GATE D31

Prime economy and the residue coset theorem: PARTIAL

partialEvidence I2Verify: doublePublic claim: No2026-07-20

What changed

D31 asks whether a prime portfolio can accumulate enough modulus bits for the coupling to become non-vacuous while the count of nonzero-residue primes, the local-cover complexity, and the exact evaluator cost all stay within the caps. The measured obstruction is a seeding count, the product over primes of the number of distinct admissible top-coefficient residues, which for the dense case stays far above the state ceiling for every portfolio that reaches the coupling onset, and using fewer larger primes raises it rather than lowering it. Phase 1 explains this with a proved structural theorem. Because the local admissible cover family is closed under multiplication of all coordinates by any unit and the top coefficient scales by the twenty-sixth power, the nonzero admissible residue set is a union of cosets of that power subgroup, hence the full quadratic-residue subgroup away from the exceptional residue class and a union of at most thirteen cosets on it. The nonzero residue set is therefore of linear size in the prime, so no structured family reaches a constant-size set at scale. LRC(13) remains open.

Evidence

check 1Coset theorem proved: covers are closed under unit scaling and the top coefficient transforms by the twenty-sixth power, so the nonzero admissible residue set is a coset union; away from the exceptional class it is the full quadratic-residue subgroup, dense at (p-1)/2

check 2Structured small-residue families are foreclosed at scale for the nonzero class: the minimum nonzero residue set is linear in the prime; a constant-size set requires the prime below 27. Sparse large-prime portfolios stay blocked under the dense model but are not refuted in general

check 3D31-PARTIAL with a proved PASS-THEORY coset result; the explicit Cartesian seeding is blocked-compute in the declared architecture; symbolic representation of the full constraint remains the open route. No BASE-SURVIVOR, no GLOBAL-WITNESS. LRC(13) OPEN

Verification: double · I2 · 11 artifacts (reports, verifiers, corruption suites, manifest with SHA-256)

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