GATE D9
Fourier tail closure & inverse structure
What changed
A rigorous barrier result: the absolute-value tail route is INSUFFICIENT. The absolute relation mass is essentially vector-independent, about the L1-norm of the Fourier coefficients to the k-th power over p, which is roughly 10,000 times the main term at k=13. So obstructions are improper through SIGNED cancellation, not small terms. In exchange, a verified inverse extractor turns every obstruction into a bounded additive-structure certificate, the tight class is recovered as additive rank 1 (the D5 anchor).
Evidence
check 1Absolute-tail route proven INSUFFICIENT (ABS mass ~ ||Ghat||1^k/p, ~10^4x main, vector-independent)
check 2Inverse extractor verified: every obstruction ⟹ bounded structure cert, tight class = rank 1
check 3Two verifiers ACCEPT, 24/24 corruption rejected, open: signed-cancellation closure
Verification: double · I2 · 14 artifacts (reports, verifiers, corruption suites, manifest with SHA-256)