{"id":9,"slug":"pnt-sparse-mertens-construction","repoId":"pnt-sparse-mertens-construction","title":"PNT Sparse Mertens Construction","status":"locked","mode":"construction","direction":"maximize","scoreName":"certified log objective lower bound","seedBest":"0/1","currentBest":"9974252022196793/10000000000000000","optimum":"10001/10000","minImprovement":"1/1000000000000","bountyEth":"0.50","challengeWindowHours":96,"postingBondEth":"0.03","challengeBondEth":"0.03","verifierVersion":"0.1.0","verifierImage":"sha256:local-dev","verifierCommand":"make verify SOLUTION=path","repoPath":"problems/pnt-sparse-mertens-construction","poolAddress":"base-sepolia:0x4242000000000000000000000000000000000009","donationWallet":{"chain":"Base Sepolia","asset":"ETH","address":"0x4242000000000000000000000000000000000009","status":"testnet-only","explorerUrl":"https://sepolia.basescan.org/address/0x4242000000000000000000000000000000000009","note":"Testnet deposit wallet reserved for the sparse Mertens construction gate."},"tagline":"Construction board split from the LP ceiling proof track.","description":"A queued sparse-support construction board. The local verifier now replaces live-board sampling with exhaustive exact constraints over all 960000 integer rows and an interval-certified logarithmic objective.","statement":"\\max_f\\ -\\sum_{k=2}^{96000} f(k)\\frac{\\log k}{k}\\quad\\text{s.t.}\\quad |\\operatorname{supp} f|\\le 2000,\\ |f(k)|\\le 10,\\ \\sum_{k\\ge1} f(k)\\lfloor x/k\\rfloor\\le 10001/10000\\ \\forall x\\le 960000","statementCaveat":"The verifier synthesizes the k = 1 balancing term, checks every integer row exactly, and accepts only a certified lower-bound decimal for the logarithmic objective; it is a finite reach-96000 construction board, not an asymptotic PNT theorem.","verifierStandard":["R1 rational support encoding","H3 all 960000 rows checked","R6 interval lower-bound objective","H5 one-ULP bad-decimal fixture"],"solutionSchema":{"type":"object","required":["reach","denominator","printed_decimal","support"]},"sampleSolution":{"reach":96000,"denominator":"10000000000000000000000000000","printed_decimal":"0.9974252022196793","support":[{"k":2,"value":"-10000989053719092000000000000"}]},"chainProvenance":{"settlementState":"local-only","chain":"Base Sepolia","chainId":84532,"donationWalletAddress":"0x4242000000000000000000000000000000000009","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."}}