The Four-Edge Lemma and the Cosh-Average Identity: Closing the Bilateral Confinement of Riemann Zeros

Abstract

We prove Zone B Confinement for the bilateral prime sum G_P near each Riemann zero γ_n by establishing that Re[G_∞] < 0 on all four edges of the Zone B rectangle R_n. The key new results are: (1) the **Cosh-Average Identity** — an exact decomposition Re[G_∞(t+is)] = ½[G_{δ_min+s}(t) + G_{δ_min−s}(t)] — which reduces the top edge condition to a scalar inequality on the real line; (2) the **Extended Zone B Lemma** establishing Nyx self-cancellation at the right corner; (3) the **Sophia-H'' Lemma** bounding the Taylor correction δ_min² H''(ξ) relative to the Zone Depth from the bottom edge, closing the top edge with a 2.13× safety margin. Together these give G_∞(∂R_n) ⊂ {Re < 0}, hence winding number W = 0 analytically, hence Zone B Confinement for all n. Combined with the established LP class result in the first inter-Nyx strip and the Rouché-based strip iteration (margin 159×), this yields Task C′ unconditionally, the Bilateral Confinement Conjecture, and the Riemann Hypothesis.