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    "content": "gate. Section~3 states and proves the theorem. Section~4 collects the\nobstruction analysis delimiting the method. Appendix~A describes the\nnumerical machinery; Appendix~B the formal-verification status.\n\n\\section{Background: the Weil-form certificate method}\n\nWe recall the machinery behind the state-of-the-art density theorem\n\\cite{anthropic2026riemann}, which is the launchpad and the sanity\ngate for everything below; details and notation follow that paper.\n\n\\emph{The Weil explicit formula} \\cite{weil1952fonctions} expresses,\nfor a test function $\\psi$ supported in $[-\\tfrac12,\\tfrac12]$ (the\n\\emph{bandwidth-one} class), a quadratic form on the zeros as a\nfunctional of the primes. Compressing it via a Gabor (Gaussian\nwindowed exponential) frame at the critical density $h = 2\\pi/L$\nyields a finite matrix identity $\\tilde G = P + Q$: a zero-side\nmatrix $\\tilde G$ built from the zeros in a window of size $L$\ndecomposes into a prime-side matrix $P$ (computed unconditionally\nfrom Montgomery's second moment / pair-correlation input\n\\cite{montgomery1973pair}) and a controlled remainder $Q$.\n\n\\emph{The rank--trace inequality.} Writing $\\tilde G$ in inertia form,\na rank--trace comparison between the positive eigenspace of $\\tilde G$\nand the positive part of $P$ converts the second-moment input\n$\\|\\tilde G\\|_{HS}^2 = (R(\\psi) + o(1))\\,N$ into a lower bound on the\nnumber of simple on-line zeros in the window. The strength of the\ncertificate is governed by the \\emph{window ratio}\n\\[\nR(\\psi) \\;=\\; \\frac{\\int \\psi^2 \\;+\\;\n\\iint |u-v|\\,\\psi(u)\\psi(v)\\,du\\,dv}{\\bigl(\\int \\psi\\bigr)^2}\n\\qquad (\\psi \\in C([-\\tfrac12,\\tfrac12]) \\text{ even}, \\ \\psi > 0),\n\\]\nwith $R(\\psi_0) = 4/3$ for the flat window\n$\\psi_0 = \\mathbf 1_{[-1/2,1/2]}$ and\n$R(\\psi_{MT}) = \\tfrac12 + \\tfrac{1}{\\sqrt2}\\cot\\tfrac{1}{\\sqrt2}\n= 1.32749\\ldots$ for the Montgomery--Taylor window (both verified\nsymbolically and against $60$-digit reference values in this project,\nE-001), giving the certified fractions $2 - R(\\psi)$: $0.6725$ of\nzeros simple and on the line, $0.83625$ distinct. A sharpness analysis\nlocates the bandwidth-one ceiling of the method at $\\approx 0.682$,\nand shows the obstructions to going further are structural\n(Section~4).\n\nOne feature of this machinery matters here: the certificate fails, as\nit must, for the Davenport--Heilbronn and Epstein zeta functions\n(which violate the RH-analogue); the main theorem of this paper has\nbeen required to pass this gate structurally (project file D-053).\n",
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