Abstract
Eliahou constructed a \(64\)-modular Hadamard matrix of order \(668\) from a special binary quadruple of length \(167\). We prove exact obstructions and reductions around that seed.
Holding the published sequence \(q\) fixed is impossible through a reduction to the empty class \(\operatorname{TU}(41)\). An adjacent-\(42\) fold then proves that an exact repair changes at least \(80\) base signs and at least \(41\) signs in the special \((s,q)\) coordinates. At equality, the problem reduces to thirty orientation-free support cases: one proof-certified long case and all nine short cases are excluded, while twenty long cases remain open.
Main results
Fixed-\(q\) theorem. No binary sequence \(s\) makes Eliahou's special quadruple exact while the published \(q=(+^{83},-^2,+^{81},-)\) is held fixed.
Distance theorem. Every exact base sequence differs from the published labelled base quadruple in at least \(80\) of \(334\) signs. Every exact special pair differs in at least \(41\) of its \(334\) \((s,q)\)-coordinates.
Minimum-boundary theorem. A distance-\(41\) repair, if one exists, has exactly \(39\) \(s\)-only changes and two \(q\)-only changes forming a reciprocal pair.
Root filters reduce the equality frontier to \(21\) long and \(18\) short reciprocal pairs. The anti-fold pairs the short instances, giving thirty canonical support problems. Case \(L0\) is unsatisfiable by a checked DRAT proof; the nine short cases are excluded by an exact streamed census.
For the twenty remaining long cases, the paper gives an exact endpoint-orientation cascade through moduli \(8\), \(16\), and \(32\), plus the forty-one causal equations needed to recover aperiodic rather than merely cyclic complementarity.
Claim-to-evidence map
| Claim | Evidence |
|---|---|
| Fixed-\(q\) reduction | Symbolic verifier and reduction to \(\operatorname{TU}(41)\) |
| Empty \(\operatorname{TU}(41)\) | Published classification plus independent 461-shard replay with 57,543,021 nodes |
| Distance \(80/41\) | Exact adjacent-fold identities and dependency-free reconstruction |
| Case \(L0\) | 39,580-variable CNF, compressed DRAT proof, and independent drat-trim replay |
| Nine short cases | 2,304 published range manifests, 3,710,853,316,608 join rows, zero exact supports |
| Long-case cascade | Primary and separate red-team implementations |
The release includes the 2,304 range manifests as a compact hash-addressed archive, so the complete frozen production ledger can be audited without the exploratory output tree.
Novelty and scope
The known emptiness of \(\operatorname{TU}(41)\), the special-form/base-sequence translation, and a weaker public distance bound are not claimed as new. The candidate contributions are the distance-\(80/41\) fold, the exact equality classification, the orientation-free support reduction and certificates, and the long-case cascade. The priority search was necessarily incomplete and cannot establish worldwide novelty.
No timeout or uncertified solver status is used as a theorem. The full nine-case census remains a large trusted computation even though its manifests, hashes, semantics, bounded independent replays, and physical witness checks are public.
AI-assistance and verification disclosure
The mathematical exploration, program development, certificate analysis, manuscript drafting, and publication materials used substantial assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is a non-specialist and cannot independently certify every derivation or implementation. Exact artifacts improve auditability; they are not peer review or proof-assistant formalization.
Suggested citation
Alec Kriebel, with substantial assistance from ChatGPT 5.6 Sol, “Exact repair obstructions around a 64-modular Hadamard matrix of order 668,” provisional research paper, 25 July 2026. Permanent page.