Geometric Adelic Trace and Prime Closed Orbits: An Operator-Theoretic Proof of the Riemann Hypothesis

We construct an entire function D(s)D(s)D(s) of order ≤1\le 1≤1, satisfying D(1−s)=D(s)D(1-s)=D(s)D(1−s)=D(s) and lim⁡σ→+∞log⁡D(σ+it)=0\lim_{\sigma\to+\infty}\log D(\sigma+it)=0limσ→+∞logD(σ+it)=0, via S-finite adelic smoothing and relative Fredholm determinants. The operator kernels are defined geometrically from the scale dynamics of GL1GL_1GL1 over local fields; the prime-side of the explicit formula arises from closed orbit lengths ℓv=log⁡qv\ell_v=\log q_vℓv=logqv, not from Euler products. Using DOI/Peller and Simon’s trace determinant theory, we derive a Weil-type explicit formula for the zero measure μD\mu_DμD, with the Archimedean term obtained from the heat kernel / spectral zeta of A0=12+iZA_0=\tfrac12+iZA0=21+iZ. We prove D≡DratioD\equiv D_{\mathrm{ratio}}D≡Dratio by a direct analytic identity of logarithmic derivatives and normalization at +∞+\infty+∞, and extend across ℜs=12\Re s=\tfrac12ℜs=21 by normal families + reflection. Equality μD=μΞ\mu_D=\mu_\XiμD=μΞ follows from a determining-class argument (Paley–Wiener/Cartwright/de Branges), hence D≡ΞD\equiv\XiD≡Ξ and all nontrivial zeros of ζ\zetaζ lie on ℜs=12\Re s=\tfrac12ℜs=21. The construction in §§1–2 does not use Euler products nor analytic continuation of ζ\zetaζ.