GATE D29

Pre-reconstruction cross-prime coupling: PARTIAL

partialEvidence I2Verify: doublePublic claim: No2026-07-20

What changed

D29 asks whether any label-free, sign-invariant invariant couples local cover choices across primes before full reconstruction. Phase 1 proves a local content reduction: the complete label-free sign-invariant content of a folded local cover is exactly the squared multiset, equivalently the elementary-symmetric coefficient residues, so every candidate local carrier factors through one coefficient object. Phase 2 identifies the first affirmative cross-prime coupling in the program. The top coefficient equals the square of the speed product, hence a bounded perfect square, and requiring its reconstructed residue to admit a bounded square root excludes local coefficient states that independent membership admits, with an exact and replayable certificate, on complete finite domains. Phase 3 locks this predicate as a theorem: soundness, one-global-scale semantics, an exact modular-root structure, and a general non-vacuity counting theorem. The exact evaluator that decides it is blocked-compute in the useful regime under the frozen caps, so the coupling is mathematically established but not yet campaign-usable. LRC(13) remains open.

Evidence

check 1Phase 1 PASS-REDUCTION: local content reduction proved; every label-free sign-invariant local observable factors through the coefficient object. V1 15/15, V2 10/10 independent, corruption 400/400

check 2Phase 2 COUPLING-SURVIVOR: the top-coefficient square-lift predicate is sound and non-vacuous on complete domains, excluding 62 to 78 percent of admissible states with exact certificates, in the pre-reconstruction regime between B13 and B13 squared

check 3Phase 3 PASS-Q13-THEOREM: soundness, one-global-scale semantics, modular-root structure and a general non-vacuity counting theorem proved and double-verified; the exact evaluator is blocked-compute in the useful regime. No BASE-SURVIVOR, no GLOBAL-WITNESS. LRC(13) OPEN

Verification: double · I2 · 24 artifacts (reports, verifiers, corruption suites, manifest with SHA-256)

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