A structural identification of the two-clock projection constant cos(π/8) with the T-gate magic-state overlap and the symmetric CHSH optimum

We give a conditional structural identification, at the level of single-qubit Pauli algebra, between three a priori independent appearances of the half-angle π/8: (i) the two-clock laboratory projection constant of Quantum Traction Theory (QTT), Iclk = cos (π/8) ; (ii) the modulus of the standard one-qubit T-gate magic-state overlap, |⟨+|T|+⟩| = cos (π/8) ; and (iii) the symmetric Clauser–Horne–Shimony–Holt (CHSH) variational angle β = π/8 at which the Tsirelson bound 2√2 is saturated. Assumptions. We adopt three structural inputs from QTT (Attar 2025): (A1) a positive-lapse two-clock geometry with a canonical clock-plane refoliation angle θ* = π/4; (A4) a real-dial primitive generator J (J² = −1) with single-qubit Pauli algebra; and (A7) a 2π modular bundle closure. The half-angle in the spinor lift is not assumed: it is derived from the real-dial Pauli commutators by the adjoint action of the clock-plane rotor (Lemma 1, Theorem 1). Comparison with established mathematics. The identifications |⟨+|T|+⟩| = cos (π/8) and the CHSH variational optimum at β = π/8 are exact results of standard quantum-information theory and are well known. The structural content of the present article is that, conditional on (A1) + (A4) + (A7), the QTT clock projection and the magic-state overlap are the same real projection of the same centered spinor rotor inside the same real-dial algebra, and that the CHSH variational saturation is implemented by Clifford-conjugate centered analyzer rotors on the same local qubit Hilbert geometry, with centered spinor exponent magnitude π/8 (Lemma 2, Theorem 4). Bell decomposition. We further establish the Jarrett–Shimony decomposition of QTT Bell consistency (Proposition 1): under the same axiomatic core (A1, A5, A6, A7), measurement independence and parameter independence (no-signaling) are preserved — PI established by projector completeness conditioned on the worldcell-bundle hidden variable λ = (w, ψw) — while outcome independence fails through the standard singlet conditional-probability calculation, with A7-contextuality supplying the QTT ontological reading (distinguished here explicitly from single-qubit Kochen–Specker contextuality, which would not apply at the qubit dimensional level). QTT accommodates the Tsirelson saturation at β = π/8 without violating parameter independence, on the same local qubit algebra that produces Iclk. As an immediate consequence, the CHSH-game quantum winning probability satisfies PwinQM = cos² (π/8) = Iclk² (Corollary 1), expressing the same structural identity at the probability level: the entire Tsirelson saturation falls out of squaring the QTT clock projection. Limitations and falsifier. The identification is a conditional algebraic theorem within QTT, not an experimental validation of QTT's other physical predictions. The structural claim is falsified inside its stated scope by (a) any inconsistency in the spinor-lift derivation given the real-dial Pauli algebra, or (b) a robust, unconditioned (non-postselected) experimental observation of CHSH |S| > 2√2, which would simultaneously falsify the standard local qubit-Hilbert ceiling and the QTT geometric+capacity ceiling. This article is a focused technical companion to the QTT framework manuscript (Attar 2025, doi: 10. 5281/zenodo. 17527179) and to the QTT neutrino mass-squared ratio paper (Attar 2026, doi: 10. 5281/zenodo. 19960814). It does not by itself derive the QTT axioms, and it does not experimentally validate any of QTT's other physical claims.