Contract-Relative Quantum Realisations of Classical Processes

Abstract

Contract-Relative Quantum Realisations of Classical Processes 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. 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 independent stochastic-process families. Periodically modulated decay processes exhibit an unbounded finite separation, with exact predictive identity capacity remaining 2 while exact causal-state transparency requires capacity N. 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. 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 four, while specified recovery-complete LANDING and GUIDANCE contracts require exact minimum Hilbert dimension 64. 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. 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. This work is Paper 6 in the TUS OS quantum-operational math research programme. Supporting mathematical papers This paper is the sixth paper in the TUS OS quantum-operational research programme. Papers 1–5 constitute the TUS OS discovery, mathematical-foundation and system-recognition corpus from which the present contract-relative resource theory develops. Together, they establish the geometric, prepare–measure, global-representation, complete-future, process, recovery, native-synthesis and system-level results used as the TUS OS foundation case in the present work. 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, qubit-SIC representation and emitted trine structure. 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 independently specified deterministic finite construction. 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 common static Atlas, its exact minimum Hilbert dimension 4, carrier reconstruction structure and the complete deterministic predictive-state minimum 64. Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusEstablishes the frozen complete-future experiment language, exact statistical-separation structure, parity-resolved discrimination results and the 33-dimensional rational probability module. Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic SynthesisEstablishes support-orthogonality propagation, common-CPTP process bounds, exact LANDING and GUIDANCE instrument/readout results, the 64-dimensional recovery-complete minima, the exact quantum-classical hybrid construction, and the native structural/predictive quotient correspondence. 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 declared canonical 64-input corpus, elevates the inherited structural/predictive correspondence into the Predictive Synthesis Identity Theorem, and composes the inherited recognition, predictive, resource and downstream-conservation results into the TUS OS Anchored Analog Quantum System Theorem. Paper 6 — Contract-Relative Quantum Realisations of Classical Processes — present workPredictive Compression, Vanishing Worst-Case Causal-State Transparency, and Finite-Dimensional BreakdownExtends the programme 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 strict finite, unbounded finite and finite-dimensional-breakdown regimes, including the TUS OS 4→64, PMD 2→N, and dual-Poisson finite-dimensional-breakdown cases. The present Paper 6 does not replace those proofs or re-establish the TUS OS quantum-operational construction. It changes the level of analysis. Holding the native system fixed while varying the operational access contract, admitted realisation category and tolerated error, it formalises the achievable-resource object Ach(S,C;M,ε), and proves that exact representation cost is contract-relative rather than a property of the native system alone. TUS OS supplies the finite strict-separation case, while the PMD and dual-Poisson families establish the unbounded-growth and finite-dimensional-breakdown regimes required for the general resource law. Contract_Relative_Quantum_Realisations_of_Classical_Processes_v1.0.pdfVersion: v1.0Release date: 14 August 2026SHA-256: 911f691bb36a8f5a373abec4fa302e45c4c37135a345548513dfdeb2a126ea03 © 2026 Mark Whitlock. All rights reserved.TUS OS® is a registered trademark of Mark Whitlock.