The problem, in plain language
Can \(668\) mutually orthogonal sign-vectors of length \(668\) be written down explicitly?
A Hadamard matrix is a square matrix \(H\) with entries \(+1\) and \(-1\) satisfying \(HH^{\mathsf T}=nI_n\). Hadamard's conjecture predicts one at every positive order divisible by four. Current public construction tables still list \(668\) as the smallest unresolved order; an explicit matrix would make \(716\) the smallest unknown case.
Shalom Eliahou's 2025 construction gives a \(64\)-modular matrix at order \(668\). Its rows are orthogonal modulo \(64\), and every row is exactly orthogonal to \(641\) of the other \(667\) rows. The underlying special Golay quadruple has only thirteen nonzero aperiodic correlation lags. That unusually sharp near-miss supplied one of the three routes studied here.
What survived the integrity audit
1. Exact obstructions around Eliahou's seed
Local repair frontier. Holding Eliahou's sequence \(q\) fixed is impossible: an exact repair would produce a Turyn sequence in the empty class \(\operatorname{TU}(41)\). More generally, any exact repair changes at least \(80\) of the \(334\) base signs and at least \(41\) signs in the natural \((s,q)\) coordinates.
At special distance \(41\), every possible repair has exactly \(39\) changes in \(s\) and one reciprocal pair of changes in \(q\). An orientation-free fold reduces the boundary to thirty canonical support problems. One checked DRAT certificate excludes the first long case, and an exact \(3{,}710{,}853{,}316{,}608\)-row census excludes all nine short cases. The other twenty long cases remain open.
2. A fixed-compression chart for Legendre pairs of length 333
Profile classification and norm gate. In the prescribed order-three compression chart, exact shell descent leaves five shell-two profile orbits and eighteen dense compressed-profile orbits. On every physical shell-two image, both recombined channels are units in all six primitive prime-\(167\) factors. Star pairing then gives three exact norm keys in \(\mathbb F_{167^3}^{*}\), and every solution must match all three between channels.
The three-key equality splits into nine character conditions of orders \(2\), \(83\), and \(28{,}057\). This is a new exact search architecture, but not a solution. Every audited one-coordinate image is full, every triple image has affine rank three, and exact small-slice joins contract at random-model rates. The result is strictly scoped to one prescribed-compression chart inside the public-open order-three multiplier subgroup \(\langle10\rangle\); it does not classify unrestricted Legendre pairs.
3. Semiregular \(C_{37}\) conference quotients
Complete quotient layer. Requiring a semiregular cyclic action of order \(37\) on a putative \(333\)-vertex conference graph gives exactly \(625\) integral quotient classes. They collapse to two characteristic-two parity types up to complement and obey a universal \(6/3\) diagonal incidence law.
Explicit exact-margin binary supports exist for both parity types and satisfy every graph equation modulo two, but neither reaches the next modulo-four layer. Constant formal generators through rank three and two natural first-nonconstant rank-two families are exactly excluded. This neither constructs nor excludes a semiregular conference graph.
Reproducibility and exact scope
The release contains source, frozen JSON certificates, SHA-256 manifests, the complete 2,304-range Eliahou production ledger, the complete 729-shard dense-shell ledger, independent field bridges, bounded replay programs, and the three paper sources. A verifier is credited only for the finite statement it encodes. Solver timeouts and interrupted searches are not treated as theorems.
| Checkpoint | Exact result | Largest recorded memory |
|---|---|---|
| Modern \(\operatorname{TU}(41)\) replay | 461/461 shards empty | Low-memory exhaustive enumeration |
| Eliahou short boundary | 9/9 canonical short cases excluded | Resumable streamed census |
| Shell-two primitive units | 90/90 factor audits nonvanishing | 1.86 GB |
| Pair-resultant field bridge | 45/45 compiled/repository probes agree | About 4 MB for the character audit |
| Conference quotient census | 625 permutation classes | About 60 MB |
The release page gives short verification commands and links to the complete production instructions. The expensive historical searches are not required merely to check the paper-level certificates.
Why the headline search is paused
The last restart was required to produce a genuinely new algebraic contraction. It did: the three pair-resultant norms reduce the six primitive factors to an exact triple key. The operational gate nevertheless failed.
A \(54\)-trit exact full-phase diagnostic also remained UNKNOWN after \(300\) seconds and \(3{,}270{,}456\) branches. That is a solver diagnostic, not mathematical evidence for nonexistence. More raw slices, isolated character witnesses, and longer stochastic solver runs would consume time without crossing a proof or construction gate.
Headline work should resume only if an implicit character-sum method computes the complete three-key join, a proof-producing decomposition closes a whole shell-two profile or Eliahou long case, a materially broader construction theorem appears, or exact external-memory hardware changes the full join into a bounded reproducible computation.
Research log
The detailed repository log records every promoted result, correction, failed gate, and resumption command. This is the short public chronology.
- 21 July 2026 — seed recovery and first exact obstruction. Reconstructed Eliahou's special length-\(167\) seed, verified its thirteen residual correlations, translated the fixed-\(q\) repair problem to \(\operatorname{TU}(41)\), and opened the prescribed-compression LP(333) chart.
- 22 July 2026 — independent lanes and certificate discipline. Developed variable-\(q\), cyclic-SDS, good-matrix, and Legendre-profile formulations; separated bounded local exclusions from global claims; froze the first resumable checkpoint and priority audit.
- 23 July 2026 — folds, phase algebra, and exact finite models. Derived the adjacent-\(42\) and anti-fold reductions, certified one minimum-boundary support case, completed labelled order-three lifts, and factored the prime-\(167\) phase algebra into six primitive components.
- 24 July 2026 — shell descent and complete scoped classifications. Excluded the high shells in the fixed-compression chart, classified five shell-two orbits and eighteen dense orbits in their stated scopes, completed their lift geometry, and launched the exact nine-short-case Eliahou census.
- 25 July 2026 — final theorems and pause gate. Completed all nine short-case exclusions, classified \(625\) semiregular conference quotients, proved primitive nonvanishing and the three pair-resultant norm keys, measured the infeasible remaining join, and paused the headline search.
Read the full timestamped research log or read the exact pause and restart criteria.
Three provisional papers
- Exact repair obstructions around a 64-modular Hadamard matrix of order 668 — fixed-\(q\), distance \(80/41\), the minimum boundary, and the long-case orientation cascade.
- Exact shell descent and finite-field norm gates in a prescribed-compression chart for Legendre pairs of length 333 — shell descent, five and eighteen profile orbits, primitive units, and the pair-resultant norm gate.
- Semiregular \(C_{37}\) conference lifts at order 334: quotient classification, modular realizations, and low-rank obstructions — the 625-class quotient theorem, modular realizations, and low-rank obstructions.
All three papers are unreviewed. Their novelty assessments are provisional, and each prominently states that it neither constructs nor excludes \(H(668)\).
Published context
- Shalom Eliahou, “A 64-modular Hadamard matrix of order 668”, Australasian Journal of Combinatorics 93(2) (2025), 422–427.
- Arthur F. Ramos, David B. Hulak, and Ruy J. G. B. de Queiroz, “Multiplier obstructions for Legendre pairs of length 333”, arXiv:2607.20765 (2026). Their theorem excludes common fixed multiplier groups of order at least nine; it explicitly leaves the low-order subgroup studied in our fixed-compression chart open.
- Matteo Cati and Dmitrii V. Pasechnik, “A database of constructions of Hadamard matrices”, arXiv:2411.18897.
- Epoch AI's Hadamard problem record, accessed 25 July 2026.
- Genet M. Edmondson, Jennifer Seberry, and Malcolm R. Anderson, “On the existence of Turyn sequences of length less than 43”, Mathematics of Computation 62 (1994), 351–362.
Scope and AI-assistance disclosure
The exploratory mathematics, programs, certificate design, independent-code audits, literature review, manuscripts, and publication materials were developed with heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is a complete amateur and cannot independently validate the mathematics. No outside individual was contacted, and no external expert has reviewed this work. Exact verification is evidence about the encoded finite statements; it is not peer review and cannot establish worldwide priority.