{"claim":{"id":"c60adf65-c5da-48f1-b447-9a2b240d7146","submission_id":"95a8d0d4-72cb-4be7-ad90-57599324b2a9","submission_revision_id":"d80c0c3a-62bc-4ec5-bae0-b1fd73d58589","builder":"builder-01","status":"active","claimed_at":"2026-10-15T14:55:24.405028Z","delivered_at":null,"ended_at":null,"delivery_note":null,"end_reason":null},"task":{"submission_id":"95a8d0d4-72cb-4be7-ad90-57599324b2a9","submission_revision_id":"d80c0c3a-62bc-4ec5-bae0-b1fd73d58589","revision_version":2,"submission_status":"under_review","submitted_at":"2026-08-08T22:33:49.241189Z","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":"2026-08-08T22:33:49.241189+00:00","industry_domain":"Applied Mathematics","task_short_name":"dde_period_doubling_point_1","operating_system":"Linux","task_description":"Objective: Locate a period-doubling (flip) bifurcation point of a delay\ndifferential equation - the critical delay tau* at which an unstable periodic\norbit's dominant nontrivial Floquet multiplier passes through -1 - deliver the\norbit at that parameter value, and then switch onto the doubled branch and\ndeliver the period-doubled orbit born at the bifurcation, evaluated at a fixed\nevidence delay tau_ev = 14.0. Continuation to a codimension-one bifurcation and\nbranch switching are the central workflow of numerical bifurcation analysis and\nprecisely what DDE-BIFTOOL exists to do; they underpin stability analysis in\nlaser dynamics, machining chatter, physiological control, and networked-control\nmodels.\n\nStarting State: No data files. The task is fully specified by the equation\nx'(t) = beta*x(t-tau)/(1+x(t-tau)^n) - gamma*x(t) with beta = 0.21,\ngamma = 0.105, n = 9.8 and tau free. The complete statement is in\n`problem_spec.md`, the only input material.\n\nInstructions & Rules: Deliverable (a) is tau* in [10, 20] at which the dominant\nnontrivial Floquet multiplier of the periodic orbit equals -1, together with\nthat orbit. Deliverable (b) is the period-doubled orbit at tau_ev = 14.0.\nWrite `output/solution.npz` containing `tau_star` (float64 scalar), `period`\n(float64 scalar T), `x` (float64, shape (4096,), samples at t_j = j*T/4096),\n`period2` (float64 scalar) and `x2` (float64, shape (4096,), the doubled\norbit's samples over one period); and `output/results.json` with keys\n`tau_star`, `period`, `resid_linf`, `mu_re`, `mu_im`, `period2`,\n`resid_linf_2`, reporting your own values truthfully. Acceptance requires each\norbit to satisfy the delay equation to 1e-8 (spectral residual on the\ntrigonometric interpolant), to be positive, fundamental and resolved, and - the\ncrux for (a) - the grader's own recomputation of the dominant nontrivial\nmultiplier of YOUR orbit at YOUR tau* must satisfy |mu + 1| <= 1e-4. Beware in\n(b): the period-1 orbit written at double period is an exact solution of the\nsame equation at the same delay and is separated from a genuine doubled orbit\nonly by the fundamentality condition. Use only Python 3 with NumPy and SciPy;\nSciPy has no delay-equation support, so the nonlinear integrator, the periodic\nboundary-value solver and the linearised monodromy operator must all be built\nfrom scratch. No network access, no package installation, write only under\n`output/`. Budget: 3 hours on 4 vCPUs / 8 GB RAM.\n\nWhy it is hard: four measured obstacles stack. (1) The orbit is unstable, so\nforward simulation actively repels it - it must be found by recurrence mining\nor continuation plus a collocation Newton solve. (2) Its stability requires\ndiscretising the monodromy operator of a delay equation, a linear map on the\ninfinite-dimensional history space C([-tau,0]), with correct identification of\nthe trivial eigenvalue mu = 1 that time-translation always contributes. (3) The\nbifurcation itself must then be located by continuation in tau and a root-solve\non mu(tau) + 1, so every error in layers 1 and 2 propagates into the answer;\nsince dmu/dtau is about -0.168 along this branch, the tolerance corresponds to\npinning tau* to about 6e-4, and a single accurate orbit is not an answer. (4)\nThe branch switch guards a real trap: the decoy orbit has a measured residual\nof 4.3e-14, well inside the 1e-8 gate, and is rejected only by the\nfundamentality gap (measured 4.4e-16 for the decoy versus 0.191 for the genuine\ndoubled orbit); a Newton continuation that is not re-seeded slides onto it\nsilently. In the measured self-test, a GPT-5-based coding agent scored 0.0 on\nall three shots: its fixed-point attempt failed the amplitude and\nfundamentality requirements, its smooth nontrivial guesses missed the DDE\nresidual by many orders of magnitude and did not meet the multiplier condition,\nand no genuine doubled branch was produced at all.\n","software_licensing":"Free/Open Source","evaluation_criteria":null,"verification_method":"A ready-made grader (`evaluate.py`, with `bif_common.py` alongside it) is\nincluded and should be run as `evaluate(task_dir, output_dir)`. It recomputes\neverything from the agent's own files, so no comparison against the reference\narrays is needed and no hidden reference value exists: \"the dominant nontrivial\nFloquet multiplier equals -1\" is an intrinsic property of the submitted\n(tau*, orbit) pair. Any correct bifurcation point passes regardless of how it\nwas found; a wrong one fails regardless. Strict rules, in the order the grader\napplies them:\n\n1. Structure. `output/solution.npz` must contain float64 `tau_star`, `period`\n   and `x` of shape (4096,), all finite, with `tau_star` in [10, 20] and\n   `period` > `tau_star`. `output/results.json` must be present and must\n   contain `tau_star` and `resid_linf`.\n2. Truthfulness gate. The `results.json` `tau_star` must match `solution.npz`\n   to 1e-9 relative, and the claimed `resid_linf` must lie within a factor of\n   10 of the grader's recomputed residual (with floors at 1e-16 and 1e-13 so\n   machine-precision residuals are not penalised). A submission cannot pass by\n   reporting flattering numbers.\n3. Orbit at tau*, computed on the trigonometric interpolant of the 4096 samples\n   (derivative via rfft multipliers 2*pi*i*k/T with the Nyquist derivative mode\n   zeroed; delayed values via multipliers exp(-2*pi*i*k*(tau_star mod T)/T)):\n   min(x) > 0; amplitude max(x) - min(x) >= 0.4; relative norm of the top 1/8\n   of Fourier modes <= 1e-10; fundamentality gap, the minimum over k in 2..6 of\n   max_j |x_j - (T/k-shift of x)_j|, >= 0.1; DDE residual max_j |r_j| <= 1e-8.\n4. Bifurcation condition. The grader rebuilds the monodromy operator of the\n   linearised equation at the submitted tau*, at two resolutions\n   (nh = 280 with 700 steps per delay, and nh = 400 with 1400), Richardson-\n   extrapolates the dominant nontrivial eigenvalue, and requires\n   |mu + 1| <= 1e-4 for full credit or <= 1e-2 for the partial band.\n5. Doubled orbit at the fixed evidence delay tau_ev = 14.0, with the same\n   spectral machinery: `period2` in [70, 95]; min(x2) > 0; amplitude >= 0.4;\n   spectral tail <= 1e-10; DDE residual at tau_ev <= 1e-8; and fundamentality\n   gap >= 0.1, which is the rule that rejects the period-1 orbit written at\n   double period.\n\nScoring map: 1.0 when rules 1-4 hold at |mu + 1| <= 1e-4 and rule 5 holds in\nfull. 0.5 when rules 1-3 hold and |mu + 1| <= 1e-2 but rule 5 fails, is a\ndecoy, or `period2` / `x2` are missing. 0.0 otherwise. Grading is deterministic\n- fixed resolutions, pure float64 NumPy on the submitted arrays, no data files\n- so re-runs are bitwise identical; measured grader wall time is about 90 s.\n\nCalibration [measured]: every threshold above is set by the grader's own\ncertified accuracy, not by ambition. (a) The |mu + 1| <= 1e-4 bound: at the\nbifurcation orbit the grader's two-resolution Richardson estimator shows a\nspread of 2.7e-6 between resolutions, and its trivial-multiplier canary - the\nexact eigenvalue mu = 1, which any correct discretisation must reproduce -\nsits at 2.3e-5, so 1e-4 is the tightest bound the grader can honestly vouch\nfor. The reference clears it by seven orders of magnitude at 1.6e-11, giving\nample headroom on the accepted side, and because dmu/dtau is about -0.168 along\nthis branch the bound still pins tau* to roughly 6e-4 absolute. (b) The 1e-8\nresidual and 1e-10 spectral-tail bounds: the reference orbits sit at residual\n8.8e-14 and 4.0e-14 and spectral tails 3.0e-17 and 3.4e-17, four to six orders\nbelow the gates, so the gates separate genuine solutions from\nunder-resolved ones without being knife-edge. (c) The fundamentality gap\n>= 0.1: measured 0.333 for the orbit at tau* and 0.191 for the genuine doubled\norbit, against 4.4e-16 for the decoy, whose residual of 4.3e-14 would otherwise\npass every other check - this single threshold is what makes deliverable (b)\nmeaningful. (d) The amplitude >= 0.4 bound: reference amplitudes are 0.615 and\n0.768, so the gate rejects near-constant fixed-point solutions without\nexcluding genuine orbits. (e) The [70, 95] window on `period2`: the reference\ndoubled period is 83.4264, while the period-1 orbit at tau_ev has period\n40.3823, so the window admits the doubled branch and excludes the single one.","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":"a2bbdd2e-108e-48f2-9fa1-fe5d22a44edb","filename":"problem_spec.md","category":"input_materials","content_type":"text/markdown","size_bytes":4165},{"file_id":"0cd51b39-483d-438d-a30b-62e2539f1407","filename":"results.json","category":"reference_output","content_type":"application/json","size_bytes":227},{"file_id":"e0e59ced-30a1-4bc3-a1a9-5af14de3966d","filename":"doubled_orbit_report.json","category":"reference_output","content_type":"application/json","size_bytes":321},{"file_id":"44b62ea9-409b-4f6d-a163-67345f640452","filename":"submission.npz","category":"reference_output","content_type":"application/octet-stream","size_bytes":66790},{"file_id":"772a061d-8f59-4197-af49-2d7fcfba2705","filename":"solver_report.json","category":"reference_output","content_type":"application/json","size_bytes":488},{"file_id":"66f22c75-77d3-4176-a4c7-10d62fc04512","filename":"trap_report.json","category":"reference_output","content_type":"application/json","size_bytes":284},{"file_id":"4c7c6a2d-c54e-4c75-a147-d68fe45c7a0c","filename":"doubled_orbit.npz","category":"reference_output","content_type":"application/octet-stream","size_bytes":33522},{"file_id":"8f771ec8-dc2a-48bc-be38-06aa2c3cf0ae","filename":"solution.npz","category":"reference_output","content_type":"application/octet-stream","size_bytes":33532},{"file_id":"79b84a75-28d2-44f1-a6ce-a53c7bc8b7e7","filename":"calibration.md","category":"reference_output","content_type":"text/markdown","size_bytes":3810},{"file_id":"9dd97050-9f36-4477-b468-1a01eef60a3a","filename":"evaluate.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":2397},{"file_id":"ebb5f5b6-fc9e-4476-b3ff-1ccef5b2330e","filename":"bif_common.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":10608},{"file_id":"3203a947-dc2e-4133-9985-2adbc8d7b7ef","filename":"find_bifurcation.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":5018},{"file_id":"798f80bf-bbef-4614-93ad-6ddbf56ce225","filename":"find_cascade.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":7513},{"file_id":"f713ba21-b83d-4e29-acb4-4dafbb02cd5e","filename":"make_doubled_ref.py","category":"reference_output","content_type":"text/x-python-script","size_bytes":1172},{"file_id":"af5e8c77-6e30-4fd8-96ad-def089a54f1a","filename":"DIFFICULTY.md","category":"difficulty_self_test_evidence","content_type":"text/markdown","size_bytes":14899}],"deep_review":{"overall_verdict":"Accept","review_text":"**Overall Verdict:** Accept — the task is complete, objectively self-certifying, and feasible, with only nonblocking reproducibility and enforcement inconsistencies in auxiliary materials.\n\n**Completeness Check:**\n- Instruction: Complete — the equation, parameters, deliverables, sampling convention, constraints, and success criteria are explicit.\n- Inputs: Complete — `problem_spec.md` fully defines the task; no external data is required.\n- Toolchain: Complete — Python, NumPy, and SciPy are legally obtainable and compatible with the stated Linux resources and runtime.\n- Reference output: Complete — `submission.npz` contains both required orbits with correct float64 schemas; component archives and reports are internally consistent.\n- Rubric: Complete — `evaluate.py` and `bif_common.py` provide deterministic structural, residual, fundamentality, resolution, Floquet, truthfulness, and partial-credit checks. Five declared reporting keys and original input dtypes are not strictly enforced, but this does not weaken the substantive scientific gates.\n\n**File and Task Reachability:**\nAttachment classification: `partially_reachable`. Task classification: `reachable`.\n\nInspected `problem_spec.md`, `results.json`, `doubled_orbit_report.json`, `submission.npz`, `solver_report.json`, `trap_report.json`, `doubled_orbit.npz`, `solution.npz`, `calibration.md`, `evaluate.py`, `bif_common.py`, `find_bifurcation.py`, `find_cascade.py`, `make_doubled_ref.py`, and `DIFFICULTY.md`. All manifest byte counts matched; JSON and text were readable; the NumPy archives were unencrypted, CRC-valid ZIP/NPY containers with finite float64 payloads and safe member names. The combined archive matches the component orbit arrays byte-for-byte.\n\nNo attachment content was left unreviewed. Uploaded programs were statically inspected and not executed. The numerical residual and Floquet claims were therefore not independently rerun, but their evaluator implementation, schemas, reports, and cross-file consistency were reviewed. The auxiliary `find_bifurcation.py` references a seed orbit absent from the manifest; this does not block the task because the instruction requires solving from the equation, the evidence documents a seed-free continuation route, and evaluation depends only on the submitted outputs.\n\nThe end-to-end path is clear: construct and continue the periodic orbit, locate the multiplier crossing, switch to the doubled branch, serialize both sampled orbits, and run the deterministic evaluator. Reported reference runtimes fit comfortably within the stated single-machine budget. Reference artifacts expose exact answers and must remain evaluator-only; the manifest correctly separates them from the sole solver-visible input.\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:**\nNone. No affirmative evidence supports a screening label; public identifiability was not externally checked within the permitted evidence boundary.\n\n**Recommended Changes:**\nNo material changes are required. Optional cleanup would be to include the cited seed-free turnkey solver instead of the seed-dependent auxiliary generator, and make the evaluator enforce every declared `results.json` key and original float64 dtype.","reviewed_at":"2026-08-15T00:27:30.299620Z"},"claim":{"id":"c60adf65-c5da-48f1-b447-9a2b240d7146","submission_id":"95a8d0d4-72cb-4be7-ad90-57599324b2a9","submission_revision_id":"d80c0c3a-62bc-4ec5-bae0-b1fd73d58589","builder":"builder-01","status":"active","claimed_at":"2026-10-15T14:55:24.405028Z","delivered_at":null,"ended_at":null,"delivery_note":null,"end_reason":null}},"revision_is_current":true}