Paper 134EE: Quantitative Positivity, Spatial Spectral Limits, and Optimal Arithmetic Completion

This paper develops quantitative positivity and spectral limits for the complete arithmetic Weilform, retaining its gamma, prime-power, polar, and mixed contributions. Explicit positive familiesarise from small spatial support and from controlled real or complex rate budgets. The singularspectrum of the inverse derivative and exact profile estimates give quantitative margins. Reciprocalrate mass then organizes the passage from localized positive spaces to compact spatial Weil operators,persistent model spaces, and recovery of every possible negative direction. The associated spectral,resolvent, and heat-trace limits specify what these restricted positive constructions can recover.The optimal arithmetic damping threshold is the supremal transverse zero width Θ. Support-growth arguments determine the negative exponent 2Θ in logarithmic Fourier and positive Sobolevnorms; the ordinary spatial norm requires an additional upper estimate. For that problem, exactprime-shift geometry, density cancellation, and separated-phase bounds lead to a decompositionof the entire frequency axis into finite bands with total interaction error O(A). This reduces themissing ordinary-growth upper bound to a uniform estimate inside the high bands.At A = 2, directed certificates control every prescribed local window and the complete first60,000 modes. An exact weighted Schur bound gives a lower floor 0.46 on the infinite remaining tailand a conservative floor −4.37 for arbitrary mixtures. The tighter head–tail estimate remains open.A quantitative bound for omitted tail columns specifies a finite certification route to that question.Separate countermodels show why counting accuracy, prime-power support, or positive smoothingalone does not supply the required signed control. The unrestricted undamped sign, the originalresponse Schur obligation, and the Riemann hypothesis are not established