P42 Prizes · Register of recordsLocked · construction · minimize

Hadamard 668 Defect

verifier 0.1.0 · 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

The chain does not trust this page.

Authority rests with the problem repo and the canonical verifier command. A revealed solution must reproduce the same exact VerdictReport for every honest runner; 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