Restricted Weil Positivity at the Prime-Free Boundary and a Local Prime-Active Extension

Research Note 51 in the "Geometry of the Critical Line" programme.

This note establishes two results on restricted Weil positivity:

Theorem A (restricted archimedean positivity): At the maximal prime-free support parameter a₂ = ½ log 2, the Weil functional is non-negative on the admissible class. For every G ∈ L²([−a₂, a₂]) satisfying Ĝ(±i/2) = 0, the archimedean Weil form W[G] ≥ 0. The proof constructs a codimension-one coercive estimate on the even Paley–Wiener space using Slepian concentration theory: the archimedean Toeplitz operator has exactly one negative eigenvalue, and the admissibility constraint projects out the corresponding eigendirection. The coercive gap is c_gap ≥ 0.063 > 0. An explicit min-max computation proves positive definiteness (not just semidefiniteness) on the admissible hyperplane. This coincides with the archimedean positivity theorem of Connes and Consani (2021, Selecta Math.); the proof here is independent, using Toeplitz decomposition and Slepian concentration rather than Sonin space compression.

Theorem B (local prime-active extension): In the first prime-active window a ∈ [a₂, a₃) with a₃ = ½ log 3, the Weil functional reduces to a single Toeplitz form with modified symbol Ψ(t) = φ(t) − √2 log 2 cos(t log 2). There exists δ* > 0 such that restricted Weil positivity persists for all a ∈ [a₂, a₂ + δ*]. The proof is by contradiction: after transport to a fixed Hilbert space L²([−1,1]), Hilbert–Schmidt compactness of bounded-window Fourier restriction and the positive tail of Ψ exclude any sequence of negative admissible minimisers accumulating at a₂. The extension is nonquantitative: no explicit value for δ* is produced. Full positivity on Window I is not claimed. The Riemann Hypothesis is not claimed.