{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:hadamard-matrix-of-order-668",
  "title": "Hadamard Matrix of Order 668",
  "canonical_statement": "There exists a Hadamard matrix of order $668$: a matrix\n\n$$\nH\\in\\{-1,1\\}^{668\\times668}\n$$\n\nsuch that\n\n$$\nHH^{\\mathsf T}=668I_{668}.\n$$\n\nEquivalently, the $668$ rows of $H$ are pairwise orthogonal.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:hadamard-matrix-of-order-668:event",
      "problem_id": "vibemathed:hadamard-matrix-of-order-668",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-12T00:00:00.000Z",
      "title": "Hadamard Matrix of Order 668",
      "summary": "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.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:hadamard-matrix-of-order-668:attempt"
      ],
      "method_family_ids": [
        "vibemathed:hadamard-matrix-of-order-668:method"
      ],
      "source_assertion_ids": [
        "vibemathed:hadamard-matrix-of-order-668:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:hadamard-matrix-of-order-668:attempt",
      "problem_id": "vibemathed:hadamard-matrix-of-order-668",
      "model": "Claude (version undisclosed)",
      "provider": "Anthropic",
      "attempted_at": "2026-08-12T00:00:00.000Z",
      "human_collaborators": [
        "Levent Alpöge",
        "Philippe Voinov",
        "Saul Reynolds-Haertle"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The announcement itself is a bare sign string, but Alpoge's thread carries a credit line: \"weekend fun w @tehwalris, Saul Reynolds-Haertle, and of course claude:)) i only claim bad suggestions!!\" - which corroborates the three named collaborators (@tehwalris is Philippe Voinov) and confirms Claude was part of the working group, with Alpoge playfully disclaiming the good ideas. Which mathematical, computational or search steps were Claude's is still not stated anywhere, and no model version is given, so the tier stays at the floor the methodology prescribes for an unspecific disclosure.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Epoch AI - FrontierMath open problem (records the solve)",
          "url": "https://epoch.ai/frontiermath/open-problems/hadamard",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Eliahou, A 64-modular Hadamard matrix of order 668 (2025)",
          "url": "https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Recent Legendre-pair work on order 668",
          "url": "https://arxiv.org/abs/2607.20765",
          "kind": "paper",
          "license": null
        },
        {
          "label": "TheoremDB record for order 668",
          "url": "https://theoremdb.org/statements/hadamard-order-668/",
          "kind": "problem-record",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Hadamard Matrix of Order 668",
          "url": "https://x.com/__alpoge__/status/2087504785952182273",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Hadamard Matrix of Order 668",
          "url": "https://vibemathed.com/problem/hadamard-matrix-of-order-668",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:hadamard-matrix-of-order-668:verification",
      "problem_id": "vibemathed:hadamard-matrix-of-order-668",
      "solution_event_id": "vibemathed:hadamard-matrix-of-order-668:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "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.",
      "sources": [
        {
          "label": "VibeMathed: Hadamard Matrix of Order 668",
          "url": "https://vibemathed.com/problem/hadamard-matrix-of-order-668",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Hadamard Matrix of Order 668",
          "url": "https://x.com/__alpoge__/status/2087504785952182273",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Hadamard Matrix of Order 668",
      "url": "https://x.com/__alpoge__/status/2087504785952182273",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Hadamard Matrix of Order 668",
      "url": "https://vibemathed.com/problem/hadamard-matrix-of-order-668",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:hadamard-matrix-of-order-668:signal:watch_only",
      "problem_id": "vibemathed:hadamard-matrix-of-order-668",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}