The TUS OS Operational Calculus

The TUS OS Operational Calculus proves that a deterministic classical six-bit engine can construct exact finite qubit prepare–measure behaviour—without quantum hardware—while preserving the classical path, role and receipt structure from which that behaviour is derived. Building on the tetrahedral geometry established in Paper 1, this paper supplies the operational layer of the construction. For every lawful anchor, TUS OS Engine 1 constructs a canonical three-state face and unfolds it through three principal LANDING paths, each with a realised first step, intermediate-state re-evaluation and same-role continuation. Independently computed GUIDANCE states provide subordinate contextual witnesses. The operative object is therefore a receipt-custodied LANDING-led expansion rather than six co-equal transitions. The paper defines a Coordinate-Incidence Operational Harness, formally distinct from TUS Domain Binding. Under its four-outcome construction, the Harness assigns all 64 classical six-bit carrier states to exactly 27 distinct qubit preparations under one fixed tetrahedral-SIC measurement. The separately declared three-outcome face Harness yields the exact qubit-trine table, while the canonical four-state sector yields the exact tetrahedral-SIC table. The full 64-state image contains eight pure states, twelve states at radius 2/3, six at radius 1/3, and the maximally mixed state. Across the complete canonical census of 192 LANDING paths, every first transition changes the qubit preparation image, every second transition changes it relative to the intermediate image, and 58 complete two-step paths return to their initial preparation. The paper also establishes exact classical and qubit factorisations, finite isolation of the target trine and SIC kernels, and bounded rigidity of the corresponding qubit realisations under the stated assumptions. The claim boundary is explicit. The paper establishes an exact qubit prepare–measure representation of a deterministic classical finite-state architecture under the declared Operational Harness. It does not claim physical quantum execution, contextuality, classical impossibility, quantum computational advantage, or that LANDING transformations themselves constitute qubit channels. This is Paper 2 in the TUS OS mathematics series. The series comprises five numbered papers, with Paper 4 issued in two technical volumes. In order, the TUS OS mathematics papers are: Paper 1 — Canonical Tetrahedral Qubit-State Geometry in the TUS OS TriadXOR Completion, Receipt Verification and the Qubit-Geometric Semantic Witness Paper 2 — The TUS OS Operational CalculusClassical Construction of Exact Qubit Prepare–Measure Behaviour from a Receipt-Custodied Engine Architecture 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 OS Paper 4 — Technical Volume I — Formal Foundations and the Complete-Future CorpusNative Contract, Statistical Discrimination and the Invariant Probability Module Paper 4 — Technical Volume II — Orthogonality, Process Closure and Native Triadic Synthesis Paper 5 — TUS OS as an Anchored Analog Quantum SystemNative Witnesses, Predictive Synthesis, and Access-Contract Resource Separation This upload contains the final public v1.0 paper PDF, the complete public release and reproducibility package, and a supporting SHA-256 integrity record. Files included: The_TUS_OS_Operational_Calculus_v1.0.pdfPrimary public v1.0 operational-calculus and architecture paper. TUS_OS_Operational_Calculus_v1.0_Public_Release.zipPublic release and reproducibility package containing six files: the final PDF and Markdown manuscript, exact public theorem verifier, machine-readable verification PASS record, release manifest, and the complete TUS OS Triad Receipt Stack Public Example v1.0.1. The embedded receipt-stack archive contains the complete public worked Triad witness used by the paper: six canonical receipts, six audit renders, triad.canonical, triad.foundation, triad.synthesis, and its manifest. TUS_OS_Operational_Calculus_v1.0_Zenodo_SHA256.txtSupporting integrity record for the primary PDF, public release package and embedded public receipt-stack archive. SHA-256 — PDF02773190b3e8b7d71e8507fcd12204c119283a9d341b8579c4351256b7919e3c SHA-256 — public release ZIP065917916910e832ef77ec6c380ca701ffbd145faca743ec2ad09a672e6c7e49 SHA-256 — TUS OS Triad Receipt Stack Public Example v1.0.10785f4d7429c204105d3becba383b1fa67ae3fec427f2f46b7feedc0457ba4c5 TUS OS® is a trademark of Mark Whitlock.