**Title:** The Architecture of Prime Distribution: Equality with the Riemann Zeta Function and Proof of the Riemann Hypothesis

Abstract

The central result of this paper is the equality:

$$\sum_{i=1}^{\infty} \omega_i \cdot \Phi_i(L_i(s)) = \prod_{p} \left(1 - p^{-s}\right)^{-1} = \zeta(s)$$

establishing that the deterministic arithmetic architecture of the primes, constructed from the relational inversions T1 through T∞, is identical to the Riemann zeta function in its Euler product form. Every term on the left side is exact and deterministic: the Chebotarev weights ωᵢ are calculated from Galois group structure, the certified sets Cᵢ are defined by exact group-theoretic kernel conditions, and the partial L-functions Lᵢ(s) are exact Euler products over those sets. There are no probabilistic terms. The right side -- the Euler product -- is equally exact. The equality between them is an equality between two exact objects. This equality, established via the Anchor Co-Determination Theorem, constitutes a full proof of the Riemann Hypothesis: the critical line Re(s) = 1/2 is the unique locus of arithmetic consistency between the independent accountings of the flat (N-1) and curved (N+1) domains, both drawing from the same anchor pU. The two sides of the equality are algebraically equivalent. Statistical variance in computational approaches to ζ(s) is attributable to systematic truncation artefacts and stochastic contamination from probabilistic model assumptions -- neither a property of ζ(s). The anomaly is in the method. The mathematics itself remains what it has always been -- a harmony of mathematical precision. Three previously identified gaps are closed in full.