Unit Disk Covering Benchmark · n = 13
n=13 · gpt-6 astra/max test report
# n=13 · gpt-6 astra/max test report
[Back to n=13](n13.html) · [Test record index](records.html)
This test took place on 2026-10-01 (UTC), with an original 12-hour limit. **No complete valid result was obtained by the deadline: not completed, score 0.** Mathematical computation stopped before the deadline. Reporting, model identification and publication were separate follow-ups. This revision summarizes the original record; it does not rerun the mathematical proof or supply the missing minimal polynomial and final proof package.
The original researcher was **gpt-6 astra/max**. Following explicit retrospective user authorization, this thread's runtime metadata was checked: all 14 original-Goal turn_context records identify model=gpt-6-astra and effort=max, with matching configuration fields. The previously published codex/unknown label was incorrect and is corrected. Later reporting models do not replace the researcher. This retrospective check does not imply permission to read chats during research.
### Existing proof reused and work performed in this test
**This test used existing online n=13 configuration, full-disk coverage and global lower-bound computer-assisted proof materials. The configuration, general-configuration classification and exclusion certificates came from external work, not independent discovery by this model. Historical replay was actually recorded, but required algebraic data and final delivery were not completed.**
The source was [VonEquinox / DiskCoveringSolve, fixed-version n=13 materials](https://github.com/VonEquinox/DiskCoveringSolve/tree/089584973f41e6374ce110f6e5b5ee456df19352/cover13), commit 089584973f41e6374ce110f6e5b5ee456df19352. It provided 119-variable candidate-root data, continuous coverage, a Voronoi reduction for general configurations, combinatorial enumeration, metric exclusions, local rigidity certificates and verification code. The repository's claim was not itself accepted as proof; the research reviewed reductions and verification scope and replayed the programs.
The source was found by the T+00:05:00 progress message. T+00:18:05 recorded the four-class enumeration audit, and a full-replay exit code of 0 was received at T+00:39:06. The first 39 minutes also included acquisition, review, enumeration dependency changes, algebraic exploration and reacquisition after a temporary directory disappeared. They are not pure verifier runtime. The replay itself was recorded as about 12.59 seconds.
Added work concerned symmetry and geometric relations, a 38-variable system and its connection to the original root box, parametrizations, elimination, finite-field and integer-relation searches, and implementation controls. T+01:29:10 progress reported a rational root certificate and interval connection passed. No acceptable radius minimal polynomial was recovered. Geometric replay and local root work do not establish complete task success.
### Task and timing
| Timing basis / basic field | Record |
|---|---|
| Record ID / independent runs | n13-codex-20261001 / 1 |
| Original model / reasoning effort | gpt-6-astra / max, checked from original-thread metadata |
| Target | Optimal 13-disk covering; complete radius minimal polynomial, irreducibility, root isolation, exact centers, full-disk coverage, general lower bound and actual proof materials |
| Displayed scoring observation window | **12 h (43200 s)**; a fixed failure observation window, not valid completion time |
| Searches confirmed stopped / worker closed | T+11:56:47 / T+11:58:38 |
| Natural elapsed time through Goal blocking | 12 h 1 min 59 s (43319 s); not marked complete |
| Main-task system accumulated time | **7 h 39 min 15 s (27555 s)**, ending at Goal blocking |
| Original report evidence cutoff | T+12:30:29 (45029 s), retained from the first report |
| Revised evidence cutoff | **T+14:38:17 (52697 s)**; includes post-research gaps and reporting work, not continuous solve time |
| Valid completion time / valid result by deadline | None, null / no |
| Original deadline / status / score | Full requirements not met / not_completed / **0** |
| Money budget / actual cost | Not recorded / not recorded |
| Codex Goal cumulative counter | 2778511; see the accounting definition below; not billed tokens or monetary cost |
| Opening benchmark version / hardware snapshot | Not frozen or recorded / unavailable |
| Benchmark version checked for the first report | b6cde6540abcf4dfc54a77979e5a7517d3f971cb |
| Answer / scoring protocols | diskcover-minpoly-v1 / diskcover-log-score-v1 |
| Formal ranking | None claimed; a unified comparison contract was not frozen |
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.
T+00:00:00 is original Goal creation. Natural elapsed time includes waits and parallel work; the system counter is separate. Their full difference is not explained by available records, so no pause totals are invented. The 119 seconds after the deadline were clock, failure-status and blocker checks, not resumed mathematical work. The later cutoff identifies retrospective review scope, not valid solve time.
### Computing environment
| Item | Record |
|---|---|
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics; 16 cores, 32 logical processors |
| Memory | DDR5, 32 GiB (34359738368 bytes) |
| Execution location | Key mathematics in local Ubuntu WSL; no remote problem solution |
| Configuration evidence | System queries during the first report, not a task-start snapshot |
Memory query SMBIOSMemoryType=34=0x22 was interpreted using [DMTF SMBIOS 3.8.0, Table 77](https://www.dmtf.org/sites/default/files/standards/documents/DSP0134_3.8.0.pdf). Configuration is not inferred from speed or substituted from other tests.
| Tool provision | Recorded tool / version | Actual use |
|---|---|---|
| Preinstalled | SageMath 10.7; original Sage Python version unverified | Exact integer, rational, polynomial and finite-field arithmetic; approximate high-precision roots |
| Preinstalled | Singular, version unrecorded | Intermediate ideal reductions; no complete n=13 elimination certificate |
| Obtained and compiled during research | flatter, commit d2b8026f29b4a69e987b15d4b240f8a5053275d3 | Integer-lattice reduction for candidate discovery |
| Obtained and compiled during research | msolve 0.10.1 | Polynomial systems and finite-field Gröbner-basis exploration |
| Written or modified during research | F4 intermediate-state, memory-management and SBA programs; no independent release | Small exact controls and intermediate problem calculations; no complete final basis certificate |
| Executed for the first report | Ubuntu WSL Python 3.12.3 | In-memory official scoring using argparse, json, math and decimal |
[flatter](https://github.com/keeganryan/flatter/tree/d2b8026f29b4a69e987b15d4b240f8a5053275d3) reduces integer lattices; [msolve](https://github.com/algebraic-solving/msolve) solves polynomial systems. Core work used Sage arithmetic modules; full dependency versions were not archived and are not inferred from other tests. Reporting Python is not an opening research snapshot.
A public Magma interface ran a control example. Actual problem-equation submission was denied and not performed; Magma is not counted as the problem's remote solving environment.
### Input prompt
The original input required the webpage's test prompt and stipulated: **public internet and WSL tools are allowed; local materials, skills, chats and archived chats are prohibited, as is access to stored WSL materials or files. Any violation fails the task. The limit is 12 hours; no complete correct result by then also means failure.**
[Download the exact original Goal input, in Chinese](reports/n13-codex-main-prompt.original.txt), SHA256 dfc86a8ff759f6c54ceda6cdaaef5b6f5165b307b00b3bf0eedf1d57ec48fe74. The complete opening webpage version was not frozen. The statement below is the public task at the version checked for the first report, not an independently preserved opening-page snapshot. Progress at T+01:06:30 explicitly reconfirmed that the minimal polynomial of the radius itself was required.
<details><summary>Expand the complete statement at the checked version</summary>
### Checked-version 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.
[Fixed-version source](https://github.com/pikaaa345/disk-covering-benchmark/blob/b6cde6540abcf4dfc54a77979e5a7517d3f971cb/tasks/n13.en.md).
</details>
### Main stages
T+00:00:00 is original Goal creation. Endpoints come from timestamped progress messages and tool results, with durations rounded to seconds; a message timestamp is not substituted for an unrecorded exact discovery or process-exit time.
| Relative natural time | Interval length (approx.) | Main work and status at the endpoint |
|---|---|---|
| T+00:00:00–T+00:39:06 | 0 h 39 min 06 s | Acquire and review external proof; source found by minute 5 and enumeration audit recorded by minute 18. Adjust dependencies and reacquire fixed inputs after a temporary-directory loss; reduced-system work overlaps. Full-replay exit 0 received, minimal polynomial missing. |
| T+00:39:06–T+01:29:10 | 0 h 50 min 04 s | Reflection symmetry, eleven rhombus relations, constraint mapping and 38-variable rational root certificate. Progress reports certification and interval connection to the 119-variable root box passed; degree-128/256 searches yield no acceptable polynomial. |
| T+01:29:10–T+03:53:22 | 2 h 24 min 12 s | Blocked/refined lattice searches, parametrizations, degenerate-branch replacement and memory experiments. Known-quartic controls pass, problem work incomplete; stored-log/script reading denied at the endpoint. |
| T+03:53:22–T+06:46:29 | 2 h 53 min 07 s | Memory input/live output, system reduction and degeneracy exclusions under memory limits. T+06:35:16 records actual-problem Magma submission denial. Input controls reveal unexpanded-parenthesis multiplication error; affected msolve runs discarded. |
| T+06:46:29–T+08:11:13 | 1 h 24 min 44 s | Fully expanded input, finite-field/p-adic work and exact controls. Inspected branches not associated with target; a degree-512 search fails extra-precision screening. T+08:04:22 reports memory-worker crash; restore public input and adjust arithmetic threading. |
| T+08:11:13–T+09:37:55 | 1 h 26 min 42 s | Rebuild precision and launch final degree-512 search; compare elimination/SBA controls. Reduce jobs after high-memory launch denial. Endpoint records 332 reconstructed auxiliary relations passing numerical screening and guarded elimination; not exact problem certificates. |
| T+09:37:55–T+11:53:41 | 2 h 15 min 46 s | Main/auxiliary searches, coordinate-recovery controls and intermediate reduction. All 129 auxiliary degree-128 candidates fail holdout screening. Endpoint main-search return 124, 705 rows, 513 rejected candidates; no minimal polynomial. |
| T+11:53:41–T+11:58:38 | 0 h 04 min 57 s | Final auxiliary degree-256 continuation times out without basis output. Searches confirmed stopped at T+11:56:47; worker closed at T+11:58:38, exit 0. No complete proof package. |
| T+11:58:38–T+12:00:00 | 0 h 01 min 22 s | Final response at T+11:59:18 reports failure; keep computation stopped. No complete valid result at deadline. |
| T+12:00:00–T+12:01:59 | 0 h 01 min 59 s | Clock, failure-status and blocker checks only; Goal blocked, not complete, research not resumed. |
Intervals are not exclusive CPU categories. Full replay took about 12.59 s. The final degree-512 search at 1572864-bit search precision ran about 3 h 32 min (12718 s); all 513 candidates from 705 rows failed an additional 32768-bit holdout. Auxiliary degree-128 search took about 14 min 52 s (892 s), with 129 candidates failing an additional 16384-bit check. Degree-256 continuation timed out after about 8 min (480 s), no basis output. Concurrent runtimes are not added. Main-wrapper return 124 is not successful exit 0; output was for candidate screening.
### Report and model-identification follow-ups
The first separate request to generate a report and score took approximately **16 min 44 s (1004.270 s)** from request to report response, recovered from this thread's message timestamps. The gap after research is excluded, and reporting is not original scoring time. An independently verified accumulated system counter for that follow-up is unavailable.
This request to revise using the n=11 Astra format starts a separate V+00:00:00; the evidence cutoff is **V+00:21:14**. This is a review cutoff, not the completed-publication duration. The user authorized retrospective model/timeline inspection only in this thread; no other chats, archived chats or local mathematical materials were searched.
No follow-up completed the missing polynomial. Reporting, model identification and uploading do not constitute successful late mathematical completion.
### Verification, delivery and limits
| Acceptance requirement | Result by the original deadline |
|---|---|
| Complete radius minimal polynomial, irreducibility, isolation and exact association | **Not completed**, no acceptable full coefficient candidate or univariate association proof |
| Exact uniquely specified ordered centers | Root-box/reduced-system connection records exist; no integrated final submission package |
| Continuous full-disk coverage | External certificates reused with historical replay; not rerun here |
| General configurations and degeneracies in the global lower bound | External classification/exclusion/rigidity reviewed and reused; not integrated with every algebraic requirement |
| Complete actual code, inputs, certificates, environment and results | **Incomplete delivery**; intermediate work and external materials are not a complete final proof |
Historical records include 28 public-manifest checks, full-replay exit 0 and 308198 topology orbits: 306437 metric exclusions, 1760 non-candidates and one candidate. This revision checks existing messages/receipts; it does not rerun mathematics or constitute independent external review.
Known-quartic 81x⁴−2 relation controls and cyclic-5 Gröbner comparisons passed. Numerical controls of 84 geometric/stress identities, high-precision auxiliary screening and intermediate bases are not exact n=13 certificates. Search failure does not prove a higher degree, algebraic impossibility or transcendence. Input errors, worker crashes and unfinished reductions are implementation limits; their total losses or hypothetical success after fixes cannot be quantified.
Automatic review denied stored-log/script reads and actual-problem Magma submission; no additional authorization was received and those denied actions were not continued. Denied large-memory launch was replaced with smaller jobs and own-process guards. No full OS access trace exists, so comprehensive resource compliance is not certified. Later clarifications for other tests are not retroactive opening authorization. **The missing minimal polynomial and full proof already determine score 0.**
The first report executed pinned official scoring with exit code 0:
~~~text
python official-score.py --n 13 --reference-seconds 21600 --elapsed-seconds 43200 --budget-seconds 43200 --status not_completed
~~~
The not_completed branch returns **0** directly. The 43200-second failure window is not success-formula completion time. Required CLI reference 21600 uses the default demonstration value, not a claimed frozen opening n=13 benchmark; failure score is independent of it. [Scoring implementation](https://github.com/pikaaa345/disk-covering-benchmark/blob/b6cde6540abcf4dfc54a77979e5a7517d3f971cb/tools/score.py), snapshot SHA256 4b2a1eb4640e6524772c8d8592bf6b41ba292dc3e3520829b679016d3fb32977.
Revision checks cover model metadata, timeline, report/data consistency, site generation and tool self-checks. Synthetic self-checks do not solve the problem. No n=13 reference fingerprint is published. With no minimal-polynomial coefficients, no answer check was run; coefficients, answer hash, matching state and completion time remain absent/null, not a fingerprint mismatch.
Public deliverables are bilingual reports, machine records, original input, existing scoring output and sources, not a complete mathematical answer or proof package. Original-report SHA256: 67d0074efb0b1028023a631ee146def6ecb8a5ab5a8a1f26f6b8e81352e3605f. This revision follows the [n=11 Astra report](record-n11-astra-20260930.html), corrects the model and adds supported stages and reporting duration. Opening version/hardware and costs remain unavailable.
[Back to the top](#)
## Downloadable records
[Machine-readable record](record-n13-codex-20261001.json) · [Report Markdown](record-n13-codex-20261001.en.md) · [Full Chinese report](record-n13-codex-20261001.zh.html) · [Original input](reports/n13-codex-main-prompt.original.txt) · [Actual score output](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-score-output.json) · [Sources](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-report-sources.json)
# n=13 · gpt-6 astra/max test report
[Back to n=13](n13.html) · [Test record index](records.html)
This test took place on 2026-10-01 (UTC), with an original 12-hour limit. **No complete valid result was obtained by the deadline: not completed, score 0.** Mathematical computation stopped before the deadline. Reporting, model identification and publication were separate follow-ups. This revision summarizes the original record; it does not rerun the mathematical proof or supply the missing minimal polynomial and final proof package.
The original researcher was **gpt-6 astra/max**. Following explicit retrospective user authorization, this thread's runtime metadata was checked: all 14 original-Goal turn_context records identify model=gpt-6-astra and effort=max, with matching configuration fields. The previously published codex/unknown label was incorrect and is corrected. Later reporting models do not replace the researcher. This retrospective check does not imply permission to read chats during research.
### Existing proof reused and work performed in this test
**This test used existing online n=13 configuration, full-disk coverage and global lower-bound computer-assisted proof materials. The configuration, general-configuration classification and exclusion certificates came from external work, not independent discovery by this model. Historical replay was actually recorded, but required algebraic data and final delivery were not completed.**
The source was [VonEquinox / DiskCoveringSolve, fixed-version n=13 materials](https://github.com/VonEquinox/DiskCoveringSolve/tree/089584973f41e6374ce110f6e5b5ee456df19352/cover13), commit 089584973f41e6374ce110f6e5b5ee456df19352. It provided 119-variable candidate-root data, continuous coverage, a Voronoi reduction for general configurations, combinatorial enumeration, metric exclusions, local rigidity certificates and verification code. The repository's claim was not itself accepted as proof; the research reviewed reductions and verification scope and replayed the programs.
The source was found by the T+00:05:00 progress message. T+00:18:05 recorded the four-class enumeration audit, and a full-replay exit code of 0 was received at T+00:39:06. The first 39 minutes also included acquisition, review, enumeration dependency changes, algebraic exploration and reacquisition after a temporary directory disappeared. They are not pure verifier runtime. The replay itself was recorded as about 12.59 seconds.
Added work concerned symmetry and geometric relations, a 38-variable system and its connection to the original root box, parametrizations, elimination, finite-field and integer-relation searches, and implementation controls. T+01:29:10 progress reported a rational root certificate and interval connection passed. No acceptable radius minimal polynomial was recovered. Geometric replay and local root work do not establish complete task success.
### Task and timing
| Timing basis / basic field | Record |
|---|---|
| Record ID / independent runs | n13-codex-20261001 / 1 |
| Original model / reasoning effort | gpt-6-astra / max, checked from original-thread metadata |
| Target | Optimal 13-disk covering; complete radius minimal polynomial, irreducibility, root isolation, exact centers, full-disk coverage, general lower bound and actual proof materials |
| Displayed scoring observation window | **12 h (43200 s)**; a fixed failure observation window, not valid completion time |
| Searches confirmed stopped / worker closed | T+11:56:47 / T+11:58:38 |
| Natural elapsed time through Goal blocking | 12 h 1 min 59 s (43319 s); not marked complete |
| Main-task system accumulated time | **7 h 39 min 15 s (27555 s)**, ending at Goal blocking |
| Original report evidence cutoff | T+12:30:29 (45029 s), retained from the first report |
| Revised evidence cutoff | **T+14:38:17 (52697 s)**; includes post-research gaps and reporting work, not continuous solve time |
| Valid completion time / valid result by deadline | None, null / no |
| Original deadline / status / score | Full requirements not met / not_completed / **0** |
| Money budget / actual cost | Not recorded / not recorded |
| Codex Goal cumulative counter | 2778511; see the accounting definition below; not billed tokens or monetary cost |
| Opening benchmark version / hardware snapshot | Not frozen or recorded / unavailable |
| Benchmark version checked for the first report | b6cde6540abcf4dfc54a77979e5a7517d3f971cb |
| Answer / scoring protocols | diskcover-minpoly-v1 / diskcover-log-score-v1 |
| Formal ranking | None claimed; a unified comparison contract was not frozen |
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.
T+00:00:00 is original Goal creation. Natural elapsed time includes waits and parallel work; the system counter is separate. Their full difference is not explained by available records, so no pause totals are invented. The 119 seconds after the deadline were clock, failure-status and blocker checks, not resumed mathematical work. The later cutoff identifies retrospective review scope, not valid solve time.
### Computing environment
| Item | Record |
|---|---|
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics; 16 cores, 32 logical processors |
| Memory | DDR5, 32 GiB (34359738368 bytes) |
| Execution location | Key mathematics in local Ubuntu WSL; no remote problem solution |
| Configuration evidence | System queries during the first report, not a task-start snapshot |
Memory query SMBIOSMemoryType=34=0x22 was interpreted using [DMTF SMBIOS 3.8.0, Table 77](https://www.dmtf.org/sites/default/files/standards/documents/DSP0134_3.8.0.pdf). Configuration is not inferred from speed or substituted from other tests.
| Tool provision | Recorded tool / version | Actual use |
|---|---|---|
| Preinstalled | SageMath 10.7; original Sage Python version unverified | Exact integer, rational, polynomial and finite-field arithmetic; approximate high-precision roots |
| Preinstalled | Singular, version unrecorded | Intermediate ideal reductions; no complete n=13 elimination certificate |
| Obtained and compiled during research | flatter, commit d2b8026f29b4a69e987b15d4b240f8a5053275d3 | Integer-lattice reduction for candidate discovery |
| Obtained and compiled during research | msolve 0.10.1 | Polynomial systems and finite-field Gröbner-basis exploration |
| Written or modified during research | F4 intermediate-state, memory-management and SBA programs; no independent release | Small exact controls and intermediate problem calculations; no complete final basis certificate |
| Executed for the first report | Ubuntu WSL Python 3.12.3 | In-memory official scoring using argparse, json, math and decimal |
[flatter](https://github.com/keeganryan/flatter/tree/d2b8026f29b4a69e987b15d4b240f8a5053275d3) reduces integer lattices; [msolve](https://github.com/algebraic-solving/msolve) solves polynomial systems. Core work used Sage arithmetic modules; full dependency versions were not archived and are not inferred from other tests. Reporting Python is not an opening research snapshot.
A public Magma interface ran a control example. Actual problem-equation submission was denied and not performed; Magma is not counted as the problem's remote solving environment.
### Input prompt
The original input required the webpage's test prompt and stipulated: **public internet and WSL tools are allowed; local materials, skills, chats and archived chats are prohibited, as is access to stored WSL materials or files. Any violation fails the task. The limit is 12 hours; no complete correct result by then also means failure.**
[Download the exact original Goal input, in Chinese](reports/n13-codex-main-prompt.original.txt), SHA256 dfc86a8ff759f6c54ceda6cdaaef5b6f5165b307b00b3bf0eedf1d57ec48fe74. The complete opening webpage version was not frozen. The statement below is the public task at the version checked for the first report, not an independently preserved opening-page snapshot. Progress at T+01:06:30 explicitly reconfirmed that the minimal polynomial of the radius itself was required.
<details><summary>Expand the complete statement at the checked version</summary>
### Checked-version 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.
[Fixed-version source](https://github.com/pikaaa345/disk-covering-benchmark/blob/b6cde6540abcf4dfc54a77979e5a7517d3f971cb/tasks/n13.en.md).
</details>
### Main stages
T+00:00:00 is original Goal creation. Endpoints come from timestamped progress messages and tool results, with durations rounded to seconds; a message timestamp is not substituted for an unrecorded exact discovery or process-exit time.
| Relative natural time | Interval length (approx.) | Main work and status at the endpoint |
|---|---|---|
| T+00:00:00–T+00:39:06 | 0 h 39 min 06 s | Acquire and review external proof; source found by minute 5 and enumeration audit recorded by minute 18. Adjust dependencies and reacquire fixed inputs after a temporary-directory loss; reduced-system work overlaps. Full-replay exit 0 received, minimal polynomial missing. |
| T+00:39:06–T+01:29:10 | 0 h 50 min 04 s | Reflection symmetry, eleven rhombus relations, constraint mapping and 38-variable rational root certificate. Progress reports certification and interval connection to the 119-variable root box passed; degree-128/256 searches yield no acceptable polynomial. |
| T+01:29:10–T+03:53:22 | 2 h 24 min 12 s | Blocked/refined lattice searches, parametrizations, degenerate-branch replacement and memory experiments. Known-quartic controls pass, problem work incomplete; stored-log/script reading denied at the endpoint. |
| T+03:53:22–T+06:46:29 | 2 h 53 min 07 s | Memory input/live output, system reduction and degeneracy exclusions under memory limits. T+06:35:16 records actual-problem Magma submission denial. Input controls reveal unexpanded-parenthesis multiplication error; affected msolve runs discarded. |
| T+06:46:29–T+08:11:13 | 1 h 24 min 44 s | Fully expanded input, finite-field/p-adic work and exact controls. Inspected branches not associated with target; a degree-512 search fails extra-precision screening. T+08:04:22 reports memory-worker crash; restore public input and adjust arithmetic threading. |
| T+08:11:13–T+09:37:55 | 1 h 26 min 42 s | Rebuild precision and launch final degree-512 search; compare elimination/SBA controls. Reduce jobs after high-memory launch denial. Endpoint records 332 reconstructed auxiliary relations passing numerical screening and guarded elimination; not exact problem certificates. |
| T+09:37:55–T+11:53:41 | 2 h 15 min 46 s | Main/auxiliary searches, coordinate-recovery controls and intermediate reduction. All 129 auxiliary degree-128 candidates fail holdout screening. Endpoint main-search return 124, 705 rows, 513 rejected candidates; no minimal polynomial. |
| T+11:53:41–T+11:58:38 | 0 h 04 min 57 s | Final auxiliary degree-256 continuation times out without basis output. Searches confirmed stopped at T+11:56:47; worker closed at T+11:58:38, exit 0. No complete proof package. |
| T+11:58:38–T+12:00:00 | 0 h 01 min 22 s | Final response at T+11:59:18 reports failure; keep computation stopped. No complete valid result at deadline. |
| T+12:00:00–T+12:01:59 | 0 h 01 min 59 s | Clock, failure-status and blocker checks only; Goal blocked, not complete, research not resumed. |
Intervals are not exclusive CPU categories. Full replay took about 12.59 s. The final degree-512 search at 1572864-bit search precision ran about 3 h 32 min (12718 s); all 513 candidates from 705 rows failed an additional 32768-bit holdout. Auxiliary degree-128 search took about 14 min 52 s (892 s), with 129 candidates failing an additional 16384-bit check. Degree-256 continuation timed out after about 8 min (480 s), no basis output. Concurrent runtimes are not added. Main-wrapper return 124 is not successful exit 0; output was for candidate screening.
### Report and model-identification follow-ups
The first separate request to generate a report and score took approximately **16 min 44 s (1004.270 s)** from request to report response, recovered from this thread's message timestamps. The gap after research is excluded, and reporting is not original scoring time. An independently verified accumulated system counter for that follow-up is unavailable.
This request to revise using the n=11 Astra format starts a separate V+00:00:00; the evidence cutoff is **V+00:21:14**. This is a review cutoff, not the completed-publication duration. The user authorized retrospective model/timeline inspection only in this thread; no other chats, archived chats or local mathematical materials were searched.
No follow-up completed the missing polynomial. Reporting, model identification and uploading do not constitute successful late mathematical completion.
### Verification, delivery and limits
| Acceptance requirement | Result by the original deadline |
|---|---|
| Complete radius minimal polynomial, irreducibility, isolation and exact association | **Not completed**, no acceptable full coefficient candidate or univariate association proof |
| Exact uniquely specified ordered centers | Root-box/reduced-system connection records exist; no integrated final submission package |
| Continuous full-disk coverage | External certificates reused with historical replay; not rerun here |
| General configurations and degeneracies in the global lower bound | External classification/exclusion/rigidity reviewed and reused; not integrated with every algebraic requirement |
| Complete actual code, inputs, certificates, environment and results | **Incomplete delivery**; intermediate work and external materials are not a complete final proof |
Historical records include 28 public-manifest checks, full-replay exit 0 and 308198 topology orbits: 306437 metric exclusions, 1760 non-candidates and one candidate. This revision checks existing messages/receipts; it does not rerun mathematics or constitute independent external review.
Known-quartic 81x⁴−2 relation controls and cyclic-5 Gröbner comparisons passed. Numerical controls of 84 geometric/stress identities, high-precision auxiliary screening and intermediate bases are not exact n=13 certificates. Search failure does not prove a higher degree, algebraic impossibility or transcendence. Input errors, worker crashes and unfinished reductions are implementation limits; their total losses or hypothetical success after fixes cannot be quantified.
Automatic review denied stored-log/script reads and actual-problem Magma submission; no additional authorization was received and those denied actions were not continued. Denied large-memory launch was replaced with smaller jobs and own-process guards. No full OS access trace exists, so comprehensive resource compliance is not certified. Later clarifications for other tests are not retroactive opening authorization. **The missing minimal polynomial and full proof already determine score 0.**
The first report executed pinned official scoring with exit code 0:
~~~text
python official-score.py --n 13 --reference-seconds 21600 --elapsed-seconds 43200 --budget-seconds 43200 --status not_completed
~~~
The not_completed branch returns **0** directly. The 43200-second failure window is not success-formula completion time. Required CLI reference 21600 uses the default demonstration value, not a claimed frozen opening n=13 benchmark; failure score is independent of it. [Scoring implementation](https://github.com/pikaaa345/disk-covering-benchmark/blob/b6cde6540abcf4dfc54a77979e5a7517d3f971cb/tools/score.py), snapshot SHA256 4b2a1eb4640e6524772c8d8592bf6b41ba292dc3e3520829b679016d3fb32977.
Revision checks cover model metadata, timeline, report/data consistency, site generation and tool self-checks. Synthetic self-checks do not solve the problem. No n=13 reference fingerprint is published. With no minimal-polynomial coefficients, no answer check was run; coefficients, answer hash, matching state and completion time remain absent/null, not a fingerprint mismatch.
Public deliverables are bilingual reports, machine records, original input, existing scoring output and sources, not a complete mathematical answer or proof package. Original-report SHA256: 67d0074efb0b1028023a631ee146def6ecb8a5ab5a8a1f26f6b8e81352e3605f. This revision follows the [n=11 Astra report](record-n11-astra-20260930.html), corrects the model and adds supported stages and reporting duration. Opening version/hardware and costs remain unavailable.
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## Downloadable records
[Machine-readable record](record-n13-codex-20261001.json) · [Report Markdown](record-n13-codex-20261001.en.md) · [Full Chinese report](record-n13-codex-20261001.zh.html) · [Original input](reports/n13-codex-main-prompt.original.txt) · [Actual score output](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-score-output.json) · [Sources](https://github.com/pikaaa345/disk-covering-benchmark/blob/main/results/n13-report-sources.json)