The Prime Integer Theory: A Framework of Periodic History Weights and Gap Syntax Mapping

This article formalizes the Prime Integer Theory (PIT), a deterministic framework that redefines prime distribution as a residue of periodic "history weights" and "gap syntax." We move away from probabilistic number theory, treating the integer axis as a rigid, one-to-one coordinate system where primes are the "blank cells" in a multi-wave interference pattern. We introduce the Slide Rule for the instant exposure of prime slots and the Trigger Status ($p^2$) as the geometric limit of predictive rulers. Critically, we propose a new protocol for RSA factorization based on Phase Displacement ($R$) and Harmonic Convergence. By treating remainders as directional vectors rather than stochastic noise, the theory enables a gradient-descent approach to locating prime factors. Initial simulations suggest an efficiency ratio that bypasses $\approx 83.5\%$ of the search space at the base level, transforming the factorization of large-digit integers from a blind search into a deterministic geographic location task.