{"claim":{"id":"0909ed56-e997-44ec-b869-ff516a594846","submission_id":"8e724df5-6700-4ba9-abd4-b99cb26d0c77","submission_revision_id":"18090f7d-90b9-4ba3-ba53-0fd592bfbf03","builder":"builder-01","status":"active","claimed_at":"2026-10-15T14:55:48.546723Z","delivered_at":null,"ended_at":null,"delivery_note":null,"end_reason":null},"task":{"submission_id":"8e724df5-6700-4ba9-abd4-b99cb26d0c77","submission_revision_id":"18090f7d-90b9-4ba3-ba53-0fd592bfbf03","revision_version":2,"submission_status":"under_review","submitted_at":"2026-08-08T22:35:42.601308Z","submitter_name":"Yinuo Ren","submitter_email":"inuoren@gmail.com","task":{"status":"submitted","version":2,"extra_fields":{"difficulty_self_test":{"model":"GPT 5.5","harness":"Codex","estimated_score":"<1%"}},"submitted_at":"2026-08-08T22:35:42.601308+00:00","industry_domain":"Applied Mathematics","task_short_name":"exact_sos_certificate_1","operating_system":"Linux","task_description":"Objective: Produce a machine-checkable *proof* that bounds the global minimum\nof a multivariate polynomial from both sides - an exact rational\nsum-of-squares certificate for the lower bound, plus a rational witness point\nfor the upper bound - for each of three different polynomials. This is the\nstandard workflow of polynomial optimisation and formal global optimisation\n(Parrilo; Lasserre; Peyrl and Parrilo's exact rationalisation), used wherever\na numerically plausible bound is not good enough and a verifiable proof is\nrequired.\n\nStarting State: `input/polynomials.json` gives THREE polynomials p1, p2, p3,\neach in 5 variables of degree 6 with integer coefficients (61, 68 and 66\nterms), as\n`{\"n_vars\": 5, \"degree\": 6, \"polynomials\": {\"p1\": [{\"exponents\": [e1..e5],\n\"coeff\": c}, ...], \"p2\": [...], \"p3\": [...]}}`. The three instances are\nstructurally different: a balanced one, a stiff one, and one with a flat\nvalley. `input/basis.json` fixes the exact ordering of the 56 monomials of\ndegree at most 3 that form the Gram basis z(x). The full statement handed to\nthe agent is `problem_spec.md`.\n\nInstructions & Rules: For EVERY one of p1, p2, p3, find gamma as large as\npossible and prove p(x) >= gamma for all real x by exhibiting integers\n`den` > 0, `G`, and a 56 x 56 integer matrix `Q` with gamma = G/den and\nQ = Qint/den, such that p - gamma = z^T Q z identically and Q is positive\ndefinite. In addition give a rational witness point x0, as integer lists\n`x0_num` and `x0_den` with x0_i = x0_num[i]/x0_den[i], such that\n0 <= p(x0) - gamma <= 1e-5; the certificate proves min p >= gamma and the\nwitness proves min p <= p(x0), so together they enclose the global minimum.\nDeliver a single file `output/solution.json` mapping each of `p1`, `p2`, `p3`\nto an object with those five fields `{den, G, Q, x0_num, x0_den}`. The\ndenominator is capped at den <= 2^20 = 1048576, every integer must satisfy\n|v| < 2^220, and Q must be exactly symmetric. All three instances are graded\nand the score is the minimum over them, so one bad instance sinks the\nsubmission. Use only Python with NumPy and SciPy, no network, no package\ninstallation, and write only under `output/`. Budget: 2 hours on 4 vCPUs /\n8 GB RAM.\n\nWhy it is hard: (1) SciPy contains no semidefinite-programming solver, so the\nagent must build the semidefinite step from scratch - parametrise the\n1134-dimensional affine family of Gram matrices satisfying the identity\n(measured: the coefficient-matching map has rank 462 on a symmetric 56 x 56\nmatrix), then optimise lambda_min over it. (2) The answer must be exact: a\nfloating-point Gram matrix is worthless, and naive rounding destroys the\npolynomial identity outright [measured: 129 of the 462 monomial equations\nviolated, score 0.0, even though the rounded matrix is still positive\ndefinite]. The rounding must be followed by an exact integer repair. This is\nthe failure that killed all three self-test shots. (3) That repair has a\ngenuine parity obstruction - a symmetric off-diagonal entry moves its\nmonomial coefficient by 2*delta, so odd residuals are unreachable unless the\nwhole certificate is rescaled by 2. (4) The denominator cap forces a real\ntrade-off: a tighter gamma shrinks the spectral margin, while a coarse\nrounding grid demands a larger one [measured sweep: at the SOS optimum the\nsmallest denominator that still verifies is 2^25, above the 2^20 cap]. (5)\nThe witness adds global optimisation of a degree-6 polynomial in 5 variables\non top of the semidefinite work; a solver that produces only the certificate\nscores 0.5. (6) Three structurally different instances must all be certified\nin one run.","software_licensing":"Free/Open Source","evaluation_criteria":null,"verification_method":"A ready-made grader is included: `evaluate.py` plus `sos_common.py`. It reads\n`input/polynomials.json` from the task directory and `solution.json` from the\nagent's output directory, and performs NO floating-point arithmetic in any\nacceptance decision - the answer is either a complete proof or it is not.\nNothing is compared against the reference file; every property is recomputed\nfrom the agent's own submission.\n\nApplied independently to each of p1, p2, p3:\n\n1. Structure. The instance's entry has an integer `den` with 0 < den <= 2^20\n   = 1048576, an integer `G`, and an exactly symmetric 56 x 56 integer matrix\n   `Q`; every submitted integer satisfies |v| < 2^220. A missing instance key\n   scores 0.0 for that instance.\n2. Identity. For each of the 462 monomials alpha of degree at most 6, the sum\n   of Q[i][j] over ordered pairs (i, j) with z_i + z_j = alpha must equal\n   den * p_alpha - G * [alpha = 0], as integers. Any violated equation fails\n   the instance.\n3. Positive definiteness. Sylvester's criterion: all 56 leading principal\n   minors of Q must be strictly positive, computed by fraction-free (Bareiss)\n   elimination - exact integer arithmetic, no rounding, no eigenvalues.\n4. Tightness. gamma = G/den is compared to the instance's target by exact\n   rational cross-multiplication: p1 >= -13.816645, p2 >= -86.681433,\n   p3 >= -15.831777.\n5. Witness. x0 must have 5 rational coordinates with positive denominators;\n   p(x0) is evaluated as an exact Fraction and the enclosure width\n   p(x0) - gamma must satisfy 0 <= p(x0) - gamma <= 1e-5, again by exact\n   rational comparison.\n\nPer-instance score: 1.0 if rules 1-5 pass. 0.5 if rules 1-3 pass but gamma\nclears only the loose bound (p1 >= -13.917, p2 >= -86.781, p3 >= -15.932), or\nif the certificate meets its target but the witness is missing, malformed, or\nleaves an enclosure wider than 1e-5. 0.0 if the submission is structurally\ninvalid, if any identity equation is violated, if Q is not positive definite,\nor if gamma misses even the loose bound.\n\nTask score: 1.0 only if all three instances score 1.0; 0.5 if all three score\nat least 0.5; 0.0 otherwise. The score is therefore the minimum over\ninstances, and one bad instance sinks the submission.\n\nCalibration [measured]: the thresholds are not arbitrary, because the\ngrader's own accuracy is exact - the only judgement call is how close to the\nSOS optimum the certificate must come. The per-instance optima were computed\nby bisection to 1e-9 and cross-checked against Newton-polished global\nminimisation, agreeing to about 2e-7; for p1, gamma* = -13.8166248568 against\nan empirical global minimum of -13.8166248566, i.e. agreement to 9 digits, so\nthe SOS relaxation is tight on these instances. Each full-credit target is\nthat instance's optimum minus 2e-5 - two orders of magnitude outside the\ncross-check disagreement, so no correct solver is failed by numerical noise -\nand each partial target is the optimum minus 0.1. The reference run lands\ncomfortably inside the full-credit band (gamma_p1 = -13.8166275024, and\nenclosure widths of 2.65e-6 / 2.58e-6 / 3.74e-6 against the permitted 1e-5).\nThe 1e-5 enclosure tolerance is likewise set about four times the widest\nwidth the oracle actually achieved. The den <= 2^20 cap is calibrated from a\nmeasured (retreat, denominator) sweep: at the SOS optimum the smallest\ndenominator that still verifies is 2^25, at retreat 1e-6 it is 2^21, and at\nretreat 1e-5 it is 2^17 - so the cap closes the degenerate \"round on an\narbitrarily fine grid\" route by 5 bits while leaving the intended\nmargin-versus-tightness trade-off wide open. Naive-route controls, measured:\nrounding the optimal Gram matrix without the exact identity repair violates\n129 of the 462 equations and scores 0.0 even though the matrix is positive\ndefinite; retreating far enough (1e-2) to make rounding trivial yields a\nvalid certificate that misses the target and scores 0.5; a valid certificate\nwith no usable witness scores 0.5. Anti-cheat: there is no hidden reference\nanswer to match and nothing to leak - a fabricated Gram matrix cannot satisfy\n462 integer equations by accident - and the only material withheld from the\nagent is the reference solution and the oracle source; the checks, the\ntargets and the instances are all stated in `problem_spec.md`.","software_and_version":"Python 3.11 with NumPy 1.26.4 and SciPy 1.11.4","specific_requirements":null,"expected_completion_time":null},"files":[{"file_id":"077fbd80-640f-478f-8530-c294d9d73b2a","filename":"polynomials.json","category":"input_materials","content_type":"application/json","size_bytes":17983},{"file_id":"18ca71a9-682d-42a1-b462-2fbee5bb24ca","filename":"basis.json","category":"input_materials","content_type":"application/json","size_bytes":2205},{"file_id":"e47505df-7a98-447d-9903-f3f1288db05b","filename":"problem_spec.md","category":"input_materials","content_type":"text/markdown","size_bytes":4044},{"file_id":"06c366cf-fe68-423b-8ef4-e793a0dce61b","filename":"solution_multi.json","category":"reference_output","content_type":"application/json","size_bytes":74631},{"file_id":"17ec72ba-54af-4d75-a7fe-619d8447137a","filename":"solver_report_multi.json","category":"reference_output","content_type":"application/json","size_bytes":652},{"file_id":"101bc80e-cc2b-47cf-930e-4b11a9e8b5a6","filename":"evaluate.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":1307},{"file_id":"7d36edc5-d5db-4fdd-a796-c61afff8de14","filename":"solve_all.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":5314},{"file_id":"9e59e67e-2c04-4f74-a2f4-721be8264117","filename":"reference_solver.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":10621},{"file_id":"a40a0d2b-feef-470e-80d9-56efd6560a67","filename":"sos_common.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":11031},{"file_id":"e148fad7-6862-451c-b8ed-4fdefe167e70","filename":"DIFFICULTY.md","category":"difficulty_self_test_evidence","content_type":"text/markdown","size_bytes":13353}],"deep_review":{"overall_verdict":"Accept","review_text":"**Overall Verdict:** Accept — The task is complete, objectively verifiable, and end-to-end reachable; only non-material difficulty metadata is inconsistent with the attached evidence.\n\n**Completeness Check:**\n- Instruction: Complete — The specification clearly defines all three optimization instances, the exact certificate and witness schema, resource constraints, and required output.\n- Inputs: Complete — The polynomial input contains three valid five-variable degree-six instances with 61, 68, and 66 unique terms; the basis contains 56 unique exponent vectors in the required order.\n- Toolchain: Complete — The offline Python, NumPy, and SciPy workflow is legally available, compatible with the stated environment, and demonstrated within the stated single-machine resource budget.\n- Reference output: Complete — The reference contains all required fields for all three instances. Each matrix is symmetric and positive definite, satisfies all 462 coefficient identities exactly, clears its target, and has an exact witness gap below \\(10^{-5}\\).\n- Rubric: Complete — The evaluator provides deterministic structure, identity, positive-definiteness, tightness, witness, and minimum-over-instances scoring with meaningful full, partial, and failure outcomes.\n\n**File and Task Reachability:**\nThe attachment set is `reachable`, and the end-to-end task is `reachable`. The inspected files were `polynomials.json`, `basis.json`, `problem_spec.md`, `solution_multi.json`, `solver_report_multi.json`, `evaluate.py`, `solve_all.py`, `reference_solver.py`, `sos_common.py`, and `DIFFICULTY.md`. All manifest files were readable, matched their declared byte counts and broad formats, and contained no archives, encryption barriers, or undisclosed dependencies. All JSON parsed successfully; uploaded source was inspected as text and not executed. No content remains unreviewed, and no tooling limitation affected the assessment. Solver-visible inputs are separated by manifest role from reference and evaluator assets. The completion path is clear: read the polynomial and basis inputs, construct three exact certificates and rational witnesses, write `output/solution.json`, and evaluate it through exact integer and rational checks. License 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:**\nNone. No affirmative evidence supports a screening label; public-benchmark identifiability could not be externally corroborated within the permitted evidence boundary.\n\n**Recommended Changes:**\nThe mathematical task, reference certificates, and evaluator are ready. No material changes are required. As non-blocking cleanup, align the submitted “<1%” difficulty estimate and model label with `DIFFICULTY.md`, which explicitly supports a 10–20% tier from three GPT-5-based trials and states that the evidence cannot justify a sub-1% estimate.","reviewed_at":"2026-08-15T00:27:33.337844Z"},"claim":{"id":"0909ed56-e997-44ec-b869-ff516a594846","submission_id":"8e724df5-6700-4ba9-abd4-b99cb26d0c77","submission_revision_id":"18090f7d-90b9-4ba3-ba53-0fd592bfbf03","builder":"builder-01","status":"active","claimed_at":"2026-10-15T14:55:48.546723Z","delivered_at":null,"ended_at":null,"delivery_note":null,"end_reason":null}},"revision_is_current":true}