A Geometric Proof of the Riemann Hypothesis and the Twin Prime Conjecture via Sieve Epoch Boundary Operators

This paper presents a definitive, deterministic proof of both the Riemann Hypothesis and the Twin Prime Conjecture utilizing the discrete spatial geometry of the odd-integer matrix. By constraining combinatorial sieve evaluations strictly within Sieve Epochs—localized intervals bounded between consecutive prime squares [pₖ², pₖ₊₁²) —we demonstrate that the classical O (2ᵏ) explosion of the Sieve Error Term is algebraically neutralized. First, we prove that the active divisor variance is strictly bounded by the polynomial volume of the local epoch (Δₖ = 2pₖgₖ + gₖ²). Second, by applying the Chinese Remainder Theorem to the primorial lattice, we prove that this minimum expanding volume (4pₖ) strictly outpaces the maximum localized architectural voids (O (ln² pₖ) ) for all pₖ ≥ 5, establishing the infinite recurrence of twin primes via footprint inheritance. Finally, we establish a Boundary Operator B̂ that maps these discrete spatial limits directly onto the Riemann-von Mangoldt explicit formula for ψ (x). We demonstrate that any deviation of a non-trivial zero ρ = σ + iγ off the critical line (σ > 1/2) generates a continuous wave drift variance of O (pₖ^ (2σ) ). This required analytical variance introduces an unresolveable contradiction, as it mathematically exceeds the absolute physical capacity of the localized geometric sieve error bound (O (pₖ ln pₖ) ). Because the underlying integer coordinates are rigidly fixed, this excess variance cannot manifest. The spatial boundaries of the Sieve Epoch act as an absolute containment field, pinning the real part of all non-trivial zeros strictly to Re (s) = 1/2. We mathematically demonstrate that Riemann's continuous wave function acts as a macroscopic envelope traversing the exact perimeter of the discrete Sieve Epoch, proving the spatial framework to be structurally optimal. MSC Codes: 11N35, 11N05, 11A41, 11M26