Source authenticated

Hadamard Matrix of Order 668

Explicit construction of a Hadamard matrix of order 668, the smallest previously unresolved order, verified exactly by this site from the announcement plus its decoder reply. The same post encodes matrices for all twelve previously-open admissible orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964), and this site verified every one of them. The entry records the order-668 existence question, which this fully resolves; the general Hadamard conjecture - existence for ALL admissible orders - remains open, with the smallest unknown order now 2004 or beyond.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 12, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: site-confirmed. Publication: announcement. AI contribution: ai-assisted. Imported under CC BY 4.0.

Canonical aliases: Hadamard Matrix of Order 668 · Hadamard 668

Confidence: Not scored

Registry verification: site confirmed · announcement · resolved

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Levent Alpöge
human · human collaborator

Philippe Voinov
human · human collaborator

Saul Reynolds-Haertle
human · human collaborator

Claude (version undisclosed)
model · ai model contributor · Anthropic

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Hadamard Matrix of Order 668

specialist registry · passed

Fully reproduced here on 12 August 2026, in two passes. The announcement is a single X long-post holding 23,828 characters of "+" and "-": no prose, no separators. The first pass scanned the raw string for seed shapes and found three Goethals-Seidel quadruples (orders 892, 1132, 1244), verified exactly, but no order-668 seed - correctly, since the payload is not a seed list. What it missed is that Alpoge's own reply to the post is a decoder: a sed-obfuscated shell script declaring twelve records and five builder routines. This site reimplemented the sed transformation in Python, read the decoded script before executing anything (pure sed/sh, no network, writes only under /tmp), and ran it. It emits twelve sign blocks, and its header table independently names the four orders the raw scan had already found, cross-validating both decodings. Every block was then checked in exact integer arithmetic: entries in $\{-1,+1\}$ and $HH^T = nI$ on the nose, for $n$ = 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948 and 1964 - all twelve previously-open admissible orders below 2000, exactly as the thread claims. The order-668 matrix has diagonal 668 everywhere and maximum absolute off-diagonal entry 0. The submitter reports an equivalent reproduction, done separately from this one. No independent expert review or published write-up exists yet, so site-confirmed records this site's own exact-arithmetic reproduction, not community acceptance.

Source ↗

Hadamard Matrix of Order 668

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Epoch AI - FrontierMath open problem (records the solve)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Eliahou, A 64-modular Hadamard matrix of order 668 (2025)

paper · pending

paper link supplied by the VibeMathed registry.