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 \( L(s) = ∑ ε_n n⁻ˢ \) with coefficients \( ε_n = ± 1 \) of period \( 6 \). Constructing a discrete Hamiltonian \( H_N \) 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 \( ζ(1/2 + iλ) = 0 \). Spectral convergence theorems guarantee that the eigenvalues of \( H_N \) 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 \( π(x) = li(x) + O(x1/2log x) \).