GATE D8
Structural elimination, Fourier criterion
What changed
A strategic pivot from enumeration to STRUCTURAL elimination. A finite-Fourier identity M(u) = G0^k + relation-mass is proved: a vector with no additive relation carrying enough Fourier mass is proper, eliminating the entire relation-sparse (generic) class in one theorem. Measured across k=3,4,5: every one of 13,000+ obstruction vectors is relation-rich (a short relation with |m| at most 3), with ZERO relation-sparse counterexamples. The tight class is the rank-1 anchor already closed by D5.
Evidence
check 1Fourier identity proved + validated, criterion relation-mass < G0^k ⟹ proper
check 2Infinite class eliminated: relation-sparse vectors are proper (two verifiers)
check 3Dichotomy measured: 0 relation-sparse obstructions in 13,000+ (open: uniform tail bound)
Verification: double · I2 · 13 artifacts (reports, verifiers, corruption suites, manifest with SHA-256)