{
  "schema": "diskcover-public-problem-v2",
  "problem_id": "diskcover-n27",
  "n": 27,
  "title": "Optimal Covering of the Unit Disk with 27 Disks",
  "language": "en",
  "page_url": "n27.html",
  "prompt_url": "prompts/n27.en.md",
  "alternate_pages": {
    "zh-CN": "n27.zh.html",
    "en": "n27.html"
  },
  "protocol": "diskcover-minpoly-v1",
  "protocol_url": "protocol.en.txt",
  "fingerprint_script_url": "diskcover_answer.py",
  "report_prompt_url": "report-prompt.en.txt",
  "reference_sha256": "c31dce1e3cac5f00d12105435f9f7c61a0456844a01867efbc69011f90473b55",
  "reference_status": "published",
  "reference_note": "Reference fingerprint published; a match does not verify a proof.",
  "reference_time_seconds": "21600",
  "scoring_version": "diskcover-log-score-v1",
  "score_formula": "round_half_up(200 + 100 * log2(reference_time_seconds / elapsed_seconds))",
  "score_rounding": "decimal_ROUND_HALF_UP_integer",
  "score_direction": "higher_is_better",
  "time_limit_seconds": "43200",
  "deadline_inclusive": true,
  "timeout_score": "0",
  "late_completion_score": "0",
  "not_tested_score": null,
  "scoring_rule": "Time limit: 12 hours (43200 seconds). If no valid result meeting this problem's acceptance criteria is obtained within 12 hours, the score is 0; completion after the deadline also scores 0. Completion within 12 hours, including the deadline, is scored as S = round(200 + 100 × log₂(reference time / elapsed time)); higher is better. An untested run is not scored as 0.",
  "scoring_status": "demo_reference_not_frozen",
  "public_record_submissions": "closed",
  "answer_disclosure": "fingerprint_only",
  "configuration": {
    "image": "assets/n27.svg",
    "caption": "Covering illustration: non-optimal configuration, display only"
  },
  "records_note": null,
  "records": [
    {
      "model": "gpt-6 astra/max",
      "model_id": "gpt-6-astra",
      "reasoning_effort": "max",
      "elapsed_seconds": "43200",
      "score": "0",
      "runs": 1,
      "status": "not_completed",
      "timing_scope": "fixed_deadline_observation_window_not_completion_time",
      "test_date_utc": "2026-10-07",
      "score_scope": "benchmark_failure_score",
      "acceptance_status": "failed_no_complete_valid_result_by_deadline",
      "model_identity_status": "verified_original_test_turn_context",
      "verification_status": "partial_candidate_verification; unconditional_global_optimality_incomplete; no_complete_valid_result_by_deadline",
      "report_url": "record-n27-astra-20261007.html",
      "report_url_en": "record-n27-astra-20261007.html",
      "within_time_limit": true,
      "score_source": "not_completed_zero"
    },
    {
      "model": "gpt-6.1 sol/max",
      "model_id": "gpt-6.1-sol",
      "reasoning_effort": "max",
      "elapsed_seconds": "43200",
      "score": "0",
      "status": "not_completed",
      "acceptance_status": "incomplete_at_deadline",
      "timing_scope": "natural_elapsed_deadline_cutoff_not_successful_completion",
      "model_identity_status": "verified_original_test_turn_context",
      "verification_status": "incomplete_at_deadline_no_complete_mathematical_submission",
      "score_scope": "failure_score",
      "test_date_utc": "2026-10-07",
      "report_url": "record-n27-sol-20261007.html",
      "report_url_en": "record-n27-sol-20261007.html",
      "source_report": "record-n27-sol-20261007.json",
      "runs": 1,
      "within_time_limit": true,
      "score_source": "not_completed_zero"
    }
  ],
  "best_recorded_run": {
    "run_id": "optimized-32-03",
    "elapsed_seconds": "882.4296346960001",
    "score": "661",
    "logical_cores": 32,
    "proof_engine": "DGC v62",
    "minpoly_engine": "MinPoly v8",
    "sha256": "c31dce1e3cac5f00d12105435f9f7c61a0456844a01867efbc69011f90473b55",
    "hash_matches": true,
    "score_scope": "benchmark_score",
    "label": "DGC v62 + MinPoly v8",
    "report_url": "record-n27-algorithm-run-20261005.html",
    "report_url_en": "record-n27-algorithm-run-20261005.html",
    "note": "882.43 s including a fresh build; 32 logical cores shared by phase. Inherited DGC source-admission and finite-classification assumptions remain; a fingerprint match does not verify the global proof.",
    "note_en": "882.43 s including a fresh build; 32 logical cores shared by phase. Inherited DGC source-admission and finite-classification assumptions remain; a fingerprint match does not verify the global proof."
  }
}