{"id":10,"slug":"hadamard-668-defect","repoId":"hadamard-668-defect","title":"Hadamard 668 Defect","status":"locked","mode":"construction","direction":"minimize","scoreName":"defect","seedBest":"222778/1","currentBest":"55444/1","optimum":"0/1","minImprovement":"1/222778","bountyEth":"0.75","challengeWindowHours":72,"postingBondEth":"0.03","challengeBondEth":"0.03","verifierVersion":"0.1.0","verifierImage":"sha256:local-dev","verifierCommand":"make verify SOLUTION=path","repoPath":"problems/hadamard-668-defect","poolAddress":"base-sepolia:0x4242000000000000000000000000000000000010","donationWallet":{"chain":"Base Sepolia","asset":"ETH","address":"0x4242000000000000000000000000000000000010","status":"testnet-only","explorerUrl":"https://sepolia.basescan.org/address/0x4242000000000000000000000000000000000010","note":"Testnet deposit wallet reserved for the full-size Hadamard defect gate."},"tagline":"The full finite construction board behind the Hadamard Mini pilot.","description":"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.","statement":"\\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}","statementCaveat":"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.","verifierStandard":["R1 integer row-pair dot products","R3 compact sign-matrix encoding","H3 all 222778 row pairs checked","H5 claimed-score trap","N-host runtime gate required"],"solutionSchema":{"type":"object","required":["n","encoding","rows"]},"sampleSolution":{"n":668,"encoding":"hex-row-bits-v1","rows":["668 lowercase hex rows"]},"chainProvenance":{"settlementState":"local-only","chain":"Base Sepolia","chainId":84532,"donationWalletAddress":"0x4242000000000000000000000000000000000010","poolAddress":null,"registryAddress":null,"problemRegistryId":null,"verifierImageHash":"sha256:local-dev","admissionMatrixHash":null,"deploymentCommit":null,"indexedThroughBlock":null,"reconciliationOk":false,"source":"static-portal-data","note":"Phase 0 portal state only: no deployed registry manifest or indexer reconciliation is attached to this problem yet."}}