The Riemann Hypothesis via Scattering Geometry and Deterministic Sieve Dynamics

We develop a geometric framework for prime distribution based on deterministic sieve dynamics, and prove the Riemann Hypothesis in the form ψ(x) = x + O(x^{1/2} (log x)^A) for an absolute constant A ≥ 1, where ψ(x) = Σ_{n≤x} Λ(n) is the Chebyshev function. The proof exploits the scattering geometry of struck sets—the rigid intersection patterns of pℤ with short intervals—together with a disjoint-strike identity that provides exact combinatorial bookkeeping for semiprimes. The key components are: (1) a Cubic Cutoff eliminating Ω(n) ≥ 3 survivors; (2) a first-hit identity removing each semiprime exactly once; (3) we use the sieve dynamics only to control survivor fluctuations, via a quadratic mean-square functional (a harmonically weighted L² norm, the ℋ-norm) and a damped prime-step recursion, yielding RH-strength pointwise control at a calibrated depth; (4) a base-case dyadic mean-square bound at dyadic depth (Appendix C) via Cesàro means of a truncated singular series combined with a phase-to-dyadic transfer; and (5) a prime-only depth calibration identifying survivors with primes on the relevant range and extracting the main term. The analytic inputs are Chebyshev's bounds, the prime harmonic sum, Mertens' product estimates, and the classical large sieve inequality.