Unit Disk Covering Benchmark · n = 13
n=13 · gpt-6.1 sol/max test report
# n=13 disk covering and minimal polynomial: 48-hour research report This test did not complete the full problem. The saved run records contain exact center definitions, verification of coverage of the closed disk, and a connection to an external global optimality proof. The radius's minimal polynomial and the required checks of that polynomial were not obtained. Computation stopped and the task was paused at the user's cumulative 48-hour Goal limit. The process below is described in broad terms. The detailed report, original conversation, and selected computational evidence remain in the separate archive. Preparing and publishing this report did not resume mathematical computation. ## Test information | Item | Record | | --- | --- | | Stable test ID | `T_20261016T173348Z_01a145c6` | | Problem | [n=13 disk covering and minimal polynomial](n13.html?lang=en) | | Model | `gpt-6.1-sol / max` | | Objective | Solve the covering problem and supply the radius's minimal polynomial with the required exact verification | | Source restrictions | No preexisting local materials, skills, or other local answers. Public web sources were allowed. The user later clarified that files newly generated for this task were allowed | | Time allowance | Initially unlimited; later changed to a cumulative 48-hour Goal limit, with a pause if unfinished | | Final state | **Incomplete and paused** | | Website display | **0 (Incomplete)** under the site's existing rule for unfinished runs; this extended research run is not a comparison under identical test conditions | | Paid cost | Unavailable | The original instructions and later clarifications remain in the separate archive. This extended run differs from the project's usual 12-hour tests. The score script handles the declared incomplete status; it does not review the mathematics or turn either time counter into a successful completion time. ## Time and resources | Measurement | Recorded value | | --- | --- | | Cumulative Goal time | **48 hours 2 minutes 48 seconds**. The final 168 seconds beyond 48 hours were used to check shutdown and save the work | | Natural elapsed time | **89 hours 27 minutes 17 seconds**, including pauses, waits, resumptions, and gaps between sessions; not continuous compute time | | Goal usage | 27,056,095 tokens, under the Goal counter only | | Thread cumulative usage | Input 607,259,143, including cached input 585,713,664; output 6,355,222; total 613,614,365. Kept separate from Goal usage and not added to it | | Report and archive work | A separate task after the test was paused; excluded from the original test duration | The CPU was **AMD Ryzen 9 8945HX with Radeon Graphics**, with **32 GiB DDR5** physical memory. The full CPU name and memory details came from a read-only query during archiving; historical computation logs also recorded the same CPU model. This was not a complete hardware snapshot at the start of the test. Computation used local Windows and Ubuntu/WSL, Python, SageMath, and local Wolfram tools. Some specialist tools were obtained from public sources during the task. A complete software version inventory was not verified, and no evidence establishes that a critical computation used a remote machine. Memory and other resource limits affected several attempts. Program durations were not simply added together as the task's total duration. ## Research process T+00:00 is the first task request. The following stages use natural elapsed time and overlap. | Stage | Broad description | | --- | --- | | T+00:03–00:15 | Searched public sources, found an existing external proof, checked its provenance, and ran verification | | T+00:22–about 04:00 | Built and revised models; tried variable reduction, elimination, and computational tests. Some attempts ended at resource limits | | From about T+04:00 | Continued numerical work and result checking in search of a verifiable algebraic result. Higher numerical precision did not supply the missing minimal polynomial | | From about T+39:41 | Further reduced variables, tested different routes, and optimized algorithms; attempted to reduce memory use, save progress, and resume computation | | Later stages through about T+89:23 | Continued addressing computational scale and memory limits and checking existing results. The required minimal polynomial was still missing | | T+89:03–89:17 | Supplemented and verified the exact geometric results and checked their connection to the external proof | | T+89:23–89:27 | Ended computation, checked that related background programs had stopped, saved results and resumption records, and paused the task | Tool invocation, resource allocation, interruptions, and timing judgments also caused difficulties and added overhead. This report uses the final saved records. Search limits were not treated as established mathematical conclusions, and intermediate results were not presented as a complete answer. ## Results and verification scope | Requirement | State at the pause | | --- | --- | | Thirteen exact centers | Concrete algebraic definitions and original verification records saved | | Coverage of the entire closed unit disk | Exact verification results saved | | Connection to global optimality | Correspondence with the external public proof and verification records saved; dependent on that external work | | Radius's minimal polynomial and all integer coefficients | **Incomplete** | | Irreducibility and isolation of the intended root of that polynomial | **Incomplete** | The external work is [VonEquinox's DiskCoveringSolve project](https://github.com/VonEquinox/DiskCoveringSolve/tree/main/cover13/proof_bundle). This test obtained, checked, and connected to that work; it did not independently discover the global proof from scratch. The model's additional modeling, computation, optimization, and geometric verification are recorded separately from the external contribution. “Verified” here refers to the programs and computational records saved during the original test. The later archive checks concerned files, metadata, and retrieval, and were not a fresh independent mathematical review. High-precision decimals, implicit equations, and degree bounds used in a search do not replace the required minimal polynomial. The main unresolved bottleneck was obtaining a minimal polynomial that could be rigorously checked. Repeated modeling, elimination, testing, and optimization did not finish that step, while computational scale and memory limits repeatedly constrained progress. The records do not establish its degree or show that more disk space or memory would necessarily finish the task. ## Archive and current state The separate original archive contains 6,130 public message or tool items, original instructions, model and usage records, the detailed report, and selected computational evidence. Its export is limited to the declared public fields and excludes non-public internal reasoning; omissions are documented. A larger computational checkpoint remains in the original workspace, with its locator and integrity information retained in the archive. The mathematical task remains paused and its computation programs have stopped. Further research requires an explicit instruction to resume. Publishing this summary does not change the test outcome, original counters, failure records, or original archive.
# n=13 disk covering and minimal polynomial: 48-hour research report This test did not complete the full problem. The saved run records contain exact center definitions, verification of coverage of the closed disk, and a connection to an external global optimality proof. The radius's minimal polynomial and the required checks of that polynomial were not obtained. Computation stopped and the task was paused at the user's cumulative 48-hour Goal limit. The process below is described in broad terms. The detailed report, original conversation, and selected computational evidence remain in the separate archive. Preparing and publishing this report did not resume mathematical computation. ## Test information | Item | Record | | --- | --- | | Stable test ID | `T_20261016T173348Z_01a145c6` | | Problem | [n=13 disk covering and minimal polynomial](n13.html?lang=en) | | Model | `gpt-6.1-sol / max` | | Objective | Solve the covering problem and supply the radius's minimal polynomial with the required exact verification | | Source restrictions | No preexisting local materials, skills, or other local answers. Public web sources were allowed. The user later clarified that files newly generated for this task were allowed | | Time allowance | Initially unlimited; later changed to a cumulative 48-hour Goal limit, with a pause if unfinished | | Final state | **Incomplete and paused** | | Website display | **0 (Incomplete)** under the site's existing rule for unfinished runs; this extended research run is not a comparison under identical test conditions | | Paid cost | Unavailable | The original instructions and later clarifications remain in the separate archive. This extended run differs from the project's usual 12-hour tests. The score script handles the declared incomplete status; it does not review the mathematics or turn either time counter into a successful completion time. ## Time and resources | Measurement | Recorded value | | --- | --- | | Cumulative Goal time | **48 hours 2 minutes 48 seconds**. The final 168 seconds beyond 48 hours were used to check shutdown and save the work | | Natural elapsed time | **89 hours 27 minutes 17 seconds**, including pauses, waits, resumptions, and gaps between sessions; not continuous compute time | | Goal usage | 27,056,095 tokens, under the Goal counter only | | Thread cumulative usage | Input 607,259,143, including cached input 585,713,664; output 6,355,222; total 613,614,365. Kept separate from Goal usage and not added to it | | Report and archive work | A separate task after the test was paused; excluded from the original test duration | The CPU was **AMD Ryzen 9 8945HX with Radeon Graphics**, with **32 GiB DDR5** physical memory. The full CPU name and memory details came from a read-only query during archiving; historical computation logs also recorded the same CPU model. This was not a complete hardware snapshot at the start of the test. Computation used local Windows and Ubuntu/WSL, Python, SageMath, and local Wolfram tools. Some specialist tools were obtained from public sources during the task. A complete software version inventory was not verified, and no evidence establishes that a critical computation used a remote machine. Memory and other resource limits affected several attempts. Program durations were not simply added together as the task's total duration. ## Research process T+00:00 is the first task request. The following stages use natural elapsed time and overlap. | Stage | Broad description | | --- | --- | | T+00:03–00:15 | Searched public sources, found an existing external proof, checked its provenance, and ran verification | | T+00:22–about 04:00 | Built and revised models; tried variable reduction, elimination, and computational tests. Some attempts ended at resource limits | | From about T+04:00 | Continued numerical work and result checking in search of a verifiable algebraic result. Higher numerical precision did not supply the missing minimal polynomial | | From about T+39:41 | Further reduced variables, tested different routes, and optimized algorithms; attempted to reduce memory use, save progress, and resume computation | | Later stages through about T+89:23 | Continued addressing computational scale and memory limits and checking existing results. The required minimal polynomial was still missing | | T+89:03–89:17 | Supplemented and verified the exact geometric results and checked their connection to the external proof | | T+89:23–89:27 | Ended computation, checked that related background programs had stopped, saved results and resumption records, and paused the task | Tool invocation, resource allocation, interruptions, and timing judgments also caused difficulties and added overhead. This report uses the final saved records. Search limits were not treated as established mathematical conclusions, and intermediate results were not presented as a complete answer. ## Results and verification scope | Requirement | State at the pause | | --- | --- | | Thirteen exact centers | Concrete algebraic definitions and original verification records saved | | Coverage of the entire closed unit disk | Exact verification results saved | | Connection to global optimality | Correspondence with the external public proof and verification records saved; dependent on that external work | | Radius's minimal polynomial and all integer coefficients | **Incomplete** | | Irreducibility and isolation of the intended root of that polynomial | **Incomplete** | The external work is [VonEquinox's DiskCoveringSolve project](https://github.com/VonEquinox/DiskCoveringSolve/tree/main/cover13/proof_bundle). This test obtained, checked, and connected to that work; it did not independently discover the global proof from scratch. The model's additional modeling, computation, optimization, and geometric verification are recorded separately from the external contribution. “Verified” here refers to the programs and computational records saved during the original test. The later archive checks concerned files, metadata, and retrieval, and were not a fresh independent mathematical review. High-precision decimals, implicit equations, and degree bounds used in a search do not replace the required minimal polynomial. The main unresolved bottleneck was obtaining a minimal polynomial that could be rigorously checked. Repeated modeling, elimination, testing, and optimization did not finish that step, while computational scale and memory limits repeatedly constrained progress. The records do not establish its degree or show that more disk space or memory would necessarily finish the task. ## Archive and current state The separate original archive contains 6,130 public message or tool items, original instructions, model and usage records, the detailed report, and selected computational evidence. Its export is limited to the declared public fields and excludes non-public internal reasoning; omissions are documented. A larger computational checkpoint remains in the original workspace, with its locator and integrity information retained in the archive. The mathematical task remains paused and its computation programs have stopped. Further research requires an explicit instruction to resume. Publishing this summary does not change the test outcome, original counters, failure records, or original archive.