{"submission_id":"daf13c7d-b840-4f3d-8382-9aeb50bfe9fb","submission_revision_id":"54f11322-530c-4047-a0b5-e61a6b7c42d2","revision_version":1,"submission_status":"submitted","submitted_at":"2026-10-15T05:48:33.580692Z","submitter_name":"Haoyang Su","submitter_email":"hoyantsu@gmail.com","task":{"status":"submitted","version":1,"domain_key":null,"extra_fields":{"difficulty_self_test":{"model":"GPT-6 Astra","harness":"Codex","estimated_score":"<1%"}},"submitted_at":"2026-10-15T05:48:33.580692+00:00","domain_payload":null,"industry_domain":"Algebraic geometry","task_short_name":"exact_degree_twelve_quadratic_module_optimum_01","operating_system":"Linux","task_description":"Determine the exact optimum of a degree-twelve quadratic module\n\nLet x1,...,x12 be independent real indeterminates and gi=xi-xi^2. Define\n\nC* = inf { C in R : x1*x2*x3*x4*x5*x6*x7*x8*x9*x10*x11*x12 + C = sigma0 + sum(i=1,...,12) gi*sigmai,\n                   sigma0,...,sigma12 are sums of squares,\n                   degree(sigma0)<=12 and degree(sigmai)<=10 for i=1,...,12 }.\n\nDetermine C* exactly and prove that the infimum is attained. Supply matching exact primal and dual certificates.\n\nThe primal certificate must give explicit polynomial bases and positive-semidefinite Gram matrices satisfying the identity at C=C*. The dual certificate must give a linear functional L on all polynomials of degree at most twelve such that L(1)=1, L(f^2)>=0 for every polynomial f of degree at most six, L(gi*h^2)>=0 for every i and every polynomial h of degree at most five, and L(x1*x2*x3*x4*x5*x6*x7*x8*x9*x10*x11*x12)=-C*.\n\nAll coefficients and positivity claims must be exact. Real algebraic coefficients are permitted. A decimal optimum, an approximate feasible SDP point, a numerical eigenvalue calculation, unequal upper and lower bounds, or a certificate for a merely feasible constant does not solve the task. The identity must hold as an ordinary polynomial identity over the reals; imposing xi^2=xi on the primal identity is not permitted.\n\nNo polynomial bases, matrices, candidate optimum, moment values, or active constraints are provided.\n\nDeliverables: exact machine-readable primal and dual certificates, a derivation, and a verifier that checks all coefficient identities and positivity claims using exact arithmetic. The certificate representation may be chosen by the solver, as described in certificate_format.txt.\n","taxonomy_version":null,"software_licensing":"Free/Open Source","evaluation_criteria":null,"verification_method":"The exact reference optimum is C=(975752827-136955*sqrt(4561921))/242476882560, using the positive square root. Compare exact real algebraic values, not decimal strings.\n\nAward 10 points for exactly identifying C, 45 points for a complete exact primal attainment certificate at C, and 45 points for a complete exact dual lower-bound certificate at C. Each certificate criterion is all-or-nothing. Full success requires 100/100.\n\nCheck every ordinary polynomial coefficient, all degree bounds (12 for sigma0 and 10 for each sigmai), exact Gram symmetry and positive semidefiniteness, and every symmetry averaging factor. Boolean reduction cannot replace the ordinary primal identity.\n\nCheck the dual on the complete degree-twelve polynomial space: L(1)=1, L(product xi)=-C, positivity on all degree-six squares and every gi-localized degree-five square. The complete moment space has dimension 18564. A compressed representation must prove coverage of the whole space. Numerical eigenvalues, optimizer statuses and incomplete test-square subsets are insufficient.\n\nEquivalent exact representations and proofs are accepted. The reference verifier checks one representation; other representations require equally rigorous interpretation and checking. A candidate acceptance message alone is insufficient.\n","software_and_version":"Python 3.13.12; SymPy 1.14.0; NumPy 2.5.4","domain_schema_version":null,"specific_requirements":null,"expected_completion_time":null},"files":[{"file_id":"f63643c6-b7d8-49ac-95a9-ad005800d524","filename":"optimal_sos_n12_inputs.zip","category":"input_materials","content_type":"application/x-zip-compressed","size_bytes":1863},{"file_id":"e1735969-6bae-45b7-b17c-19dc2f1b68cc","filename":"optimal_sos_n12_reference.zip","category":"reference_output","content_type":"application/x-zip-compressed","size_bytes":1332216},{"file_id":"15e99c50-df06-443c-8993-e57329134c6e","filename":"optimal_sos_n12_evidence.zip","category":"difficulty_self_test_evidence","content_type":"application/x-zip-compressed","size_bytes":141204}],"deep_review":{"overall_verdict":"Strong Accept","review_text":"**Overall Verdict:** Strong Accept — The submission is complete, reachable, and supported by exact reference certificates whose substantive mathematical checks were independently confirmed.\n\n**Completeness Check:**\n\n- Instruction: Complete — Clearly defines the optimization problem, degree bounds, exactness requirements, and required deliverables.\n- Inputs: Complete — Solver-visible task and certificate-format files are sufficient and separated from reference materials.\n- Toolchain: Complete — The open-source Python workflow is suitable for a single machine and requires no external service.\n- Reference output: Complete — Includes explicit primal and dual JSON certificates, derivation, verification source, configuration, and verification log.\n- Rubric: Complete — The 10/45/45 allocation objectively distinguishes exact identification, primal attainment, and complete dual certification.\n\n**File and Task Reachability:**\n\nAttachments: `reachable`. Task: `reachable`.\n\nInspected `optimal_sos_n12_inputs.zip`, `optimal_sos_n12_reference.zip`, and `optimal_sos_n12_evidence.zip`, including the task, certificate format, both reference certificates, derivation, verification source, and trial evidence. All three archives matched their manifest byte counts, passed archive-integrity checks, and contained readable UTF-8 text or JSON without unsafe paths or encryption. All 43 members were inventoried; large exploratory numerical records received structural and representative-content inspection.\n\nIndependent exact checks confirmed the reference optimum, every ordinary primal coefficient identity, basis substitutions, degree bounds, Gram symmetry and factorizations, and positivity of all 21 primal blocks. Dual checks confirmed normalization, objective equality, all seven positivity blocks, and reconstruction of all 140 moment-matrix entry orbits. The finite moment rule and parity reduction cover the complete ordinary polynomial spaces and make every required localizer zero. Uploaded programs were inspected, not executed.\n\nThe completion path is coherent: derive certificates from the solver-visible inputs, submit exact JSON data with a derivation and verifier, and evaluate the three rubric criteria. The self-test documents one incomplete, zero-score trial; it does not establish a population success probability.\n\nLicense keys, accounts, tokens, and other credentials are not required from the contributor. Confidential, proprietary, or internal material may be evaluated as submitted evidence without requiring additional disclosure.\n\n**Special Screening:**\n\nNone. Trial evidence cites related published research, but public availability of this exact task or protected solution was not independently established.\n\n**Recommended Changes:**\n\nThe complete exact reference and transparent reporting of the unsuccessful self-test support readiness. No changes are required.","reviewed_at":"2026-10-15T05:57:20.525512Z"},"claim":null}