Unit Disk Covering Benchmark · n = 11
n=11 · gpt-6 astra/max test report
# n=11 · gpt-6 astra/max test report
[Back to n=11](n11.html) · [Test record index](records.html)
This test took place on 2026-09-30 (UTC). It missed the original 60-minute deadline and continued with authorization. The original problem required a nonzero integer polynomial; the current problem additionally requires the minimal polynomial and an irreducibility proof. This historical record is not automatically a formal pass under the current problem. This website update summarizes existing reports and logs and does not rerun the mathematical proof. The optimal radius, coefficients, isolating endpoints and proof packages remain withheld.
### Existing proof reused and work performed in this test
**This test used an existing, independently replayable computer-assisted proof package for the optimal n=11 radius, found online. The candidate configuration, full-disk coverage certificates and global optimality certificates excluding every better configuration came from external work; they were not independently discovered by this model during the test.**
The source was the [fixed-version proof bundle in VonEquinox / DiskCoveringSolve](https://github.com/VonEquinox/DiskCoveringSolve/tree/089584973f41e6374ce110f6e5b5ee456df19352/cover11/proof_bundle), commit `089584973f41e6374ce110f6e5b5ee456df19352`. It supplied a 110-variable KKT candidate-root certificate, full-disk coverage, a Voronoi reduction for general configurations and exhaustive combinatorial enumeration, together with stress, continuous-branch and local-bound certificates and verification code. This reused a complete optimality proof package, rather than merely a numerical candidate. The original repository's completion claim was not itself taken as proof; the historical record contains actual replay results.
The model's work was to find, review and replay those materials, recover an explicit radius polynomial and algebraic center data, prove their connection to the certified root, and prepare the delivery. This record therefore measures research and algebraic completion using an existing external proof; it does not demonstrate discovery of a complete optimality proof from scratch.
Astra found the source at about T+00:54:13. The approximately 1 h 9 min 45 s interval T+00:54:13–T+02:03:57 covered acquisition, review and replay of the existing proof, alongside algebraic exploration; two full replays passed. This was verification of an existing proof, not discovery of a global proof from scratch or pure verifier runtime. Subsequent main-task work chiefly supplied the explicit univariate polynomial and certified connections; the minimal polynomial was handled in a separate follow-up.
### Task and timing
| Timing basis | Record |
|---|---|
| Main task natural elapsed time | 5 h 34 min 29.011 s, from request to final response |
| Independent minimal-polynomial follow-up | 13 min 46.178 s, from follow-up request to response |
| Total used for the display | **5 h 48 min 15.189 s (20895.189 s)**, the sum of those two intervals, excluding the gap between tasks |
| Main task system accumulated time | 4 h 58 min 1 s (17881 s) |
| Follow-up system accumulated time | Not recorded |
| Original 60-minute deadline | Missed; continued work and the follow-up do not change that judgment |
Natural elapsed time includes waits, pauses and overlapping computation. It is not replaced by the system counter, and their difference is not assigned to a guessed cause.
### Computing environment
| Item | Record |
|---|---|
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics |
| Memory | DDR5, 32 GB |
| Execution location | Local Windows and local Ubuntu WSL; no recorded key remote computation |
| Configuration evidence | Windows queries during preparation of the SOL report; not a task-start hardware snapshot |
The original Astra report did not preserve a separate hardware snapshot. The configuration above uses the publisher's shared test-machine record from the SOL report, not an independent Astra task-start query.
SageMath and Python were used for exact algebra, certificate replay and independent verification. The original Astra report did not fully archive their versions. msolve, a polynomial-system solver, was obtained during the research for elimination and parametrization attempts; it is not listed as a verified preinstalled tool.
### Input prompt
The actual input also required: **do not use local materials or skills, or consult other local answers; violating this rule fails the task. The deadline is 60 minutes, and failure to finish by then counts as failure.**
[Download the exact original input, in Chinese](reports/n11-astra-main-prompt.original.txt). The statement below is an English translation, not the current revised problem.
<details><summary>Expand the complete historical problem statement</summary>
### Historical problem statement
Determine the optimal radius for covering the unit disk with eleven congruent disks.
Let
$$D=\{x\in\mathbb R^2:\|x\|_2\le1\},\qquad R_D(C)=\max_{x\in D}\min_{1\le i\le11}\|x-c_i\|_2,$$
where $C=(c_1,\ldots,c_{11})\in(\mathbb R^2)^{11}$, and define
$$r_D(11)=\inf_{C\in(\mathbb R^2)^{11}}R_D(C).$$
Determine $r_D(11)$ and meet all four requirements:
1. **Exact radius:** explicitly list every coefficient of a nonzero integer polynomial $P(t)$ and specific rational numbers $0\le a<b$. Prove that $P$ has exactly one real root $\alpha$ in $(a,b)$.
2. **Explicit configuration:** specify a fixed set of centers $C_\ast$ by radicals, polynomials with rational isolating intervals, or other explicitly listed finite algebraic data with a uniqueness proof.
3. **Coverage:** prove
$$\forall x\in D,\quad\exists i\in\{1,\ldots,11\},\quad\|x-c_i^\ast\|_2\le\alpha.$$
4. **Global optimality:** prove
$$\forall C\in(\mathbb R^2)^{11},\quad R_D(C)\ge\alpha,$$
and hence $R_D(C_\ast)=r_D(11)=\alpha$. Do not assume symmetry or a combinatorial structure without proof.
Any rigorous method is allowed. Existence, algebraicity, unfinished enumeration, a proposed algorithm, or a numerical candidate alone does not complete the task. Computer-assisted proofs must include actual results and the code, inputs and proof materials needed for rigorous verification. Numerical approximations and checks of finitely many sample points cannot establish coverage of the entire disk.
</details>
### Main stages
T+00:00:00 is the start of the main task. The intervals are natural elapsed time, including waits and parallel work.
| Relative natural time | Interval length (approx.) | Main work and status at the endpoint |
|---|---|---|
| T+00:00:00–T+00:54:13 | 0 h 54 min 13 s | Background research, candidate reconstruction and independent lower bounds; find an existing complete optimality proof package online. A candidate and coarse bounds were available but did not match; an external complete proof package was found at the end. Its global proof was not independently discovered by the model. |
| T+00:54:13–T+02:03:57 | 1 h 9 min 45 s | Acquire, review and replay the existing external full-disk coverage and global optimality proof. Reuse the DiskCoveringSolve proof, inputs and verification code; two full replays passed. Algebraic work ran alongside replay, but an explicit univariate polynomial was still missing. |
| T+02:03:57–T+03:05:27 | 1 h 1 min 30 s | Geometric reduction and elimination attempts. Geometric reduction produced new equations; several elimination and integer-relation searches failed. |
| T+03:05:27–T+03:20:24 | 0 h 14 min 57 s | Remove degenerate branches and obtain finite-field elimination data. Removing degenerate branches enabled finite-field elimination and established the recovery scale. |
| T+03:20:24–T+04:49:12 | 1 h 28 min 48 s | Parametrization, a unique-root box and rational coefficient recovery attempts. Root-box and reconstruction arguments progressed; failed automatic recovery led to a prime-by-prime route. |
| T+04:49:12–T+05:05:51 | 0 h 16 min 39 s | p-adic lifting, rational reconstruction, root isolation and exact substitution. Successive lifting recovered the coefficients; exact substitution and the unique-root connection passed. |
| T+05:05:51–T+05:34:29 | 0 h 28 min 38 s | Integrate independent verifiers, complete the arguments and package delivery. Independent checks, branch selection and complete delivery were finished. |
These intervals were reconstructed from timestamped progress nodes. Durations are rounded to seconds and marked approximate. They are not exclusive CPU-time categories; program runtimes inside them must not be added again.
### Minimal-polynomial follow-up
The additional instruction was: “Simplify the polynomial to the minimal polynomial of $alpha$.” The [exact original follow-up input](reports/n11-astra-followup-prompt.original.txt) is retained separately.
The follow-up runs from F+00:00:00 to F+00:13:46.178, measured from its own request. This entire interval is included in the displayed total. Work included rational factorization, selecting the target-root factor, an irreducibility certificate, a standalone Python verifier and delivery. The initial screening/factorization script took about 0.661 s internally; the independent verifier took about 18.005 s. Those program runtimes do not replace the 13 min 46.178 s follow-up duration.
### Verification, delivery and limits
Historical results record successful execution of 28 global tasks, exact algebraic connections and the subsequent minimal-polynomial verification. These historical executions are distinct from this website update. The global certificate came from existing external work; this test added replay, geometric/algebraic connections, explicit data recovery and the minimal-polynomial follow-up.
The full proof and follow-up packages remain local. The website publishes an answer-withheld report, original inputs and a machine-readable timeline. This update does not repeat the full historical proof replay or treat a status field as formal acceptance of the current problem.
Source: `cover11-work-report.md`, original-report SHA256: `d2aa68c72384dc0ffb9732e076e0446f68406d0d093c3f17b1aa9a8a78ec1592`. The original contains answers and is not published verbatim.
[Back to the top](#)
## Downloadable records
[Machine-readable record for this test](record-n11-astra-20260930.json) · [Report Markdown](record-n11-astra-20260930.en.md)
# n=11 · gpt-6 astra/max test report
[Back to n=11](n11.html) · [Test record index](records.html)
This test took place on 2026-09-30 (UTC). It missed the original 60-minute deadline and continued with authorization. The original problem required a nonzero integer polynomial; the current problem additionally requires the minimal polynomial and an irreducibility proof. This historical record is not automatically a formal pass under the current problem. This website update summarizes existing reports and logs and does not rerun the mathematical proof. The optimal radius, coefficients, isolating endpoints and proof packages remain withheld.
### Existing proof reused and work performed in this test
**This test used an existing, independently replayable computer-assisted proof package for the optimal n=11 radius, found online. The candidate configuration, full-disk coverage certificates and global optimality certificates excluding every better configuration came from external work; they were not independently discovered by this model during the test.**
The source was the [fixed-version proof bundle in VonEquinox / DiskCoveringSolve](https://github.com/VonEquinox/DiskCoveringSolve/tree/089584973f41e6374ce110f6e5b5ee456df19352/cover11/proof_bundle), commit `089584973f41e6374ce110f6e5b5ee456df19352`. It supplied a 110-variable KKT candidate-root certificate, full-disk coverage, a Voronoi reduction for general configurations and exhaustive combinatorial enumeration, together with stress, continuous-branch and local-bound certificates and verification code. This reused a complete optimality proof package, rather than merely a numerical candidate. The original repository's completion claim was not itself taken as proof; the historical record contains actual replay results.
The model's work was to find, review and replay those materials, recover an explicit radius polynomial and algebraic center data, prove their connection to the certified root, and prepare the delivery. This record therefore measures research and algebraic completion using an existing external proof; it does not demonstrate discovery of a complete optimality proof from scratch.
Astra found the source at about T+00:54:13. The approximately 1 h 9 min 45 s interval T+00:54:13–T+02:03:57 covered acquisition, review and replay of the existing proof, alongside algebraic exploration; two full replays passed. This was verification of an existing proof, not discovery of a global proof from scratch or pure verifier runtime. Subsequent main-task work chiefly supplied the explicit univariate polynomial and certified connections; the minimal polynomial was handled in a separate follow-up.
### Task and timing
| Timing basis | Record |
|---|---|
| Main task natural elapsed time | 5 h 34 min 29.011 s, from request to final response |
| Independent minimal-polynomial follow-up | 13 min 46.178 s, from follow-up request to response |
| Total used for the display | **5 h 48 min 15.189 s (20895.189 s)**, the sum of those two intervals, excluding the gap between tasks |
| Main task system accumulated time | 4 h 58 min 1 s (17881 s) |
| Follow-up system accumulated time | Not recorded |
| Original 60-minute deadline | Missed; continued work and the follow-up do not change that judgment |
Natural elapsed time includes waits, pauses and overlapping computation. It is not replaced by the system counter, and their difference is not assigned to a guessed cause.
### Computing environment
| Item | Record |
|---|---|
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics |
| Memory | DDR5, 32 GB |
| Execution location | Local Windows and local Ubuntu WSL; no recorded key remote computation |
| Configuration evidence | Windows queries during preparation of the SOL report; not a task-start hardware snapshot |
The original Astra report did not preserve a separate hardware snapshot. The configuration above uses the publisher's shared test-machine record from the SOL report, not an independent Astra task-start query.
SageMath and Python were used for exact algebra, certificate replay and independent verification. The original Astra report did not fully archive their versions. msolve, a polynomial-system solver, was obtained during the research for elimination and parametrization attempts; it is not listed as a verified preinstalled tool.
### Input prompt
The actual input also required: **do not use local materials or skills, or consult other local answers; violating this rule fails the task. The deadline is 60 minutes, and failure to finish by then counts as failure.**
[Download the exact original input, in Chinese](reports/n11-astra-main-prompt.original.txt). The statement below is an English translation, not the current revised problem.
<details><summary>Expand the complete historical problem statement</summary>
### Historical problem statement
Determine the optimal radius for covering the unit disk with eleven congruent disks.
Let
$$D=\{x\in\mathbb R^2:\|x\|_2\le1\},\qquad R_D(C)=\max_{x\in D}\min_{1\le i\le11}\|x-c_i\|_2,$$
where $C=(c_1,\ldots,c_{11})\in(\mathbb R^2)^{11}$, and define
$$r_D(11)=\inf_{C\in(\mathbb R^2)^{11}}R_D(C).$$
Determine $r_D(11)$ and meet all four requirements:
1. **Exact radius:** explicitly list every coefficient of a nonzero integer polynomial $P(t)$ and specific rational numbers $0\le a<b$. Prove that $P$ has exactly one real root $\alpha$ in $(a,b)$.
2. **Explicit configuration:** specify a fixed set of centers $C_\ast$ by radicals, polynomials with rational isolating intervals, or other explicitly listed finite algebraic data with a uniqueness proof.
3. **Coverage:** prove
$$\forall x\in D,\quad\exists i\in\{1,\ldots,11\},\quad\|x-c_i^\ast\|_2\le\alpha.$$
4. **Global optimality:** prove
$$\forall C\in(\mathbb R^2)^{11},\quad R_D(C)\ge\alpha,$$
and hence $R_D(C_\ast)=r_D(11)=\alpha$. Do not assume symmetry or a combinatorial structure without proof.
Any rigorous method is allowed. Existence, algebraicity, unfinished enumeration, a proposed algorithm, or a numerical candidate alone does not complete the task. Computer-assisted proofs must include actual results and the code, inputs and proof materials needed for rigorous verification. Numerical approximations and checks of finitely many sample points cannot establish coverage of the entire disk.
</details>
### Main stages
T+00:00:00 is the start of the main task. The intervals are natural elapsed time, including waits and parallel work.
| Relative natural time | Interval length (approx.) | Main work and status at the endpoint |
|---|---|---|
| T+00:00:00–T+00:54:13 | 0 h 54 min 13 s | Background research, candidate reconstruction and independent lower bounds; find an existing complete optimality proof package online. A candidate and coarse bounds were available but did not match; an external complete proof package was found at the end. Its global proof was not independently discovered by the model. |
| T+00:54:13–T+02:03:57 | 1 h 9 min 45 s | Acquire, review and replay the existing external full-disk coverage and global optimality proof. Reuse the DiskCoveringSolve proof, inputs and verification code; two full replays passed. Algebraic work ran alongside replay, but an explicit univariate polynomial was still missing. |
| T+02:03:57–T+03:05:27 | 1 h 1 min 30 s | Geometric reduction and elimination attempts. Geometric reduction produced new equations; several elimination and integer-relation searches failed. |
| T+03:05:27–T+03:20:24 | 0 h 14 min 57 s | Remove degenerate branches and obtain finite-field elimination data. Removing degenerate branches enabled finite-field elimination and established the recovery scale. |
| T+03:20:24–T+04:49:12 | 1 h 28 min 48 s | Parametrization, a unique-root box and rational coefficient recovery attempts. Root-box and reconstruction arguments progressed; failed automatic recovery led to a prime-by-prime route. |
| T+04:49:12–T+05:05:51 | 0 h 16 min 39 s | p-adic lifting, rational reconstruction, root isolation and exact substitution. Successive lifting recovered the coefficients; exact substitution and the unique-root connection passed. |
| T+05:05:51–T+05:34:29 | 0 h 28 min 38 s | Integrate independent verifiers, complete the arguments and package delivery. Independent checks, branch selection and complete delivery were finished. |
These intervals were reconstructed from timestamped progress nodes. Durations are rounded to seconds and marked approximate. They are not exclusive CPU-time categories; program runtimes inside them must not be added again.
### Minimal-polynomial follow-up
The additional instruction was: “Simplify the polynomial to the minimal polynomial of $alpha$.” The [exact original follow-up input](reports/n11-astra-followup-prompt.original.txt) is retained separately.
The follow-up runs from F+00:00:00 to F+00:13:46.178, measured from its own request. This entire interval is included in the displayed total. Work included rational factorization, selecting the target-root factor, an irreducibility certificate, a standalone Python verifier and delivery. The initial screening/factorization script took about 0.661 s internally; the independent verifier took about 18.005 s. Those program runtimes do not replace the 13 min 46.178 s follow-up duration.
### Verification, delivery and limits
Historical results record successful execution of 28 global tasks, exact algebraic connections and the subsequent minimal-polynomial verification. These historical executions are distinct from this website update. The global certificate came from existing external work; this test added replay, geometric/algebraic connections, explicit data recovery and the minimal-polynomial follow-up.
The full proof and follow-up packages remain local. The website publishes an answer-withheld report, original inputs and a machine-readable timeline. This update does not repeat the full historical proof replay or treat a status field as formal acceptance of the current problem.
Source: `cover11-work-report.md`, original-report SHA256: `d2aa68c72384dc0ffb9732e076e0446f68406d0d093c3f17b1aa9a8a78ec1592`. The original contains answers and is not published verbatim.
[Back to the top](#)
## Downloadable records
[Machine-readable record for this test](record-n11-astra-20260930.json) · [Report Markdown](record-n11-astra-20260930.en.md)