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

One half-angle behind the T gate, CHSH, and the QTT clock Version: 1.5 Concept DOI: 10.5281/zenodo.19979594 Author: Ali Attar Website: quantumtraction.org Main book anchor: Quantum Traction Theory: Main Book v10.01 The same half-angle appears in three familiar places. In QTT's two-clock geometry the laboratory projection is \[ I_{\mathrm{clk}}=\cos\left({\theta_*\over2}\right)=\cos{\pi\over8}, \qquad \theta_*={\pi\over4}. \] In standard quantum information, the one-qubit \(T\)-gate has the magic-state overlap \[ \left|\langle +|\mathsf T|+\rangle\right|=\cos{\pi\over8}, \] and the symmetric CHSH optimum occurs at the same variational half-angle, \[ \beta={\pi\over8}, \qquad |S|_{\max}=2\sqrt2. \] The paper's structural claim is not that the standard T-gate or CHSH facts are new. They are established Hilbert-space results. The QTT claim is that, under A1, A4, and A7, the clock projection, the magic-state overlap, and the symmetric CHSH analyzer are the same centered spinor-rotor projection read in three languages. The shared real-dial rotor is \[ R_Z(\Theta)=\exp\left(-{i\Theta Z\over2}\right), \] so a Bloch-plane rotation by \(\Theta=\pi/4\) carries the centered spinor exponent \(\pi/8\). The clock-trace identity may be read as \[ I_{\mathrm{clk}} = \cos{\pi\over8} = \left|\langle +|\mathsf T|+\rangle\right|. \] At the probability level the CHSH-game win probability is then \[ P_{\mathrm{win}}^{\mathrm{QM}} = \cos^2{\pi\over8} = I_{\mathrm{clk}}^{2}. \] The Bell-consistency statement is also explicit. QTT preserves measurement independence and parameter independence while outcome independence fails through contextual A7 closure on noncommuting projectors. The local qubit-Hilbert ceiling remains the Tsirelson ceiling: \[ |S|\le2\sqrt2. \] What to look for in the paper: the spinor half-angle derived from the real-dial Pauli algebra; the Clock-Trace Theorem identifying \(I_{\mathrm{clk}}\) with the T-gate overlap; the CHSH centered-rotor reading at \(\beta=\pi/8\); the Jarrett--Shimony decomposition of the QTT Bell mechanism; the falsifier: a robust unconditioned violation of \(|S|\le2\sqrt2\) would break both the standard qubit ceiling and the QTT capacity ceiling in this scope.