Survey

This note records a sequence of numerical and analytical checks in which the geometric ingredients of a spectral framework for gravity and matter (a scale window with a conformal so(2,1) structure, a see-saw reflection, a honeycomb substrate with a Z3 valley structure, a Planck lattice) are tested against the known routes towards the Riemann hypothesis: Hilbert–Pólya, Berry–Keating, the approximate functional equation, Bost–Connes, quantum graphs and Ihara zeta functions, the operator of Bender–Brody–Müller, the de Bruijn–Newman heat flow, Gram's law, the Selberg theory of the modular surface and the Lax–Phillips scattering semigroup. Every station ends with a number and with the point at which it stops. The central construction is the Osterwalder–Schrader reconstruction of the generator of logarithmic time from the Weil form, computed from the primes and the archimedean terms of the explicit formula without using the zeros; in a finite band its eigenvalues reproduce the first ten zeros. Version 2.1 makes the finite part of this rigorous and reduces the rest to one statement. It proves that the reconstruction is the explicit formula read through a Rayleigh–Ritz compression, and that zeros off the critical line would appear as negative squares of the Weil form and as complex eigenvalues of the reconstructed generator, at most one pair for each pair of off-line ordinates (finite-dimensional Pontryagin theorem). It shows that the remaining deviation of the reconstructed eigenvalues from the zeros is quadrature error, scaling with the square of the step. It then computes the Weil positivity profile on compact windows, prime by prime: the form is positive on all windows computed, with margins that fall faster than exponentially (7.6×10⁻³ at half-width 0.3, 4×10⁻¹³ at 0.7), in agreement with recently reported certified bounds; beyond half-width 0.41 the pole and archimedean terms alone are indefinite, and positivity is restored only by the primes with their exact von Mangoldt weights — at half-width 0.7 the admissible range of a common factor on the prime weights has width 10⁻⁷. By Weil's criterion, the Riemann hypothesis is equivalent to positivity on every window. The note does not prove it. It identifies what a proof along this route would need: not an estimate but an identity in which the prime weights enter exactly, as the Hodge index theorem does for curves over finite fields. The framework supplies the variable, the generator and the decomposition of the Weil form; it does not supply that identity, and the framework itself does not depend on the hypothesis at any point.