Unit Disk Covering Benchmark · n = 13
n=13 · gpt-6.1 sol/max test report
# n=13 · gpt-6.1 sol/max test report
[Back to n=13](n13.html) · [Test record index](records.html)
This test took place on 2026-10-01 (UTC), with a 12-hour deadline. **No valid result satisfying every requirement was delivered by the deadline; the score is 0.** The run did not supply the radius's complete minimal polynomial, its irreducibility proof and required rational root-isolation certificate, or a complete mathematical proof package. This revision follows the n=11 Astra report structure and does not restart mathematical research or change the failure outcome.
The original Goal used **gpt-6.1-sol / max**. At the user's request, this was verified from this thread's original configuration metadata: all seven retained Goal-interval configuration records agree. The earlier report's unknown model/effort fields are corrected using that evidence. Settings used for subsequent reporting and publication are recorded separately from the original solve.
### Existing proof reused and work performed in this test
**This run reused an existing online n=13 configuration, coverage arguments, classification and global exclusion materials. Those configurations and certificates came from external work; this model did not discover them from scratch during the test.**
The source was the [fixed version of VonEquinox / DiskCoveringSolve](https://github.com/VonEquinox/DiskCoveringSolve/tree/a304d38e5c885f050ebf14d6b5e61406d7e832ad/cover13), commit a304d38e5c885f050ebf14d6b5e61406d7e832ad. It supplied 119-variable candidate-root data and isolation certificates, full-disk coverage reductions and inputs, combinatorial enumeration, metric partitioning, and candidate/noncandidate branch certificates.
The model reviewed continuous coverage and the reduction of general configurations, wrote exact verification programs and replayed partial certificates, checked reflection and independent constraint gradients, formed a 38-variable system, certified high-precision root data, and attempted elimination and integer-relation searches. Radius minimal-polynomial recovery did not succeed. The external repository's completion claim, successful program exits and high-precision data do not substitute for all acceptance requirements.
At the user's explicit request, this thread's two continuous records were completely traversed to recover message, tool-dispatch and output-reception timestamps. The earlier unrecorded-time claim resulted from relying on summaries without completing that traversal, and is corrected. Repository retrieval returned at T+00:14:10.494; the cover13 proof text returned at T+00:15:20.590. Times below belong to this run, not the other n=13 record.
### Task and timing
T+00:00:00 is original second-resolution Goal creation metadata. The user instruction's recovered thread-write timestamp is T+00:00:02.918, not a separately measured network receipt. The original scoring origin and deadline remain unchanged.
| Timing basis | Record |
|---|---|
| Original deadline and scoring observation endpoint | **T+12:00:00, 43200 seconds**; incomplete, a failure observation window |
| Valid completion time | No valid completion event; null |
| Clock observation after computation stopped | T+12:01:17, 43277 seconds; own mathematical processes had stopped; result remained incomplete |
| Natural elapsed time when Goal became blocked | T+12:05:13, 43513 seconds; includes post-deadline status continuations |
| Goal system accumulated time | **12 h 5 min 14 s (43514 seconds)**, frozen at blocked status; recorded separately |
| Original report evidence cutoff | T+12:35:16, 45316 seconds; includes the gap after research and independent report preparation; problem remained incomplete |
| Original failure response thread-write timestamp | **T+12:02:44.205, 43364.205 seconds**, now recovered; not valid completion. UI rendering end was not separately measured |
| Original reporting: request to first report response | **T+12:20:56.009 → T+12:51:14.875, 1818.866 seconds (30 min 18.866 s)**, including reasoning, computation, waits and delivery logging |
| Later model audit, revisions and publication | Separately authorized follow-ups, excluded from the mathematical score and not merged with the original reporting interval |
Natural elapsed time includes thinking, computation, waits and overlapping work. Parallel program runtimes are not added together. The system counter does not replace natural time; differences are not assigned to guessed pause or wait durations. The old 561-second observation is a subinterval of reporting; its full request-to-response log span is now recovered as 1818.866 seconds.
### Computing environment
| Item | Record |
|---|---|
| Original model and reasoning setting | gpt-6.1-sol / max, consistently verified from original Goal configuration records |
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics; 16 cores, 32 logical processors |
| Memory | DDR5, 32 GiB |
| Hardware evidence | Current Windows hardware queries during original report preparation; no opening snapshot |
| Local computation | Ubuntu WSL2; newly authored memory-transferred Python programs with integer arithmetic and custom rational/interval operations |
| Python and kernel | Queries during original report preparation: Python 3.12.3, GCC 13.3.0, 6.6.87.2-microsoft-standard-WSL2; not a frozen opening environment |
| WSL-visible resources | 32 logical processors and approximately 15.23 GiB visible memory, recorded separately from host physical memory |
| Preinstalled professional tools | Not inventoried; the existing local Sage entry was not executed, and preexisting local SymPy, NumPy and SciPy were not used for core computation |
| Network computation tools | Public SageCell (SageMath 10.9 in the original record); Wolfram connector also used for the initial 38-variable numerical solution and elimination attempts. Connector kernel version, execution host and remote CPU/memory unrecorded; numerical output was not treated as proof |
| Subagents and runs | No subagents; one solve. Retries and replays are not independent tests |
| Monetary budget and cost | Unrecorded; null |
| Codex Goal cumulative counter | 3887192; see the accounting definition below; not billed tokens or monetary cost |
The Codex Goal cumulative counter is a budget-accounting field of Codex's Goal feature, not an industry-standard total-token metric. The verified [Codex 0.159.2 implementation](https://github.com/openai/codex/blob/rust-v0.159.2/codex-rs/ext/goal/src/accounting.rs#L527-L532) accumulates usage increments as input tokens minus cached input tokens plus output tokens: uncached input plus output. Output already includes reasoning tokens, which must not be added again. Historical versions follow their own implementation. This field is not the whole-chat token total, billed-token usage or monetary cost, and must not be compared directly with another platform's total-token figure.
The run temporarily obtained [flatter from flatn 0.3.5](https://files.pythonhosted.org/packages/b0/c5/d0a76fc673f0b1e02a23322350a61e8ee09e5daf2022bd3fc6da729b46cd/flatn-0.3.5-py3-none-manylinux1_x86_64.whl), a lattice-reduction tool for discovering integer relations, and [msolve 0.10.1](https://msolve.proj.lip6.fr/downloads/v0.10.1/msolve-0.10.1_linux_intel_x86.zip) for polynomial-system elimination attempts. Public downloads and native programs were transferred through anonymous memory. The package version 0.3.5 is not a separately verified flatter executable version.
DDR5 is inferred from the measured SMBIOS type 34 (0x22), using the [DMTF SMBIOS 3.4.0 memory-type table](https://www.dmtf.org/sites/default/files/standards/documents/DSP0134_3.4.0.pdf#page=104). Hardware and Python queries are explicitly distinguished from an opening snapshot.
### Input prompt
The input required the n=13 website test prompt, prohibited local materials, skills, other chats and archived chats, allowed public internet and local WSL tools, and imposed a 12-hour deadline for a complete result. A resource-rule violation or failure to finish by that deadline counts as failure.
[Download the exact original user instruction, in Chinese](reports/n13-sol-main-prompt.original.txt). The task requires the radius's complete minimal polynomial and irreducibility proof, rational root isolation, uniquely specified ordered centers, coverage of the entire closed disk, a lower bound for all configurations including degeneracies, and complete actual computation materials.
The opening website commit was not recorded. The full statement below is the post-hoc pinned version checked against its embedded page prompt; it is not described as an opening frozen contract.
<details><summary>Expand the complete checked problem statement</summary>
### Checked problem statement
Optimal Covering of the Unit Disk with 13 Disks
Let
$$
D=\{x\in\mathbb R^2:\|x\|_2\le1\},\qquad
R_D(C)=\max_{x\in D}\min_{1\le i\le13}\|x-c_i\|_2,
$$
where $C=(c_1,\ldots,c_{13})\in(\mathbb R^2)^{13}$, and define
$$
r_D(13)=\inf_{C\in(\mathbb R^2)^{13}}R_D(C).
$$
The covering disks are closed. They may overlap and extend beyond $D$; their centers are not initially restricted to lie in $D$.
Determine $r_D(13)$ and satisfy all of the following requirements:
1. **Minimal polynomial and root isolation:** Explicitly list every coefficient of the normalized integer representation $P(t)\in\mathbb Z[t]$ of the minimal polynomial of the optimal radius $r_D(13)$ over $\mathbb Q$. Require $P$ to be nonconstant and primitive, with positive leading coefficient, and prove that it is irreducible in $\mathbb Q[t]$. Give specific rational numbers $0\le a<b$, prove $P(a)P(b)\ne0$, and prove that $P$ has exactly one real root $\alpha$ in $(a,b)$. Use the configuration and proofs below to establish $r_D(13)=\alpha$.
2. **Explicit configuration:** Give a definite ordered tuple of centers $C_*=(c_1^*,\ldots,c_{13}^*)$, specified exactly by radicals, polynomials and rational isolating intervals, or other explicitly supplied finite algebraic data whose uniqueness is proved. Uniqueness here means that the submitted data uniquely specify these center coordinates; it does not require a proof that the optimal configuration is unique.
3. **Coverage:** Rigorously prove
$$
\forall x\in D,\quad\exists i\in\{1,\ldots,13\},\quad
\|x-c_i^*\|_2\le\alpha.
$$
If continuous coverage is reduced to a finite set of checks, prove that the reduction is sufficient and rigorously complete every required check.
4. **Global optimality:** Rigorously prove
$$
\forall C\in(\mathbb R^2)^{13},\quad R_D(C)\ge\alpha,
$$
and hence $R_D(C_*)=r_D(13)=\alpha$. Do not restrict symmetry, contact relations or combinatorial structure without proof. Every normalization, classification, enumeration or pruning step must be proved not to omit a potentially better configuration. Handle coincident centers, redundant disks and other relevant degeneracies.
5. **Computer-assisted proof:** If the proof relies on computation, deliver the complete code actually used, inputs, certificates, necessary environment information and actual verification results. Explain why those results imply the mathematical statements above.
**Final submission:** Explicitly list every coefficient of $P(t)$ and the rational root-isolating interval $(a,b)$, together with the configuration, complete proof and necessary verification materials. Submit the minimal polynomial of the optimal radius itself. A polynomial for its square or another auxiliary quantity, or an elimination polynomial without a proof of irreducibility, does not satisfy the requirement.
Any rigorous method is allowed. Merely proving existence or algebraicity, or supplying unfinished enumeration, a solution algorithm or a numerical candidate, does not count as completion. Successful numerical optimization, a program printing “passed”, or floating-point results without rigorous error control cannot replace a proof. Checking only finitely many sample points, the boundary, or a numerical image cannot replace a proof that the entire closed unit disk is covered.
</details>
### Main stages
Timestamps were recovered at millisecond resolution, relative to original second-resolution Goal creation. **Spans cover dispatch to output reception, or asynchronous submission to later result observation, including network, orchestration, waits and gaps rather than isolated computation time.** Results existed no later than their reception. Routes overlap and spans must not be added. Thinking and code-writing cannot be isolated from pairs of tool events.
| Record start → output reception / event | Observed span | Work and retained result |
|---|---|---|
| T+00:14:03.244 → T+00:14:10.494 | 00:00:07.250 | Read external proof repository |
| T+00:15:07.426 → T+00:15:20.590 | 00:00:13.164 | Read cover13 proof text and materials |
| T+00:31:25.737 → T+00:31:43.012 | 00:00:17.275 | 119-variable root: all 119 integer contraction rows checked |
| T+00:49:48.757 → T+00:50:00.134 | 00:00:11.377 | 34 triangular faces, 10 arcs and 62-dimensional local isolation |
| T+00:55:37.547 → T+00:55:50.498 | 00:00:12.951 | Candidate tree: 1831 nodes, 837 DUAL and 79 LOCAL leaves |
| T+00:56:57.688 → T+00:57:16.808 | 00:00:19.120 | 1760 noncandidate cases: 2908 nodes and 2334 DUAL leaves |
| T+01:01:31.316 → T+01:03:56.058 | 00:02:24.742 | Replay after first-attempt key-check correction: 308198 orbits and 1761 residual-case interfaces |
| T+01:02:35.521 → T+01:02:42.749 | 00:00:07.228 | Wolfram connector: initial numerical solution, 17 constraints and 38 equations; not proof |
| T+01:24:41.998 → T+01:24:59.930 | 00:00:17.932 | After auxiliary-point permutation correction: 119 reflected variables strictly inside original root box |
| T+01:52:45.811 → T+01:53:07.716 | 00:00:21.905 | 17 independent constraint gradients and nonzero related multipliers |
| T+05:44:37.421 → T+05:51:52.261 | 00:07:14.840 | Strict squared-radius 60000-digit floor, contraction and self-map output |
| T+08:04:53.218 | Observation only | First observed strict 330000-digit squared-radius floor output |
| T+09:47:02.551 → T+10:31:58.797 | 00:44:56.246 | Failed t search, degree bound 512, 300000 digits; program-reported 2616.76 s |
| T+09:51:50.671 → T+09:52:14.222 | 00:00:23.551 | Public SageCell: all 38 normalized symbolic identities exactly zero |
| T+10:40:35.187 → T+11:15:20.586 | 00:34:45.399 | Failed d search, degree bound 512, 220000 digits; program-reported 2050.15 s |
| T+10:52:32.894 → T+11:11:46.734 | 00:19:13.840 | Strict 580000-digit squared-radius floor and HIGH/LINK_BOX contraction and self-map output |
| T+12:00:00.000 | Deadline | Minimal polynomial and complete delivery missing; score 0 |
| T+12:02:44.205 | Response log event | Original failure response recorded |
| T+12:05:13.000 | Status event | Goal metadata marked blocked |
The first enumeration replay asserted because a count dictionary omitted zero-valued keys; the first reflection replay used an incorrect auxiliary-point permutation. Corrections preceded successful outputs above; failed attempts were not counted as passes. Elimination and integer-relation searches ran throughout these periods and later; allocation, thread-creation and runtime-limit failures were not counted as successes.
2616.76 and 2050.15 seconds are program-reported timings, different from submission-to-result-observation spans; they cannot be added or substituted.
**Actual output and invocation code for the 580000-digit stage were recovered; they were not preserved only in a summary.** At T+11:11:46.734 the recorded output included EXACT_CERT_RADIUS_SQUARED_DECIMAL_FLOOR_DIGITS 580000, strict HIGH/LINK_BOX contraction and self-map outputs, and ALL_NEW_DYADIC_ROOT_CERTIFICATION_INEQUALITIES_STRICT. No mathematical checks were rerun for this reconstruction. Complete certificate data and the proof package remain undelivered; output labels do not replace mathematical proof.
High precision does not supply the minimal polynomial coefficients. Failed heuristics do not establish degree above 512 or nonexistence. Synthetic irreducibility tests do not certify the actual radius.
[Download reconstructed timing and evidence labels](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-sol-timing.json). Only relative times, counts and selected labels are public; the raw transcript, internal paths, centers and polynomial coefficients are excluded.
### Verification, delivery and limits
| Acceptance requirement | Status at the original deadline |
|---|---|
| Complete radius minimal polynomial, irreducibility and rational root isolation | **Incomplete**; no actual polynomial. Algebraic existence or high-precision squared-radius data cannot replace it |
| Exact ordered centers and uniqueness of the specified data | External root data and unique-box checks exist; no complete final algebraic delivery with the required radius connection |
| Entire closed-disk coverage | Original review of the continuous reduction and exact face/arc checks; no complete final proof submission |
| Exhaustive global lower bound | Original combinatorial partition and branch checks; no complete final proof submission |
| Complete code, inputs, certificates, environment and results | **No complete proof package meeting all requirements submitted** |
Retained partial executions include 308198 orbits: 301689 direct exclusions, 4748 Farkas exclusions and 1761 residual cases. The candidate branch had 1831 nodes, 837 DUAL leaves and 79 LOCAL leaves, maximum depth 27. The 1760 noncandidate cases had 2908 nodes and 2334 DUAL leaves, maximum depth 6. Retained coverage/local-isolation, orbit and noncandidate check results exited 0. Those results require their mathematical reductions and full supporting materials for acceptance; counts or successful statuses alone are not a complete proof.
Publication does not rerun the mathematics or provide an external independent mathematical review. New work here verifies the original Goal model configuration at the user's request, revises report structure, and builds/checks the pages and downloads. It is not counted as pre-deadline mathematical progress.
During original report preparation, the pinned official scoring source was executed as ordinary Python in a public SageCell temporary kernel. Inputs were n=13, status not_completed, observation window 43200 seconds and budget 43200 seconds. It returned score 0 and exit code 0. The [actual scoring output](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-sol-score-output.json) preserves the full fields. This script does not certify a mathematical proof. The CLI reference parameter 21600 is the post-hoc common example scale, not an opening frozen n=13 reference time, and cannot alter the failure branch's zero score.
No n=13 reference fingerprint has been published. With no complete minimal polynomial, the answer-fingerprint checker was not run. The submitted fingerprint, match status and valid completion time remain null; neither a match nor a mismatch is claimed. The 12-hour failure score is 0, rather than the 100 that the formula would assign to valid completion exactly at the deadline.
The original solve did not read other chats, skills or preexisting local research materials. Reporting and publication use this task's generated artifacts, public repository data, and this thread's configuration metadata explicitly requested by the user. Original message bodies, internal paths, full execution logs and mathematical answers are not published. No complete OS-level access trace exists, so comprehensive forensic certification is not claimed. External-proof reuse and the single retrospective-run scope remain disclosed; no formal ranking is asserted.
[Back to the top](#)
## Downloadable records
[Machine-readable record for this test](record-n13-sol-20261001.json) · [English report Markdown](record-n13-sol-20261001.en.md) · [中文报告](record-n13-sol-20261001.zh.html) · [Original user instruction](reports/n13-sol-main-prompt.original.txt)
# n=13 · gpt-6.1 sol/max test report
[Back to n=13](n13.html) · [Test record index](records.html)
This test took place on 2026-10-01 (UTC), with a 12-hour deadline. **No valid result satisfying every requirement was delivered by the deadline; the score is 0.** The run did not supply the radius's complete minimal polynomial, its irreducibility proof and required rational root-isolation certificate, or a complete mathematical proof package. This revision follows the n=11 Astra report structure and does not restart mathematical research or change the failure outcome.
The original Goal used **gpt-6.1-sol / max**. At the user's request, this was verified from this thread's original configuration metadata: all seven retained Goal-interval configuration records agree. The earlier report's unknown model/effort fields are corrected using that evidence. Settings used for subsequent reporting and publication are recorded separately from the original solve.
### Existing proof reused and work performed in this test
**This run reused an existing online n=13 configuration, coverage arguments, classification and global exclusion materials. Those configurations and certificates came from external work; this model did not discover them from scratch during the test.**
The source was the [fixed version of VonEquinox / DiskCoveringSolve](https://github.com/VonEquinox/DiskCoveringSolve/tree/a304d38e5c885f050ebf14d6b5e61406d7e832ad/cover13), commit a304d38e5c885f050ebf14d6b5e61406d7e832ad. It supplied 119-variable candidate-root data and isolation certificates, full-disk coverage reductions and inputs, combinatorial enumeration, metric partitioning, and candidate/noncandidate branch certificates.
The model reviewed continuous coverage and the reduction of general configurations, wrote exact verification programs and replayed partial certificates, checked reflection and independent constraint gradients, formed a 38-variable system, certified high-precision root data, and attempted elimination and integer-relation searches. Radius minimal-polynomial recovery did not succeed. The external repository's completion claim, successful program exits and high-precision data do not substitute for all acceptance requirements.
At the user's explicit request, this thread's two continuous records were completely traversed to recover message, tool-dispatch and output-reception timestamps. The earlier unrecorded-time claim resulted from relying on summaries without completing that traversal, and is corrected. Repository retrieval returned at T+00:14:10.494; the cover13 proof text returned at T+00:15:20.590. Times below belong to this run, not the other n=13 record.
### Task and timing
T+00:00:00 is original second-resolution Goal creation metadata. The user instruction's recovered thread-write timestamp is T+00:00:02.918, not a separately measured network receipt. The original scoring origin and deadline remain unchanged.
| Timing basis | Record |
|---|---|
| Original deadline and scoring observation endpoint | **T+12:00:00, 43200 seconds**; incomplete, a failure observation window |
| Valid completion time | No valid completion event; null |
| Clock observation after computation stopped | T+12:01:17, 43277 seconds; own mathematical processes had stopped; result remained incomplete |
| Natural elapsed time when Goal became blocked | T+12:05:13, 43513 seconds; includes post-deadline status continuations |
| Goal system accumulated time | **12 h 5 min 14 s (43514 seconds)**, frozen at blocked status; recorded separately |
| Original report evidence cutoff | T+12:35:16, 45316 seconds; includes the gap after research and independent report preparation; problem remained incomplete |
| Original failure response thread-write timestamp | **T+12:02:44.205, 43364.205 seconds**, now recovered; not valid completion. UI rendering end was not separately measured |
| Original reporting: request to first report response | **T+12:20:56.009 → T+12:51:14.875, 1818.866 seconds (30 min 18.866 s)**, including reasoning, computation, waits and delivery logging |
| Later model audit, revisions and publication | Separately authorized follow-ups, excluded from the mathematical score and not merged with the original reporting interval |
Natural elapsed time includes thinking, computation, waits and overlapping work. Parallel program runtimes are not added together. The system counter does not replace natural time; differences are not assigned to guessed pause or wait durations. The old 561-second observation is a subinterval of reporting; its full request-to-response log span is now recovered as 1818.866 seconds.
### Computing environment
| Item | Record |
|---|---|
| Original model and reasoning setting | gpt-6.1-sol / max, consistently verified from original Goal configuration records |
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics; 16 cores, 32 logical processors |
| Memory | DDR5, 32 GiB |
| Hardware evidence | Current Windows hardware queries during original report preparation; no opening snapshot |
| Local computation | Ubuntu WSL2; newly authored memory-transferred Python programs with integer arithmetic and custom rational/interval operations |
| Python and kernel | Queries during original report preparation: Python 3.12.3, GCC 13.3.0, 6.6.87.2-microsoft-standard-WSL2; not a frozen opening environment |
| WSL-visible resources | 32 logical processors and approximately 15.23 GiB visible memory, recorded separately from host physical memory |
| Preinstalled professional tools | Not inventoried; the existing local Sage entry was not executed, and preexisting local SymPy, NumPy and SciPy were not used for core computation |
| Network computation tools | Public SageCell (SageMath 10.9 in the original record); Wolfram connector also used for the initial 38-variable numerical solution and elimination attempts. Connector kernel version, execution host and remote CPU/memory unrecorded; numerical output was not treated as proof |
| Subagents and runs | No subagents; one solve. Retries and replays are not independent tests |
| Monetary budget and cost | Unrecorded; null |
| Codex Goal cumulative counter | 3887192; see the accounting definition below; not billed tokens or monetary cost |
The Codex Goal cumulative counter is a budget-accounting field of Codex's Goal feature, not an industry-standard total-token metric. The verified [Codex 0.159.2 implementation](https://github.com/openai/codex/blob/rust-v0.159.2/codex-rs/ext/goal/src/accounting.rs#L527-L532) accumulates usage increments as input tokens minus cached input tokens plus output tokens: uncached input plus output. Output already includes reasoning tokens, which must not be added again. Historical versions follow their own implementation. This field is not the whole-chat token total, billed-token usage or monetary cost, and must not be compared directly with another platform's total-token figure.
The run temporarily obtained [flatter from flatn 0.3.5](https://files.pythonhosted.org/packages/b0/c5/d0a76fc673f0b1e02a23322350a61e8ee09e5daf2022bd3fc6da729b46cd/flatn-0.3.5-py3-none-manylinux1_x86_64.whl), a lattice-reduction tool for discovering integer relations, and [msolve 0.10.1](https://msolve.proj.lip6.fr/downloads/v0.10.1/msolve-0.10.1_linux_intel_x86.zip) for polynomial-system elimination attempts. Public downloads and native programs were transferred through anonymous memory. The package version 0.3.5 is not a separately verified flatter executable version.
DDR5 is inferred from the measured SMBIOS type 34 (0x22), using the [DMTF SMBIOS 3.4.0 memory-type table](https://www.dmtf.org/sites/default/files/standards/documents/DSP0134_3.4.0.pdf#page=104). Hardware and Python queries are explicitly distinguished from an opening snapshot.
### Input prompt
The input required the n=13 website test prompt, prohibited local materials, skills, other chats and archived chats, allowed public internet and local WSL tools, and imposed a 12-hour deadline for a complete result. A resource-rule violation or failure to finish by that deadline counts as failure.
[Download the exact original user instruction, in Chinese](reports/n13-sol-main-prompt.original.txt). The task requires the radius's complete minimal polynomial and irreducibility proof, rational root isolation, uniquely specified ordered centers, coverage of the entire closed disk, a lower bound for all configurations including degeneracies, and complete actual computation materials.
The opening website commit was not recorded. The full statement below is the post-hoc pinned version checked against its embedded page prompt; it is not described as an opening frozen contract.
<details><summary>Expand the complete checked problem statement</summary>
### Checked problem statement
Optimal Covering of the Unit Disk with 13 Disks
Let
$$
D=\{x\in\mathbb R^2:\|x\|_2\le1\},\qquad
R_D(C)=\max_{x\in D}\min_{1\le i\le13}\|x-c_i\|_2,
$$
where $C=(c_1,\ldots,c_{13})\in(\mathbb R^2)^{13}$, and define
$$
r_D(13)=\inf_{C\in(\mathbb R^2)^{13}}R_D(C).
$$
The covering disks are closed. They may overlap and extend beyond $D$; their centers are not initially restricted to lie in $D$.
Determine $r_D(13)$ and satisfy all of the following requirements:
1. **Minimal polynomial and root isolation:** Explicitly list every coefficient of the normalized integer representation $P(t)\in\mathbb Z[t]$ of the minimal polynomial of the optimal radius $r_D(13)$ over $\mathbb Q$. Require $P$ to be nonconstant and primitive, with positive leading coefficient, and prove that it is irreducible in $\mathbb Q[t]$. Give specific rational numbers $0\le a<b$, prove $P(a)P(b)\ne0$, and prove that $P$ has exactly one real root $\alpha$ in $(a,b)$. Use the configuration and proofs below to establish $r_D(13)=\alpha$.
2. **Explicit configuration:** Give a definite ordered tuple of centers $C_*=(c_1^*,\ldots,c_{13}^*)$, specified exactly by radicals, polynomials and rational isolating intervals, or other explicitly supplied finite algebraic data whose uniqueness is proved. Uniqueness here means that the submitted data uniquely specify these center coordinates; it does not require a proof that the optimal configuration is unique.
3. **Coverage:** Rigorously prove
$$
\forall x\in D,\quad\exists i\in\{1,\ldots,13\},\quad
\|x-c_i^*\|_2\le\alpha.
$$
If continuous coverage is reduced to a finite set of checks, prove that the reduction is sufficient and rigorously complete every required check.
4. **Global optimality:** Rigorously prove
$$
\forall C\in(\mathbb R^2)^{13},\quad R_D(C)\ge\alpha,
$$
and hence $R_D(C_*)=r_D(13)=\alpha$. Do not restrict symmetry, contact relations or combinatorial structure without proof. Every normalization, classification, enumeration or pruning step must be proved not to omit a potentially better configuration. Handle coincident centers, redundant disks and other relevant degeneracies.
5. **Computer-assisted proof:** If the proof relies on computation, deliver the complete code actually used, inputs, certificates, necessary environment information and actual verification results. Explain why those results imply the mathematical statements above.
**Final submission:** Explicitly list every coefficient of $P(t)$ and the rational root-isolating interval $(a,b)$, together with the configuration, complete proof and necessary verification materials. Submit the minimal polynomial of the optimal radius itself. A polynomial for its square or another auxiliary quantity, or an elimination polynomial without a proof of irreducibility, does not satisfy the requirement.
Any rigorous method is allowed. Merely proving existence or algebraicity, or supplying unfinished enumeration, a solution algorithm or a numerical candidate, does not count as completion. Successful numerical optimization, a program printing “passed”, or floating-point results without rigorous error control cannot replace a proof. Checking only finitely many sample points, the boundary, or a numerical image cannot replace a proof that the entire closed unit disk is covered.
</details>
### Main stages
Timestamps were recovered at millisecond resolution, relative to original second-resolution Goal creation. **Spans cover dispatch to output reception, or asynchronous submission to later result observation, including network, orchestration, waits and gaps rather than isolated computation time.** Results existed no later than their reception. Routes overlap and spans must not be added. Thinking and code-writing cannot be isolated from pairs of tool events.
| Record start → output reception / event | Observed span | Work and retained result |
|---|---|---|
| T+00:14:03.244 → T+00:14:10.494 | 00:00:07.250 | Read external proof repository |
| T+00:15:07.426 → T+00:15:20.590 | 00:00:13.164 | Read cover13 proof text and materials |
| T+00:31:25.737 → T+00:31:43.012 | 00:00:17.275 | 119-variable root: all 119 integer contraction rows checked |
| T+00:49:48.757 → T+00:50:00.134 | 00:00:11.377 | 34 triangular faces, 10 arcs and 62-dimensional local isolation |
| T+00:55:37.547 → T+00:55:50.498 | 00:00:12.951 | Candidate tree: 1831 nodes, 837 DUAL and 79 LOCAL leaves |
| T+00:56:57.688 → T+00:57:16.808 | 00:00:19.120 | 1760 noncandidate cases: 2908 nodes and 2334 DUAL leaves |
| T+01:01:31.316 → T+01:03:56.058 | 00:02:24.742 | Replay after first-attempt key-check correction: 308198 orbits and 1761 residual-case interfaces |
| T+01:02:35.521 → T+01:02:42.749 | 00:00:07.228 | Wolfram connector: initial numerical solution, 17 constraints and 38 equations; not proof |
| T+01:24:41.998 → T+01:24:59.930 | 00:00:17.932 | After auxiliary-point permutation correction: 119 reflected variables strictly inside original root box |
| T+01:52:45.811 → T+01:53:07.716 | 00:00:21.905 | 17 independent constraint gradients and nonzero related multipliers |
| T+05:44:37.421 → T+05:51:52.261 | 00:07:14.840 | Strict squared-radius 60000-digit floor, contraction and self-map output |
| T+08:04:53.218 | Observation only | First observed strict 330000-digit squared-radius floor output |
| T+09:47:02.551 → T+10:31:58.797 | 00:44:56.246 | Failed t search, degree bound 512, 300000 digits; program-reported 2616.76 s |
| T+09:51:50.671 → T+09:52:14.222 | 00:00:23.551 | Public SageCell: all 38 normalized symbolic identities exactly zero |
| T+10:40:35.187 → T+11:15:20.586 | 00:34:45.399 | Failed d search, degree bound 512, 220000 digits; program-reported 2050.15 s |
| T+10:52:32.894 → T+11:11:46.734 | 00:19:13.840 | Strict 580000-digit squared-radius floor and HIGH/LINK_BOX contraction and self-map output |
| T+12:00:00.000 | Deadline | Minimal polynomial and complete delivery missing; score 0 |
| T+12:02:44.205 | Response log event | Original failure response recorded |
| T+12:05:13.000 | Status event | Goal metadata marked blocked |
The first enumeration replay asserted because a count dictionary omitted zero-valued keys; the first reflection replay used an incorrect auxiliary-point permutation. Corrections preceded successful outputs above; failed attempts were not counted as passes. Elimination and integer-relation searches ran throughout these periods and later; allocation, thread-creation and runtime-limit failures were not counted as successes.
2616.76 and 2050.15 seconds are program-reported timings, different from submission-to-result-observation spans; they cannot be added or substituted.
**Actual output and invocation code for the 580000-digit stage were recovered; they were not preserved only in a summary.** At T+11:11:46.734 the recorded output included EXACT_CERT_RADIUS_SQUARED_DECIMAL_FLOOR_DIGITS 580000, strict HIGH/LINK_BOX contraction and self-map outputs, and ALL_NEW_DYADIC_ROOT_CERTIFICATION_INEQUALITIES_STRICT. No mathematical checks were rerun for this reconstruction. Complete certificate data and the proof package remain undelivered; output labels do not replace mathematical proof.
High precision does not supply the minimal polynomial coefficients. Failed heuristics do not establish degree above 512 or nonexistence. Synthetic irreducibility tests do not certify the actual radius.
[Download reconstructed timing and evidence labels](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-sol-timing.json). Only relative times, counts and selected labels are public; the raw transcript, internal paths, centers and polynomial coefficients are excluded.
### Verification, delivery and limits
| Acceptance requirement | Status at the original deadline |
|---|---|
| Complete radius minimal polynomial, irreducibility and rational root isolation | **Incomplete**; no actual polynomial. Algebraic existence or high-precision squared-radius data cannot replace it |
| Exact ordered centers and uniqueness of the specified data | External root data and unique-box checks exist; no complete final algebraic delivery with the required radius connection |
| Entire closed-disk coverage | Original review of the continuous reduction and exact face/arc checks; no complete final proof submission |
| Exhaustive global lower bound | Original combinatorial partition and branch checks; no complete final proof submission |
| Complete code, inputs, certificates, environment and results | **No complete proof package meeting all requirements submitted** |
Retained partial executions include 308198 orbits: 301689 direct exclusions, 4748 Farkas exclusions and 1761 residual cases. The candidate branch had 1831 nodes, 837 DUAL leaves and 79 LOCAL leaves, maximum depth 27. The 1760 noncandidate cases had 2908 nodes and 2334 DUAL leaves, maximum depth 6. Retained coverage/local-isolation, orbit and noncandidate check results exited 0. Those results require their mathematical reductions and full supporting materials for acceptance; counts or successful statuses alone are not a complete proof.
Publication does not rerun the mathematics or provide an external independent mathematical review. New work here verifies the original Goal model configuration at the user's request, revises report structure, and builds/checks the pages and downloads. It is not counted as pre-deadline mathematical progress.
During original report preparation, the pinned official scoring source was executed as ordinary Python in a public SageCell temporary kernel. Inputs were n=13, status not_completed, observation window 43200 seconds and budget 43200 seconds. It returned score 0 and exit code 0. The [actual scoring output](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-sol-score-output.json) preserves the full fields. This script does not certify a mathematical proof. The CLI reference parameter 21600 is the post-hoc common example scale, not an opening frozen n=13 reference time, and cannot alter the failure branch's zero score.
No n=13 reference fingerprint has been published. With no complete minimal polynomial, the answer-fingerprint checker was not run. The submitted fingerprint, match status and valid completion time remain null; neither a match nor a mismatch is claimed. The 12-hour failure score is 0, rather than the 100 that the formula would assign to valid completion exactly at the deadline.
The original solve did not read other chats, skills or preexisting local research materials. Reporting and publication use this task's generated artifacts, public repository data, and this thread's configuration metadata explicitly requested by the user. Original message bodies, internal paths, full execution logs and mathematical answers are not published. No complete OS-level access trace exists, so comprehensive forensic certification is not claimed. External-proof reuse and the single retrospective-run scope remain disclosed; no formal ranking is asserted.
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## Downloadable records
[Machine-readable record for this test](record-n13-sol-20261001.json) · [English report Markdown](record-n13-sol-20261001.en.md) · [中文报告](record-n13-sol-20261001.zh.html) · [Original user instruction](reports/n13-sol-main-prompt.original.txt)