P42 Prizes · Register of recordsLocked · construction · minimize

Hadamard 668 Defect

verifier 0.1.2 · pending admissionimage digest pending admissionrepo problems/hadamard-668-defect

A queued 668 by 668 sign-matrix defect ladder. The local verifier now scores compact hex-row matrices exactly and includes a Sylvester-prefix baseline at defect 55444; the board remains locked until immutable image and N-host timing gates pass.

§1Statement

\begin{gathered} \min_{H \in \{\pm 1\}^{668\times 668}} \operatorname{defect}(H), \qquad \operatorname{defect}(H) = \#\bigl\{(i,j) : i<j,\ \langle r_i, r_j \rangle \neq 0 \bigr\}, \ \ \tbinom{668}{2} = 222{,}778 \text{ pairs} \\[2pt] r_i = \text{row } i \text{ of } H, \qquad \operatorname{defect}(H) = 0 \iff H H^{\mathsf{T}} = 668\, I_{668} \end{gathered}

Defect 0 exhibits a Hadamard matrix of order 668 — the smallest order for which none is known. Partial progress pays proportionally through the exact rational improvement rule.

§2Verification

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Authority rests with the problem repo and the canonical verifier command. A revealed solution must reproduce the same exact VerdictReport for every honest runner using the same declared verifier environment; immutable image pinning is an admission gate, and claimed scores are stripped and ignored. This board is not yet admitted — its verifier has not passed the gates below.

Conditions precedent to admission

R1integer row-pair dot products
R3compact sign-matrix encoding
H3all 222778 row pairs checked
H5claimed-score trap
☐N-host runtime gate required

§3Solution format

{
  "type": "object",
  "required": [
    "n",
    "encoding",
    "rows"
  ]
}

Plate 1. Canonical raw solution schema. Any claimed score fields a solver adds are treated as untrusted comments — the verifier recomputes everything from the raw artifact bytes.

{
  "n": 668,
  "encoding": "hex-row-bits-v1",
  "rows": [
    "668 lowercase hex rows"
  ]
}

Plate 2. Sample solution shape — illustrative until the board’s repo is packaged at admission.

§4The record

no submissions

No award has yet been made. This board opens for submissions when its conditions precedent close.

ReproduceGET /prizes/api/leaderboard?problem_id=10

Hadamard 668 Defect

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Conditions precedent to admission

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Stone

Mark

Proof

Invitation

Protocol