Restricted Weil ground states and real Fourier zeros: A form-core criterion and a certified interval of supports

We prove a form-core criterion for the Fourier transform of a simple isolated even ground state to have only real zeros. The criterion applies to closed lower-semibounded forms whose trigonometric restriction matrices have a specified divided-difference structure; it requires neither an operator graph core nor compact resolvent. We verify the required normalization and trigonometric form density for the completed restricted Weil form at every finite support. Retained-tail spectral certificates and parity-resolved support nesting then establish the spectral hypothesis for every 0 < a ≤ 4/5, with lowest eigenvalue > 10⁻¹⁷ and spectral gap > 10⁻¹⁵. Consequently, for every 0 < a ≤ 4/5, every nonzero ground eigenfunction has an entire Fourier transform with only real zeros. The initial interval 0 < a ≤ 153/1000 has a computer-free gap > 1/500. The continuous-interval proof uses explicit rational candidates and analytic interval enclosures, without importing an external numerical certificate.