An Executable Cartography of the Local Boundary: An Anisotropic Event-Based Local Family Saturating CHSH, with an Exact Malus Response at Reference Preparation

This document presents a family of deterministic, event-based local models — an ANISOTROPIC family: its correlations depend on the absolute orientations of the settings, not only on the angular separation. The family reaches the local CHSH ceiling (S = 2, with analytic proof and numerical confirmation) while maintaining full acceptance (every generated event is counted — no detection loophole) and exhibiting, at the reference preparation (source at θ = 0), the exact Malus-law response cos²a — a result that follows from a trigonometric identity under the uniform threshold and does NOT extend to arbitrary preparations (section 4). The document records: the closed form of the sixteen base functions; the closed form of the correlation curve E(a,b) = 1 − (2/π)·d(δ(a), δ(b)), with d the smallest angular distance in radians (the phase δ is family-specific: atan2(sin a, cos 2a) for SF/SFS, atan2(sin 2a, cos a) for CF/CFS; the mixed octet requires interval analysis); the anisotropic structure (three distinct correlations at the same 22.5° separation; a half-turn counterexample); and the two-protocol design over the same emission (P sector active for single-station transmission; P sector disabled for pair correlation), with the price of the alternative measured. This version further incorporates the complete audit of the sixteen operations (all saturate a CHSH inequality under the appropriate sign convention; closed forms by root intervals in the mixed octet) and the extended conjugated decision rule, under which the pure octet exhibits, at reference preparation, station responses in two angular frequencies with phase-shifted variants — cos²a and cos²(a ± 45°) at the doubled frequency; cos²(a/2) and cos²(a/2 ± 45°) at the simple frequency (the half-angle projection) — while the mixed octet splits between constant transmissions and saturation-shaped curves. This work does NOT claim a violation of Bell’s theorem, does NOT claim a general Malus law, and does NOT claim an established physical interpretation for the analyzer parameter. The contribution is cartographic: the executable, verifiable anatomy of an anisotropic local family at the boundary.