From Finite Relational Structure to Qubit Operational Form

From Finite Relational Structure to Qubit Operational Form develops a finite local-to-global analysis of the six-dimensional Hamming-cube foundation Atlas underlying the TUS OS construction. It shows that every admissible completed foundation carries the same exact local qubit operational representation, while the complete repeated-common-effect Atlas requires complex Hilbert dimension four. The paper classifies the full ambient family of completed four-state foundations in the six-dimensional Hamming cube. Every foundation is uniquely an affine coset associated with one perfect matching of the six coordinates. There are exactly 240 unordered foundations and 5,760 ordered codebooks. Every foundation induces the same exact 27-class tetrahedral-SIC qubit operational quotient, with the same preparation-fibre and Bloch-radius structure. Each ambient chart reduces losslessly to a signed perfect matching. The chart-coloured union of two matchings admits an exact component-parity collision law, yielding the complete seven-value image spectrum over all 28,680 ambient foundation pairs. Exactly 10,800 pairs jointly reconstruct the full 64-state carrier, giving an ambient mathematical chart-cover number of two. The frozen TUS selector defines a canonical subatlas consisting of 36 distinct foundations and 64 anchor-indexed chart instances across nine matching classes. Its selected-foundation reconstruction graph is bipartite and eight-regular with 144 edges, while selector multiplicity yields exactly 400 reconstructing pairs among the 2,016 canonical chart-instance pairs. Only this canonical subatlas inherits the native LANDING and GUIDANCE structure. At the sequential level, all 384 active map/chart descent tests fail: no active LANDING map descends to a state-only transformation on a fixed 27-class local chart quotient. Under the complete principal future language, every canonical chart refines to all 64 deterministic predictive states, with maximum separating depth two. The subordinate GUIDANCE witness is also classified: its endpoint map has image size 34, eight two-cycle attractors and maximum transient depth five, while the additional GUIDANCE-selected chart raises principal path reconstruction from 138 of 192 contexts to 169 of 192. The paper also establishes an exact local-to-global quantum-dimension result. Although every individual foundation admits an exact qubit representation, no unrestricted qutrit state/common-effect family can reproduce all 4,096 Atlas probabilities while satisfying all 240 overlapping foundation POVM identities. Together with an explicit chart-independent dimension-four construction, this gives an exact minimum complex Hilbert dimension of four for the complete static Atlas. The attaining construction uses two declared qubit sectors, but unrestricted uniqueness or rigidity of minimum representations is not claimed. The scope boundary is explicit. The result does not establish a physical quantum carrier, operational access to all 240 ambient foundations, LANDING as a quantum channel, GUIDANCE as a quantum observable or instrument, contextuality, nonlocality, quantum advantage, unrestricted minimum-representation rigidity, or whole-position process synthesis. The paper therefore presents a complete finite local-to-global result at the static and deterministic predictive scopes defined in the manuscript. This is Paper 3 in the TUS OS mathematics series. 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 verification package, and a supporting SHA-256 integrity record. Files included: From_Finite_Relational_Structure_to_Qubit_Operational_Form_v1.0.pdfMain public v1.0 paper. From_Finite_Relational_Structure_to_Qubit_Operational_Form_v1.0_Public_Verification_Package.zipPublic verification package containing the Research Control Record and Detailed Proof Archive, compact mathematics companion, source snapshots, retained verifier chain, exact finite corpora and certificates, qutrit-obstruction verification materials, GUIDANCE and sequential results, literature-search custody, release-custody records, internal manifest and checksum list. From_Finite_Relational_Structure_to_Qubit_Operational_Form_v1.0_Zenodo_SHA256.txtSupporting integrity record binding the main public PDF and final public verification package. SHA-256 — main PDF313cb5c800460082a1ec02fae18a6c06ade9a7eac6b8fd1c6a34727e6e9cb49e SHA-256 — public verification package20967f277ca504adfc211e3c49c6ded9bc071b4781c0914f46f5e01c0ac4310d TUS OS® is a trademark of Mark Whitlock.