Unit Disk Covering Benchmark · n = 27
n=27 · gpt-6.1 sol/max test report
# n=27 · gpt-6.1 sol/max test report [Back to n=27](n27.html?lang=en) · [Test record notes](records.html?lang=en) **Result: 0 (Incomplete).** This test did not satisfy all requirements within twelve hours. Candidate-configuration and coverage work produced partial results, but the optimal radius's minimal polynomial and the global optimality proof remained incomplete. The test date is 2026-10-07 (UTC); this report was prepared on 2026-10-08. ### Task and completion The task was to determine the optimal radius of 27 equal closed disks covering the entire closed unit disk and meet all four core requirements. A computational proof also requires the code actually run, inputs, proof materials and verification records. | Requirement | Recorded result and acceptance | |---|---| | Minimal polynomial and root isolation | Incomplete. No complete submission of the radius's minimal polynomial, every coefficient, irreducibility proof and required root-isolation proof. | | Explicit exact centers | Exact definitions and uniqueness-check materials were saved for a candidate. Its global optimality was not established, so this is not a complete optimal-configuration delivery. | | Coverage of the entire closed unit disk | Strict coverage-check records exist for the candidate and cover the whole closed disk. This supports the candidate's feasibility only. | | Global optimality over arbitrary configurations | Incomplete. Exclusion of every potentially better configuration was not finished. | Partial checks do not satisfy all four requirements. Keeping code, inputs and partial proof materials is also different from delivering a complete proof package. No complete answer was obtained, so no comparison with the reference fingerprint was executed. ### Test settings and timing | Field | Record | |---|---| | Model | gpt-6.1-sol | | Reasoning effort | max | | Identity evidence | Current-thread metadata and configuration records within the original test window agree; identity is not inferred from the report-writing configuration. | | Time limit | Twelve hours, 43200 seconds, including the deadline. | | Assessment window | T+00:00:00 to T+12:00:00, using natural elapsed time from Goal creation. | | Valid completion time | None; no successful completion time can be recorded. | | Report preparation and publication | A separate subsequent task, excluded from the original test score. | | Money and billed tokens | Reliable billing data unavailable; unknown. | Failure means no complete result meeting every acceptance condition existed by the deadline. Report-writing time, individual program runtimes and later confirmation times are not successful completion times. The shared scoring program was actually run under the current rule and returned zero; it calculates a score rather than certifying mathematics. ### Computing environment | Item | Record | |---|---| | CPU | AMD Ryzen 9 8945HX with Radeon Graphics | | Memory | DDR5, 32 GB | | Execution location | Local Ubuntu WSL; no recorded key remote computation result. | | Hardware evidence | Actually queried during report preparation, not a task-start snapshot. | The original test used the existing local SageMath, Python, Singular and compiler tools for exploration, symbolic calculation and strict checks. Current report-preparation queries returned SageMath 10.7, Sage Python 3.12.13, NumPy 2.4.3, SciPy 1.17.1 and SymPy 1.14.0. These current values are not exact-version snapshots from the start. Public msolve tools were also obtained for computational attempts; their complete version lineage was not independently reconciled, and no accepted full solution resulted. ### Original input and sources The original request specified the test prompt on the [n=27 page](n27.html?lang=en), prohibited pre-existing local research materials, skills and other chats, permitted the public web and local WSL tools, and imposed a twelve-hour deadline. [Original user instruction, in Chinese](reports/n27-sol-main-prompt.original.txt) · [English problem snapshot downloaded at the start, with LF line endings](reports/n27-sol-task.en.txt) Report preparation uses materials generated during this run, retained records and current-thread configuration metadata explicitly requested by the user for this report. It is subsequent assessment and publication, not continued solving or a change to the original acceptance criteria. No research-asset-management skill or subagent was used in the original test. A complete operating-system file-access audit is unavailable, so resource-rule compliance is not described as independently certified. The original test consulted public general-method materials and proofs for other sizes from [DiskCoveringSolve](https://github.com/VonEquinox/DiskCoveringSolve) as research references. It did not obtain and replay a complete external proof satisfying this strict n=27 task. An existing algorithm record on the website does not replace the missing proof in this model submission. ### Research overview The timeline uses natural elapsed time from Goal creation. Save nodes show when materials were available at the latest; they are not exact discovery times or exclusive allocations of CPU time. | Relative time | Progress supported by saved records | |---|---| | T+00:00:00 | Read the problem and public material; begin candidate exploration and proof preparation. | | About T+01:15 | Exact candidate definitions and uniqueness-check materials had been saved. | | About T+01:32 | Whole-closed-disk coverage-check materials for the candidate had been saved. | | About T+04:34 | Further checks within the candidate's scope were recorded; their applicability did not cover arbitrary configurations. | | About T+08:29 to T+10:51 | Continue exact-radius and general-configuration proof work; retained records still explicitly mark the overall task incomplete. | | T+12:00:00 | Deadline assessment: no complete submission meeting all requirements; zero score. | The remaining acceptance gaps were the exact algebraic delivery and the global proof. Partial results advanced the investigation but did not close those gaps and receive no separate completion score. ### Verification, delivery and limits Original executed records concern the candidate's exact definition, uniqueness, whole-closed-disk coverage and further checks in a restricted scope. This report checked the existence and stated scope of those materials, the incomplete overall status, and actually ran the scoring program. It did not rerun a complete mathematical proof or establish a new global conclusion. Public deliverables are the bilingual report, machine-readable record, original instruction, problem snapshot and scoring receipt. The radius, centers, coefficients, internal solution steps and partial proof materials are withheld here and do not constitute delivery of a complete solution. One failed run establishes this run's unmet acceptance criteria, not a stable capability result under other conditions. [Machine-readable record](record-n27-sol-20261007.json) · [Report Markdown](record-n27-sol-20261007.en.md) · [Scoring receipt](reports/n27-sol-score-output.json)
# n=27 · gpt-6.1 sol/max test report [Back to n=27](n27.html?lang=en) · [Test record notes](records.html?lang=en) **Result: 0 (Incomplete).** This test did not satisfy all requirements within twelve hours. Candidate-configuration and coverage work produced partial results, but the optimal radius's minimal polynomial and the global optimality proof remained incomplete. The test date is 2026-10-07 (UTC); this report was prepared on 2026-10-08. ### Task and completion The task was to determine the optimal radius of 27 equal closed disks covering the entire closed unit disk and meet all four core requirements. A computational proof also requires the code actually run, inputs, proof materials and verification records. | Requirement | Recorded result and acceptance | |---|---| | Minimal polynomial and root isolation | Incomplete. No complete submission of the radius's minimal polynomial, every coefficient, irreducibility proof and required root-isolation proof. | | Explicit exact centers | Exact definitions and uniqueness-check materials were saved for a candidate. Its global optimality was not established, so this is not a complete optimal-configuration delivery. | | Coverage of the entire closed unit disk | Strict coverage-check records exist for the candidate and cover the whole closed disk. This supports the candidate's feasibility only. | | Global optimality over arbitrary configurations | Incomplete. Exclusion of every potentially better configuration was not finished. | Partial checks do not satisfy all four requirements. Keeping code, inputs and partial proof materials is also different from delivering a complete proof package. No complete answer was obtained, so no comparison with the reference fingerprint was executed. ### Test settings and timing | Field | Record | |---|---| | Model | gpt-6.1-sol | | Reasoning effort | max | | Identity evidence | Current-thread metadata and configuration records within the original test window agree; identity is not inferred from the report-writing configuration. | | Time limit | Twelve hours, 43200 seconds, including the deadline. | | Assessment window | T+00:00:00 to T+12:00:00, using natural elapsed time from Goal creation. | | Valid completion time | None; no successful completion time can be recorded. | | Report preparation and publication | A separate subsequent task, excluded from the original test score. | | Money and billed tokens | Reliable billing data unavailable; unknown. | Failure means no complete result meeting every acceptance condition existed by the deadline. Report-writing time, individual program runtimes and later confirmation times are not successful completion times. The shared scoring program was actually run under the current rule and returned zero; it calculates a score rather than certifying mathematics. ### Computing environment | Item | Record | |---|---| | CPU | AMD Ryzen 9 8945HX with Radeon Graphics | | Memory | DDR5, 32 GB | | Execution location | Local Ubuntu WSL; no recorded key remote computation result. | | Hardware evidence | Actually queried during report preparation, not a task-start snapshot. | The original test used the existing local SageMath, Python, Singular and compiler tools for exploration, symbolic calculation and strict checks. Current report-preparation queries returned SageMath 10.7, Sage Python 3.12.13, NumPy 2.4.3, SciPy 1.17.1 and SymPy 1.14.0. These current values are not exact-version snapshots from the start. Public msolve tools were also obtained for computational attempts; their complete version lineage was not independently reconciled, and no accepted full solution resulted. ### Original input and sources The original request specified the test prompt on the [n=27 page](n27.html?lang=en), prohibited pre-existing local research materials, skills and other chats, permitted the public web and local WSL tools, and imposed a twelve-hour deadline. [Original user instruction, in Chinese](reports/n27-sol-main-prompt.original.txt) · [English problem snapshot downloaded at the start, with LF line endings](reports/n27-sol-task.en.txt) Report preparation uses materials generated during this run, retained records and current-thread configuration metadata explicitly requested by the user for this report. It is subsequent assessment and publication, not continued solving or a change to the original acceptance criteria. No research-asset-management skill or subagent was used in the original test. A complete operating-system file-access audit is unavailable, so resource-rule compliance is not described as independently certified. The original test consulted public general-method materials and proofs for other sizes from [DiskCoveringSolve](https://github.com/VonEquinox/DiskCoveringSolve) as research references. It did not obtain and replay a complete external proof satisfying this strict n=27 task. An existing algorithm record on the website does not replace the missing proof in this model submission. ### Research overview The timeline uses natural elapsed time from Goal creation. Save nodes show when materials were available at the latest; they are not exact discovery times or exclusive allocations of CPU time. | Relative time | Progress supported by saved records | |---|---| | T+00:00:00 | Read the problem and public material; begin candidate exploration and proof preparation. | | About T+01:15 | Exact candidate definitions and uniqueness-check materials had been saved. | | About T+01:32 | Whole-closed-disk coverage-check materials for the candidate had been saved. | | About T+04:34 | Further checks within the candidate's scope were recorded; their applicability did not cover arbitrary configurations. | | About T+08:29 to T+10:51 | Continue exact-radius and general-configuration proof work; retained records still explicitly mark the overall task incomplete. | | T+12:00:00 | Deadline assessment: no complete submission meeting all requirements; zero score. | The remaining acceptance gaps were the exact algebraic delivery and the global proof. Partial results advanced the investigation but did not close those gaps and receive no separate completion score. ### Verification, delivery and limits Original executed records concern the candidate's exact definition, uniqueness, whole-closed-disk coverage and further checks in a restricted scope. This report checked the existence and stated scope of those materials, the incomplete overall status, and actually ran the scoring program. It did not rerun a complete mathematical proof or establish a new global conclusion. Public deliverables are the bilingual report, machine-readable record, original instruction, problem snapshot and scoring receipt. The radius, centers, coefficients, internal solution steps and partial proof materials are withheld here and do not constitute delivery of a complete solution. One failed run establishes this run's unmet acceptance criteria, not a stable capability result under other conditions. [Machine-readable record](record-n27-sol-20261007.json) · [Report Markdown](record-n27-sol-20261007.en.md) · [Scoring receipt](reports/n27-sol-score-output.json)