M21e closes a crucial logical gap in the Olympus Programme’s conditional proof of the Riemann Hypothesis by introducing and proving Critical Stasis. Earlier results (M12c, M21d) established that zeros of the rank‑parameterised zeta functions are born only on the critical line (No‑Off‑Line‑Birth) and that the Zeta Flow satisfies anti‑symmetry under s↦1−s. What remained unproved was the claim that zeros, once born on the critical line, cannot drift off it. The first main result shows exactly this. Using anti‑symmetry together with a new conjugate symmetry of the Euler product (derived from the fact that warped primes are real and positive for all R∈(1,2), M21e proves that the flow velocity at any zero on the critical line is purely imaginary. Hence the critical line is a flow‑invariant manifold: zeros may move vertically but never acquire real drift. This result, the Critical Stasis Theorem, corrects an implicit but unjustified step in M12c and makes the conditional RH proof structurally sound. The second contribution addresses the Birth Continuity Conjecture (BCC) directly. Rather than treating it as a difficult operator‑theoretic problem, the paper reformulates BCC as a normal‑family convergence problem. For every fixed R<2, the warped‑prime Euler product converges absolutely throughout the critical strip, so the completed zeta family is locally bounded. By Vitali’s theorem, BCC follows provided one additional condition holds: uniform convergence of the rank‑dependent Gamma factor as R→2−. This reduces the entire RH gap to a single, explicit analytic statement about smooth dependence of the Hyper Core scaling dimension—a problem accessible using M12b. Putting everything together, M21e yields a corrected conditional chain: BCC ⇒ Critical Stasis + No‑Off‑Line‑Birth ⇒ RH All steps except the Gamma‑factor convergence are proved. The paper ends by sharply isolating this final gap and outlining concrete verification strategies.(((contact details: pawel@garycki.com)))
Paweł Łukasz Garycki (Fri,) studied this question.