M31 The Gamma-Factor Convergence - Proof of the Birth Continuity Conjecture and the Conditional Riemann Hypothesis

M31 is the Riemann-Hypothesis culmination of the Olympus Programme. It identifies the remaining analytic gap as the uniform convergence of the rank-R Gamma factorGammaR (s) -> pi^ (-s/2) Gamma (s/2) on compact subsets of the critical strip as R -> 2^-, and proves this convergence using the explicit rank-R Gamma-factor formula, the smooth scaling dimension dR, the convergence dR -> 2, and the normalisation factor hR (s) -> 1. The new result of M31 is the proof of Proposition (V2): uniform Gamma-factor convergence. The paper then combines (V2) with the previously proved local boundedness (V1) and applies Vitali’s theorem to obtain the Birth Continuity Conjecture (BCC). The proof chain is: V2 -> BCC -> No-Off-Line-Birth + Critical Stasis -> RH where No-Off-Line-Birth comes from M21d, Critical Stasis from M21e given BCC, and the Birth-of-Zeros structure from M12c. M31 states that this proves RH conditional on the Olympus ONS spectral framework: every nontrivial zero rhoₖ of zeta (s) satisfiesRe (rhoₖ) = 1/2. The paper’s structural point is that the RH mechanism uses three ingredients: the self-dual functional equation of the completed rank-R zeta, positivity and growth of warped primes, and smooth rank-dependence of the archimedean Gamma factor. It explicitly says the proof does not use GRH, random matrix theory, Langlands, explicit zero formulae, or deep algebraic geometry. M31 also includes a BSD companion section. It proposes the elliptic analogue at rank R = 3/2, where the elliptic ONS corresponds to the AGM half-rank; the same Gamma-factor convergence method applies, but the remaining BSD gap is to connect the critical-birth count to the Mordell-Weil rank. Navigational Note - M31 Primer first The present monograph assumes familiarity with the Operational Manifold framework and its core objects (rank‑dependent ONS, Dynamic Zeta Flow, and the Gamma‑factor formalism). For readers entering the M31 material for the first time, the M31 Primer provides the minimal and sufficient introduction. It develops the necessary definitions, notation, and logical structure of the Olympus Programme in a condensed and pedagogical form, and should be read first, prior to engaging with the present text. The broader programme continues beyond the Riemann Hypothesis. While M31 treats the rank‑2 endpoint (classical zeta), the subsequent mini-monographs: M32, M33, M34 extend the same rank‑flow and Gamma‑factor framework to the half‑rank regimes, initiating the systematic treatment of the Birch–Swinnerton‑Dyer programme via the AGM / Heun / ISHE structures. M34 culminates with Yang-Mills, while the three also deal with the Hodge Conjecjure. These problems are linked, and the papers perform varying degree of narrowing of them. ( ( (contact: pawel@garycki. com) ) )