Canonical Tetrahedral Qubit-State Geometry in the TUS OS Triad

Abstract

Abstract Canonical Tetrahedral Qubit-State Geometry in the TUS OS Triad reveals an exact tetrahedral qubit-state geometry inside the deterministic six-bit construction already used to generate the TUS OS Triad. The paper proves that, for every lawful six-bit anchor, the full-state-space maximin construction produces three states with exact pairwise Hamming separation (4,4,4), and that every such triple has a unique same-space equidistant completion given by D=A⊕B⊕C. Under the normalised sign embedding, the four completed states have unit norm, pairwise inner product −1/3, zero centroid and rank three. They therefore form a regular tetrahedron. The paper then constructs an exact isometric representation of this geometry as four qubit Bloch vectors forming a symmetric informationally complete POVM, and shows that affine centring and common renormalisation of the original three-state face yields exact qubit-trine geometry. The analytical results are corroborated by exhaustive finite computation covering all 64 six-bit anchors, 249,984 ordered anchor-conditioned candidate pairs, 41,664 unordered triples, 960 mutual-distance-four triples and 240 completed four-state sets. The same geometric recovery is demonstrated against two frozen public TUS receipt packages. The paper also defines the boundary between the mathematical geometry and the surrounding TUS OS protocol architecture. Only the three states A,B,C are emitted as active Triad starts; the fourth state D is a mathematical completion and does not become an additional receipt-bearing protocol member. The resulting qubit geometry is therefore an exact representation of the finite structure, not a claim that TUS OS physically prepares qubits, performs SIC or trine measurements, exhibits nonclassicality, or obtains quantum computational advantage. This is Paper 1 in the TUS OS mathematics series, which develops the mathematical structure of the TUS OS architecture from finite Triad geometry through operational, global and process-level quantum representations. This upload contains the final public v1.0 paper PDF, the complete reproducibility package, and a supporting SHA-256 verification record. Files included: Canonical_Tetrahedral_Qubit_State_Geometry_in_the_TUS_OS_Triad_v1.0.pdfPrimary public v1.0 theorem and architecture paper. Contains the analytical proofs of six-dimensional minimality, exact maximin separation, unique XOR completion, tetrahedral Gram geometry, qubit-SIC representation and centred-face trine geometry, together with protocol linkage, exhaustive verification, pilot recovery and explicit claim boundaries. TUS_Paper_1_Reproducibility_Package_v1.0.zipSupporting reproducibility package containing 35 files, including extracted formal proofs, complete 64-anchor results, the 960-triple census, all 240 completed four-state sets, frozen Pilot 1 and Pilot 2 verification material, the primary exhaustive verifier, a second clean-room implementation, pilot-archive verification, recorded outputs, chronology material, claims/nonclaims controls and final release-validation records. TUS_Paper_1_Zenodo_SHA256.txtSupporting integrity record containing SHA-256 hashes for the public PDF and reproducibility package. SHA-256 — PDF:d78ce33170f24027857440e2a9b3cf3ab75cf6706f3776b1fd3e9a8ac7fb1bb2 SHA-256 — reproducibility package:d9d2a146318fd5fbdd78f665f1c6cff75e247df48076d26d049cfb9a41a691bd TUS OS® is a trademark of Mark Whitlock.© 2026 Mark Whitlock. All rights reserved.