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Theoretical analysis examines zero-facing slopes and results in Purple Mathematics, highlighting implications for prime detection.
Every structure of Purple Mathematics has faced Infinity: the determination slope descends toward the successor prime, the sensor watches the successor, the anticipation scan looks forward. This paper asks what changes when the slope faces Zero instead — when the signal travels from the direction of Infinity. The honest half comes first: the determination layer mirrors trivially. The mirror dig, weight, and walk are exact sign-flips of the originals, so the determination calculus is odd under pole exchange — the framework's founding x↦1/x balance symmetry, holding as a theorem, with nothing statistically new. The reception layer, by contrast, does not mirror trivially, and three new results follow. The Janus reception: over every prime gap the infinity-facing slope of the predecessor and the zero-facing slope of the successor cross in an X, so every zeta zero has two hosts; its two strike coordinates obey the exact identity t + t̃ = 1 — every zero divides its gap between its past and its future, and the shares sum to one (verified at deviation 0 for the first 497 usable zeros). The repulsion law: which slope the rising signal strikes first is decided by the height-weighted crossing point x* = (h_p·p₊ + hp₊·p)/(h_p + hp₊); measured over 497 zeros, the taller triangle repels first contact (39.6% against 59.1%), and the two opposite biases cancel to a global split of 49.9% against 50.1% — the third appearance in the program of strong hidden biases cancelling to flawless symmetry. The arrow of anticipation: with a fixed budget of zeros the detector's resolution is Δx ≈ 2πx/γ_K, so from any sensor the view toward Zero is legible and the view toward Infinity fogs at a computable rate; measured with K = 500 zeros, prime-detection accuracy falls monotonically from 97.0% near x = 100 to 83.8% near x = 1600 in lockstep with the resolution. The zeros themselves remain, as everywhere in this series, Riemann's; every claim concerns the framework's own geometry.
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Samir Hanna Safar (2026) studied this question.
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