The Micro-Architecture of Sieve Epochs: Footprint Inheritance and a Deterministic Proof of the Twin Prime Conjecture

While combinatorial sieve theory provides robust asymptotic bounds on the density of prime numbers, it fundamentally struggles to distinguish specific micro-architectures, such as bounded prime gaps. Building upon the geometric framework of Sieve Epochs—which guarantees expanding, interference-free spatial volumes between consecutive prime squares—this paper investigates the exact modular construction of prime footprints within those volumes. By analyzing the matrix of odd integers, we demonstrate a principle of "Footprint Inheritance". When a new Sieve Epoch is initiated at pₖ squared, the historical prime subset P_ (k-1) projects a perfectly symmetric, periodic matrix of un-sieved twin prime architectures, governed by the Chinese Remainder Theorem. The newly active prime, pₖ, introduces only a single linear congruence to this inherited matrix. We prove algebraically that because pₖ (where pₖ is greater than or equal to 5) can strike a maximum of two residues modulo pₖ, the survival rate of twin prime architectures strictly multiplies by (pₖ - 2). Bounded within the spatially expanding geometry of the Sieve Epoch, we eliminate reliance on probabilistic heuristics by invoking absolute minimum geometric runways and maximum modular voids. This strict intersection provides a deterministic, geometric proof of the Twin Prime Conjecture.