The exact price of local realism in CHSH experiments: a measurement-dependence-detection trade-off surface, a moiré phase-locking mechanism that saturates it, and an unmeasured fringe in the fourfold coincidence sum

For local hidden-variable accounts of the CHSH experiment that are faithful – reproducing the observed singles and coincidence rates and unbiased marginals of a polarization singlet with symmetric detector efficiency ${\eta }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ – I determine the minimal measurement dependence $M$ (Hall’s variational measure) needed to achieve a CHSH value $S$. Linear programming over all local strategies yields, to machine precision at 32 grid points, $M(S,{\eta }_{\mathrm{e}\mathrm{f}\mathrm{f}})=\max \{0,{\eta }_{\mathrm{e}\mathrm{f}\mathrm{f}}((S+2){\eta }_{\mathrm{e}\mathrm{f}\mathrm{f}}-4)\mathrm{/}6\}$, whose edges reproduce Hall’s tight bound at ${\eta }_{\mathrm{e}\mathrm{f}\mathrm{f}}=1$, the Garg-Mermin detection threshold, and the postselection ceiling $S=4\mathrm{/}{\eta }_{\mathrm{e}\mathrm{f}\mathrm{f}}-2$. The quantum point is certified exactly: $M(2\sqrt{2},9\mathrm{/}10)=(27\sqrt{2}-33)\mathrm{/}100$, with primal and dual certificates in $\mathbb{Q}(\sqrt{2})$ arithmetic. I solve the unique detection profile $D(m)=\sqrt{m}h(m)$ under which a deterministic sign model reproduces the singlet exactly, derive the $\sqrt{m}$ edge law, and prove exact quantum correlations and angle-independent coincidence rates jointly impossible for pure-detection models. Surviving local accounts trade off measurement dependence against a $\cos 4(a-b)$ modulation of the fourfold coincidence sum, of relative amplitude up to 12.4%, which vanishes at the CHSH angles and has never been bounded below 1%. A settings-torus protocol reaches $5\sigma$ sensitivity at 0.1% within hours: a flat result forces $M\gtrsim 95\mathrm{\%}$ of Hall’s floor; a fringe would contradict the flat-rate prediction of quantum mechanics. Finally I exhibit a local mechanism – moire phase locking – with deterministic phase evolution and all randomness quenched in frozen offsets and flight times; with 1024 offsets it attains the certified floor exactly at ${\eta }_{\mathrm{e}\mathrm{f}\mathrm{f}}=1$ at every tested fidelity, from 10% down to 1%, and its softening of the correlation extremes is a falsifiable fingerprint.