GATE D12
Ratio graph: sparse closes, dense low-rank
What changed
The non-tight remainder is now a finite graph with exact rational labels, not a cloud of vectors. An exact dual-deficit identity expresses the margin per exceptional edge. Sparse graphs (at most 2 exceptional edges) keep the punctured margin positive, so the vector is proper (22/22, zero counterexamples). Dense graphs carry an independent relation basis via a spanning forest, with the rank proved by exact linear algebra, not edge count, a low-rank structural handoff. Every failing vector has at least 3 exceptional edges and emits a verifiable relation basis, no unexplained residual.
Evidence
check 1Exact dual-deficit identity: margin = generic + per-edge deviations (pair + / triple −, signs exact)
check 2Sparse (≤2 exceptional edges) ⟹ margin > 0 ⟹ proper (22/22, 0 counterexamples)
check 3Dense ⟹ spanning-forest independent relation basis (Q-rank, not edge count), 2 verifiers, 32/32 corruption
Verification: double · I2 · 9 artifacts (reports, verifiers, corruption suites, manifest with SHA-256)