Finite-Window Continuation and Root Stiffness in Restricted Weil Positivity

Paper 51 in the "Geometry of the Critical Line" programme. This paper develops the general finite-window continuation framework for restricted Weil positivity. The companion note RN51 established positivity in the prime-free region a ∈ (0, a₂] and the first prime-active interval; the bridge note RN52 fixed the scalar Weil evaluator on which the present analysis acts. Building on these, we extend the fixed-window framework to arbitrary prime-power symbol windows. For any prime-power threshold, the admissible Weil form reduces to a spectral integral against a modified symbol Ψₐ with finitely many cosine perturbations. A threshold turn-on lemma proves that new prime-power terms enter with zero contribution at each boundary, ensuring continuity of the quadratic form across windows. Inside each fixed symbol window, the paper establishes the full fixed-window analysis: uniform tail control for near-minimisers, Hilbert–Schmidt compactness of the truncated operator, precompactness of near-minimising sequences, existence of exact minimisers, full continuity of the constrained minimum m (a), and a first-touch theorem guaranteeing a zero minimiser if m (a) ever reaches zero. The first-touch condition is reduced to a scalar obstruction via the bordered Euler–Lagrange equation for the full self-adjoint operator Kₐ, which has compact resolvent and purely discrete spectrum. The scalar obstruction sₐ (0) = 0 is reformulated as a weighted spectral-balance law, and norm-resolvent continuity of the operator family provides continuity of all spectral data. A three-term spectral decomposition isolates a finite-rank soft sector, whose contribution is expressed in overlap coordinates as gₛoft (a) = z (a) ᵀ M (a) ⁻¹ z (a). The paper culminates in the renormalized central-mode theory: the positive-level bordered equation yields an exact balance law for the shifted ratio ρ̃₀ (a) = w₀ (a) / (λ₀ (a) − m (a) ), which is proved bounded above and below. This gives a one-sided root stiffness bound for the constrained minimum inside the fixed-window resolvent regime. Cross-threshold numerator control and asymptotic continuation are not addressed here; they are pursued separately within the programme. The Riemann Hypothesis is not claimed.