability research programme. Building on Papers 1–5, it develops the general theory of contract-relative quantum realisations of classical processes, showing that quantum resource requirements are not intrinsic to a process alone but depend on the governing observational, recovery, and transparency contract. Paper 7 extends this into operational observability and the classification of contract-induced resource transitions; Papers 8–9 develop generic Answerability theory and calculus; Paper 10 proves the protected and executable realisation of TUS OS within that calculus; and Paper 11 closes the programme with A General Theory of Answerability, establishing Structural Identity, Recognition, the canonical mandatory protected core, and lawful Realisation Freedom. Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic WitnessEstablishes the canonical tetrahedral geometry, unique XOR completion, exact qubit-SIC representation and centred-face trine geometry. Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine ArchitectureEstablishes exact full-carrier qubit prepare–measure behaviour from the deterministic classical engine under the declared Coordinate-Incidence Operational Harness. For each lawful foundation, the 64 six-bit carriers map to exactly 27 anchor-relative qubit preparations, while the separately declared face Harness gives the exact trine table. Paper 3 — From Finite Relational Structure to Qubit Operational FormAmbient Foundations, Canonical Subatlas, Exact Global Quantum Dimension, GUIDANCE Witness, Sequential Closure and Global Representation in TUS OSEstablishes the repeated-common-effect static Atlas, its exact minimum complex Hilbert dimension 4, exact carrier-reconstruction structure and deterministic predictive-state minimum 64 for the frozen complete principal future language. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability ModuleEstablishes the frozen complete principal terminal future-experiment language, complete statistical separation of distinct carrier pairs, exact parity-resolved discrimination results, complement reachability and the 33-dimensional rational invariant probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes universal complement-support orthogonality and its propagation under exact common-CPTP dynamics, the resulting common-CPTP process bounds, recovery-complete LANDING and GUIDANCE results, exact 64-dimensional selected-line instrument/readout minima at their declared contracts, an exact classical–quantum hybrid construction, and the native structural/predictive quotient correspondence. The state-only common-CPTP minimum remains bounded within 32–64 rather than being fixed at 64. Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource SeparationDefines Analog Quantum Views and Anchored Analog Quantum Systems, proves that TUS OS is a proper Anchored Analog Quantum System on its 64-input abstract canonical corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, and composes the recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical ProcessesPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownGeneralises the access-dependent resource phenomenon beyond the TUS-specific discovery corpus by formalising the contract-relative achievable-resource object (S,C,𝔐,ε) ⟼ Ach(S,C;𝔐,ε), and establishes three exact contract-relative regimes: strict finite separation, unbounded finite separation and finite-capacity breakdown. These are instantiated respectively by the TUS OS 4→64 separation, the periodically modulated decay 2→N exact prediction-to-transparency separation, and dual-Poisson finite-capacity breakdown for every positive uniform worst-case transparency threshold. Paper 7 — Operational Observability and Contract-Induced Resource Transitions in Quantum RealisabilityStructural Classification, Family Scaling, and Finite-Resource BreakdownDevelops a structural classification theory for contract-induced representation-resource transitions, introducing the Contract Distinction System (CDS), CRC cofinal/refinement equivalence, and Certified Observability Transport Theory (COTT). It separates internal divergence from family divergence, classifies baseline-feasible strengthenings as resource-neutral, strict finite, finite-resource breakdown, or UNRESOLVED, and proves exact and asymptotic witnesses including an interface-induced 2→3 separation, the TUS OS 4→64 transition, incrementally unbounded finite separation for periodically modulated decay processes, and finite-resource breakdown for positive-threshold dual-Poisson transparency. Paper 8 — The Resource Geometry of Answerable InferenceIrreducible Answerability Structure, Minimal Sufficient Cores, and Certified Inference-Resource BoundariesDevelops the resource geometry of answerable inference. It formalises lawful-discharge semantics, source-level inferential obligations, answerability-sufficient reductions and irreducible sufficient cores, while keeping semantic structure, certified lower bounds and operational feasibility independently constituted. Gate E establishes a local answerability-to-resource threshold result; globally, exact closure identifies the Pareto-minimal inference-resource
From a finite event record to Bell correlations and their loss under recording A pair is prepared together, read at two separated instruments, and allowed to leave a material record. This paper builds the correlation law from a real source geometry in Artian's Quantum Traction Theory. Its source-to-access equation is pK(r,s∣a,b)=C(KΩ)C[(Πr(a)⊗Πs(b)⊗I)KΩ] Here C is an additive, basis-invariant event capacity, Omega is the normalized orientation-neutral two-channel source, K is a specified record contact, and the projectors select the two instrument outputs. A real quarter-turn and a qualified shared-reference composition generate the readout algebra. Capacity symmetry fixes its quadratic weight. A separately stated capacity-to-frequency condition connects that weight to laboratory counts. The correlation follows before the Bell comparison For a phase-compensated record contact with overlap g between its two normalized record profiles, the construction gives EK(a,b)=−azbz−g(axbx+ayby),0≤g≤1. The intact-source limit is E = -a dot b. The same calculation yields the fixed-setting and optimized Bell responses: S45=2(1+g),Sopt=21+g2. Fresh independent record contacts multiply their overlaps. The paper derives the repeated-contact law, a capacity-based overlap calibration, uncertainty bounds, and a protocol that separates calibration from the Bell observations being judged. A source-first account with an explicit dependency map A1 supplies event-time labels and the clock-angle correspondence; A4 supplies the real dial; A5-X supplies completed-event support, with its inherited A6/A7 conditions. The two-channel preparation, balanced composition, capacity symmetry, frequency bridge, and record coupling are printed separately. Each theorem states exactly which conditions it uses. The Access uncertainty commutator is treated as a distinct inherited algebraic premise, rather than inferred from Bell correlations. When a receiver follows the source Version 2.0 constructs a matched receiver rather than assigning its transformation by name. Preserving every probability of the stated bilinear contact forces its transport to be the conjugate of the source transport, up to an overall phase: ∣rTVTUs∣2=∣rTs∣2for all r,s⟹V=eiαU. A finite set of basis and sum probes certifies this relation. A robust extension converts imperfect probe probabilities into an operator-error bound. Finite clock and shift operations then produce the comparison loop L=M+ω(I−M),ω=e2πi/d. The matched sector cancels the comparison phase while the unmatched sector retains it. This is a joint comparison law; each local readout can remain incompatible. The paper identifies the binary loop's ambiguity, provides additional probes that distinguish it, and transfers the Bell correlations into a complete detector record when the original carrier is consumed. One contact, two observable roles A specified marker contact supplies both the Bell record overlap and a resolved weak-measurement instrument: γK0K1=sinθ,=diag(cos(θ/2),sin(θ/2)),=diag(sin(θ/2),cos(θ/2)). An independent replay executes this instrument on all 40,032 tree records in the public Feng and colleagues archive, reproducing every stored decoder success/failure score. The main five-setting batch contains 32,000 records; the complete archive has six settings. The reproduced job-cluster residual probabilities are p = 0.0188753 and p = 0.0442603, respectively. These diagnose modest tension with the ideal instrument under the stated retrospective covariance. The matched QTT and quantum representations make the same archival predictions, so this replay is not a significance test between them. The package preserves the complete source-first Bell construction and adds finite receiver certificates, physical transport-error bounds, a common-contact derivation, five vector figures, a twelve-item achievements ledger, and offline reproducibility materials. The next operational question is precise: which independently qualified physical contact fixes both the record overlap and the receiver comparison response? Dependencies: QTT Main Book; finite address-capacity counting; real source histories and composition; A1-CHSH spinor character; Observation as Access.