A Proof of the Riemann Hypothesis —— from the eZe model

We present a proof of the Riemann hypothesis based on a discrete symmetry framework, referred to as the eZe model. The framework naturally satisfies the reflection relation \ (U = 1/ (2L) \) between the upper and lower fields, and exhibits periodicity modulo \ (7 \). Extracting the modulo‑\ (7 \) pattern yields a Dirichlet series \ (L (s) = ₙ n^-s \) with coefficients \ (ₙ = 1 \) of period \ (6 \). Constructing a discrete Hamiltonian \ (HN \) from the symmetries and taking the continuum limit, we obtain the Berry–Keating operator \ (H = 12 (xp + px) \). The operator is self-adjoint under the reflection symmetry, and its eigenvalue equation reduces to \ ( (1/2 + i) = 0 \). Spectral convergence theorems guarantee that the eigenvalues of \ (HN \) converge to those of \ (H \), hence all nontrivial zeros of the Riemann zeta function lie on the critical line \ ( (s) = 1/2 \). As a corollary, the prime number theorem follows with the optimal error term \ ( (x) = li (x) + O (x^1/2 x) \).