A Spectral Program for the Riemann Hypothesis

We construct a Hermitian kernel K_η on the prime-logarithm lattice Λ = 0, log 2, log 3, log 5, … and show that the Riemann Hypothesis is equivalent to K_η being positive semi-definite on all finite subsets of Λ. The kernel arises from the Weil explicit formula evaluated on Gaussian autoconvolutions: for a packet g (x) = c₀ψ (x) + Σ cⱼψ (x − log pⱼ) with complex coefficients and an atom at the origin, the Weil functional on h = g ∗ g̃ becomes a Hermitian form W (h) = Σ cλ c̄μ K_η (λ−μ). The atom at the origin is essential — without it, no genuine diagonal exists in the prime-sum evaluation (a product of primes is never prime). The prime-sum contribution to K_η has star structure: on Λₖ, it converges to a rank-two indefinite matrix with nonzero entries only between the origin and each log pⱼ, with values (log pⱼ) /√pⱼ. The off-diagonal is killed by the Fundamental Theorem of Arithmetic and Gaussian suppression (multiplicative shell bound, proved as a standalone tool in the companion paper "Gaussian diagonal dominance for explicit formulas"). Kernel positivity reduces, via the Schur complement, to the question: does the archimedean diagonal a_η = K_η (0) dominate the star correction and the prime-to-prime off-diagonal? The archimedean diagonal is computable non-circularly via the Riemann–von Mangoldt smooth zero-counting formula: a_η ~ 10 log (1/η). The star correction |b|²/a_η grows as log log k / log (1/η) — very slowly. Numerical verification confirms Schur positivity for k ≤ 13 primes and η ∈ 0. 03, 0. 1, with comfortable margins. Independent verification via Li's criterion confirms λₙ > 0 for n ≤ 3, 300 (Maślanka 2004), with an identical prime-sum structure exhibiting single-prime dominance over higher powers (ratio |O/D| ≈ 0. 21). The density of Gaussian translates at the centers Λ follows from a completeness theorem of Liehr: Σ 1/log pⱼ = ∞. The paper also constructs alignment operators Hₖ on the k-torus whose spectral gap connects to the Weil functional via a bridge theorem. A tube variational argument shows that this spectral gap path cannot establish Weil positivity (θₑff → 0 due to rank-one kinetic structure), motivating the direct kernel approach. The Diophantine path via μ (log 2/log 3) ≤ 2 remains an independent open problem. The program does not prove RH. It maps RH to the Schur positivity of a structured Hermitian kernel, identifies the arithmetic mechanism (FTA star + Gaussian suppression) that controls the prime side, computes the archimedean diagonal non-circularly, and isolates the remaining question as a row-sum estimate on the prime-to-prime block.