Unit Disk Covering Benchmark · n = 12
n=12 · deepseek-v4-pro/max test report
# n=12 · deepseek-v4-pro/max test report
[Back to n=12](n12.html) · [Test record directory](records.html)
**Score: 0. Acceptance: failed; the submitted optimal radius is incorrect.** The test took place on October 3, 2026 (UTC), corresponding to October 2 in New York. The record ID uses the New York date. Submission was within the 12-hour limit. An independently certified twelve-disk cover has a strictly smaller radius, so the submitted root cannot be the required optimum.
Codex prepared this report from the original DeepSeek Harness session and deliverables, using assets from the designated mathematical research project. DeepSeek was not asked to write it. This public text withholds radii, polynomial coefficients, centers and counterexample coordinates. The private local assessment ZIP preserves the original answers and exact evidence. This is one run.
### Task and delivered work
The actual task demanded the optimal radius's primitive, positive-leading, irreducible integer minimal polynomial and rational isolating interval; a uniquely specified ordered algebraic configuration; rigorous coverage of the entire closed disk; a global lower bound for every twelve-center configuration; and the code, inputs, certificates and verification results actually used.
| Requirement | Finding |
|---|---|
| Minimal polynomial and isolation | The submitted cubic's normalization, irreducibility and root isolation arguments are correct. Its root is strictly greater than an exactly certified feasible radius, so it is not the optimal radius's minimal polynomial. |
| Explicit configuration | The isolated root and fixed rotation order uniquely specify the twelve centers. This does not establish optimality. |
| Continuous coverage proof | Exact contact identities are supplied, but several classification steps rely on high-precision floating point rather than certified enclosures. The relaxed-radius program output does not complete the claimed exact proof. This finding does not assert that the proposed configuration has a coverage hole. |
| Global optimality | Missing, and the claimed global lower bound at the submitted root is directly refuted by the better cover. |
| Computational materials | Actual files and outputs exist and have been preserved. They do not imply all required conclusions. |
The adopted scoring rule assigns zero to an incorrect submission. An on-time answer, a completed Goal state, or a valid polynomial for a candidate does not qualify for successful-time scoring.
### Timing
T+00:00:00 is the first `goal/change: create`, seq10. The submission endpoint is the last formal `assistant/message`, seq1193. All stages use natural elapsed time.
| Measure | Saved value |
|---|---|
| Start to final submission | **2 h 13 min 36 s (8015.612 s)** |
| Start to final turn end | 8015.654 s, about 0.04 s later |
| Sum of four turn intervals | 1 h 20 min 49 s (4849.269 s); includes tool waiting |
| Difference from natural time | About 52 min 46 s, including a transport-recovery gap and a blocked-to-user-response gap |
| System / Goal cumulative task time | Unavailable; turn intervals are not substituted |
| Goal blocked | T+01:32:26, with global optimality explicitly unfinished |
| Goal complete | T+02:13:27; a state change following the user's closing instruction, not mathematical acceptance |
| Limit | 12 hours; submission was within it, but a complete correct result was not obtained |
| Archiving and independent review | Separate follow-up work; workflow, snapshot and packaging timestamps are retained separately |
The first turn ended with a `TRANSPORT` error around T+00:53:51 and resumed around T+01:14:59. Nine retry events are saved. After the third turn, the blocked Goal waited until the user's instruction around T+02:04:57. These events do not quantify pure reasoning or provider computation. The last turn's approximately 559 seconds is not the whole task time.
### Computing conditions
| Item | Evidence |
|---|---|
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics |
| Memory | DDR5, 32 GB |
| Hardware scope | Windows CIM query during report preparation; not a test-start snapshot |
| Execution | Local Windows and local WSL; no observed remote core-computation record |
| Model | `deepseek-v4-pro`, `reasoningEffort=max`, `provider=deepseek-official` |
| Model identity | Matching original request configurations at seq22, 983 and 1108; no observed solver-model switch |
| Harness | DeepSeek Harness, `agentPreset=standard`; session title matches the supplied screenshot |
Around T+00:02:34 the original environment output confirmed WSL Python **3.12.3**, SymPy **1.14.0**, mpmath **1.3.0**, NumPy **2.5.3**, and GCC **13.3.0**. Symbolic identities, numerical geometry, high-precision solving and integer-relation search used these tools. SciPy/HiGHS supported optimization and LP; the original deliverable reports SciPy 1.18.1, without a separately verified preinstallation date. pypdf installation was attempted during the run. Image-library versions and installation dates were not fully recorded. The review's Python 3.13.13 is separate from the original solving environment.
### Tokens and cost
| Field | Sum of 161 saved assistant-message usage records |
|---|---:|
| inputTokens | 163,009 |
| cacheReadTokens | 36,592,640 |
| outputTokens | 382,409 |
| cacheWriteTokens | 0 |
| totalTokens | **37,138,058** |
| Separate Goal token counter | Unavailable |
| Invoiced monetary cost | Unavailable; not estimated |
Here inputTokens excludes cache reads; input, cache-read and output fields sum to totalTokens. Repeated context reads are accumulated input, not thirty-seven million newly generated tokens. The screenshot's rounded 37.1M agrees with these saved values; the recorded cache-input fraction is about 99.56%. Billing for title generation, searches or failed requests is not fully known. No new DeepSeek request was made for this review.
### Actual input
[Download the saved Goal text](reports/n12-deepseek-main-prompt.original.txt). Its original UTF-8 bytes and hash are preserved. The displayed version only changes reading layout.
The resource restrictions prohibit pre-existing local material, skills and chats, and permit the web and local WSL tools. Newly downloaded/generated material is distinguished from prohibited pre-existing material. No complete operating-system file-access audit was performed; mathematical refutation already determines the zero score.
<details>
<summary>Expand the actual original prompt</summary>
“Optimal Covering of the Unit Disk with 12 Disks
Let
$$
D=\{x\in\mathbb R^2:\|x\|_2\le1\},\qquad
R_D(C)=\max_{x\in D}\min_{1\le i\le12}\|x-c_i\|_2,
$$
where $C=(c_1,\ldots,c_{12})\in(\mathbb R^2)^{12}$, and define
$$
r_D(12)=\inf_{C\in(\mathbb R^2)^{12}}R_D(C).
$$
The covering disks are closed. They may overlap and extend beyond $D$; their centers are not initially restricted to lie in $D$.
Determine $r_D(12)$ 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 $r_D(12)$ 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 $r_D(12)=\alpha$.
2. **Explicit configuration:** Give a definite ordered tuple of centers $C_*=(c_1^*,\ldots,c_{12}^*)$, 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
$$
\forall x\in D,\quad\exists i\in\{1,\ldots,12\},\quad
\|x-c_i^*\|_2\le\alpha.
$$
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
$$
\forall C\in(\mathbb R^2)^{12},\quad R_D(C)\ge\alpha,
$$
and hence $R_D(C_*)=r_D(12)=\alpha$. 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.”限制：不允许使用任何本地资料、技能、chat、archived chat等资料、可以使用网络、本地的WSL当中的工具（但是绝对不能访问其中任何已经存储的资料或文件），一旦违反任何一条规则即视为任务失败。任务限时12小时，超时没有得到完整且正确的结果也视为失败。完成时间越短越好。
</details>
### Recorded stages
| Natural elapsed time | Work | Result at that stage |
|---|---|---|
| T+00:00–00:12:23 | References, environment checks, papers, image/SVG reconstruction | Acquired a known rotational construction; no complete global proof. |
| T+00:12:23–00:32:41 | Contact equations, Jacobian correction, Newton and integer relations | Obtained a cubic candidate and algebraic coordinates, with symbolic contact checks. |
| T+00:32:41–00:48:18 | Triple, boundary and antipode checks; LP; C branch-and-bound | Contact equalities were exact; classification retained floating-point steps; C used a relaxed radius. |
| T+00:48:18–01:23:18 | LP output, transport failure/recovery, further searching and first delivery | Interpreted a finite-grid LP as a continuous obstruction; acknowledged the missing global proof. |
| T+01:23:18–01:32:38 | Continued checks and closing report | Marked Goal blocked, then waited for user input. |
| T+02:04:57–02:13:36 | User continuation/closure instructions and final revision | Submitted on belief in the candidate's optimality; requirement 4 remained unproved. |
### Sources and contribution
The run consulted [Friedman's catalog](https://erich-friedman.github.io/packing/circovcir/), [Fejes Tóth's smaller-case paper](https://library.slmath.org/books/Book52/files/18fejes.pdf), and [Dounreay configuration material](https://gitlab.com/parclytaxel/Dounreay). The known external construction was not independently discovered by the model. Reconstruction, equations, numerical solving, candidate relations, symbolic identities and programs were produced during this run.
A catalog label, unsuccessful literature search, long-standing record or tight contacts do not establish global optimality or impossibility. Claims of first discovery and exhaustive literature status are not certified here. Research-project assets read afterward belong to this assessment, not to the original solver's achievements.
### Verification and rejection
**Decisive counterexample.** Twelve concrete rational centers were selected from a saved research asset, and a rational radius $\beta$ was chosen. An inspected exact consumer independently reconstructed every required critical candidate of the clipped Voronoi partition: interior triple circumcenters, all pair-bisector/unit-circle intersections, smooth-boundary maxima and relevant degeneracies. Python `Fraction` and guarded quadratic-surd comparisons checked 293 distinct candidates with 2,071 exact distance comparisons, returning `ACCEPT`. This is a complete continuous-cover reduction, not point sampling.
Exact arithmetic also gives $P(\beta)<0$. The submission correctly establishes that $P$ is strictly increasing and $P(\alpha)=0$. Therefore
$$r_D(12)\le\beta<\alpha.$$
This refutes the submitted optimum and its required universal lower bound. It does not rely on the research project's global classification or optimality claims, and does not certify the true optimum.
**Additional proof defects.** In §5.2, $m\,2\arcsin r\ge2\pi$ implies $\arcsin r\ge\pi/m$. For $m\ge9$ it does not imply the claimed stronger nine-arc bound. A proof that at most nine disks meet the boundary is missing. Rejection of this argument does not refute the bound's numerical value itself.
In §5.3, finite demand points and a finite set of allowed centers do not automatically give a fractional cover of the entire disk. A finite-grid objective below twelve does not establish the claimed continuous integrality gap or invalidate every LP/dual method.
In §4.2, high precision, tiny residuals and numerical margins are not certified enclosures. The C check in §4.3 uses a larger radius, so cannot prove the exact-radius statement. Moreover, uniformly rounding negative differences upward before taking absolute values can underestimate a claimed distance upper bound in `sup_sq`; decimal center endpoints also lack a complete exact enclosure argument. Its printed certificate is not accepted as a rigorous proof here.
**Review scope.** The original submission, directly supporting programs, model settings and usage were inspected; the better-cover certificate was actually checked. Original optimization, Newton, PSLQ, LP and C runs were not fully replayed. Original bytes, the rational counterexample, exact consumer, results and provenance are preserved privately. Prior research self-review is not promoted to independent peer review or an unconditional global theorem.
### Record files
[Machine record](record-n12-deepseek-v4-pro-max-20261002.json) · [English Markdown](record-n12-deepseek-v4-pro-max-20261002.en.md) · [中文](record-n12-deepseek-v4-pro-max-20261002.zh.html). The complete assessment ZIP contains private conversations, answers and counterexample data and remains local.
# n=12 · deepseek-v4-pro/max test report
[Back to n=12](n12.html) · [Test record directory](records.html)
**Score: 0. Acceptance: failed; the submitted optimal radius is incorrect.** The test took place on October 3, 2026 (UTC), corresponding to October 2 in New York. The record ID uses the New York date. Submission was within the 12-hour limit. An independently certified twelve-disk cover has a strictly smaller radius, so the submitted root cannot be the required optimum.
Codex prepared this report from the original DeepSeek Harness session and deliverables, using assets from the designated mathematical research project. DeepSeek was not asked to write it. This public text withholds radii, polynomial coefficients, centers and counterexample coordinates. The private local assessment ZIP preserves the original answers and exact evidence. This is one run.
### Task and delivered work
The actual task demanded the optimal radius's primitive, positive-leading, irreducible integer minimal polynomial and rational isolating interval; a uniquely specified ordered algebraic configuration; rigorous coverage of the entire closed disk; a global lower bound for every twelve-center configuration; and the code, inputs, certificates and verification results actually used.
| Requirement | Finding |
|---|---|
| Minimal polynomial and isolation | The submitted cubic's normalization, irreducibility and root isolation arguments are correct. Its root is strictly greater than an exactly certified feasible radius, so it is not the optimal radius's minimal polynomial. |
| Explicit configuration | The isolated root and fixed rotation order uniquely specify the twelve centers. This does not establish optimality. |
| Continuous coverage proof | Exact contact identities are supplied, but several classification steps rely on high-precision floating point rather than certified enclosures. The relaxed-radius program output does not complete the claimed exact proof. This finding does not assert that the proposed configuration has a coverage hole. |
| Global optimality | Missing, and the claimed global lower bound at the submitted root is directly refuted by the better cover. |
| Computational materials | Actual files and outputs exist and have been preserved. They do not imply all required conclusions. |
The adopted scoring rule assigns zero to an incorrect submission. An on-time answer, a completed Goal state, or a valid polynomial for a candidate does not qualify for successful-time scoring.
### Timing
T+00:00:00 is the first `goal/change: create`, seq10. The submission endpoint is the last formal `assistant/message`, seq1193. All stages use natural elapsed time.
| Measure | Saved value |
|---|---|
| Start to final submission | **2 h 13 min 36 s (8015.612 s)** |
| Start to final turn end | 8015.654 s, about 0.04 s later |
| Sum of four turn intervals | 1 h 20 min 49 s (4849.269 s); includes tool waiting |
| Difference from natural time | About 52 min 46 s, including a transport-recovery gap and a blocked-to-user-response gap |
| System / Goal cumulative task time | Unavailable; turn intervals are not substituted |
| Goal blocked | T+01:32:26, with global optimality explicitly unfinished |
| Goal complete | T+02:13:27; a state change following the user's closing instruction, not mathematical acceptance |
| Limit | 12 hours; submission was within it, but a complete correct result was not obtained |
| Archiving and independent review | Separate follow-up work; workflow, snapshot and packaging timestamps are retained separately |
The first turn ended with a `TRANSPORT` error around T+00:53:51 and resumed around T+01:14:59. Nine retry events are saved. After the third turn, the blocked Goal waited until the user's instruction around T+02:04:57. These events do not quantify pure reasoning or provider computation. The last turn's approximately 559 seconds is not the whole task time.
### Computing conditions
| Item | Evidence |
|---|---|
| CPU | AMD Ryzen 9 8945HX with Radeon Graphics |
| Memory | DDR5, 32 GB |
| Hardware scope | Windows CIM query during report preparation; not a test-start snapshot |
| Execution | Local Windows and local WSL; no observed remote core-computation record |
| Model | `deepseek-v4-pro`, `reasoningEffort=max`, `provider=deepseek-official` |
| Model identity | Matching original request configurations at seq22, 983 and 1108; no observed solver-model switch |
| Harness | DeepSeek Harness, `agentPreset=standard`; session title matches the supplied screenshot |
Around T+00:02:34 the original environment output confirmed WSL Python **3.12.3**, SymPy **1.14.0**, mpmath **1.3.0**, NumPy **2.5.3**, and GCC **13.3.0**. Symbolic identities, numerical geometry, high-precision solving and integer-relation search used these tools. SciPy/HiGHS supported optimization and LP; the original deliverable reports SciPy 1.18.1, without a separately verified preinstallation date. pypdf installation was attempted during the run. Image-library versions and installation dates were not fully recorded. The review's Python 3.13.13 is separate from the original solving environment.
### Tokens and cost
| Field | Sum of 161 saved assistant-message usage records |
|---|---:|
| inputTokens | 163,009 |
| cacheReadTokens | 36,592,640 |
| outputTokens | 382,409 |
| cacheWriteTokens | 0 |
| totalTokens | **37,138,058** |
| Separate Goal token counter | Unavailable |
| Invoiced monetary cost | Unavailable; not estimated |
Here inputTokens excludes cache reads; input, cache-read and output fields sum to totalTokens. Repeated context reads are accumulated input, not thirty-seven million newly generated tokens. The screenshot's rounded 37.1M agrees with these saved values; the recorded cache-input fraction is about 99.56%. Billing for title generation, searches or failed requests is not fully known. No new DeepSeek request was made for this review.
### Actual input
[Download the saved Goal text](reports/n12-deepseek-main-prompt.original.txt). Its original UTF-8 bytes and hash are preserved. The displayed version only changes reading layout.
The resource restrictions prohibit pre-existing local material, skills and chats, and permit the web and local WSL tools. Newly downloaded/generated material is distinguished from prohibited pre-existing material. No complete operating-system file-access audit was performed; mathematical refutation already determines the zero score.
<details>
<summary>Expand the actual original prompt</summary>
“Optimal Covering of the Unit Disk with 12 Disks
Let
$$
D=\{x\in\mathbb R^2:\|x\|_2\le1\},\qquad
R_D(C)=\max_{x\in D}\min_{1\le i\le12}\|x-c_i\|_2,
$$
where $C=(c_1,\ldots,c_{12})\in(\mathbb R^2)^{12}$, and define
$$
r_D(12)=\inf_{C\in(\mathbb R^2)^{12}}R_D(C).
$$
The covering disks are closed. They may overlap and extend beyond $D$; their centers are not initially restricted to lie in $D$.
Determine $r_D(12)$ 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 $r_D(12)$ 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 $r_D(12)=\alpha$.
2. **Explicit configuration:** Give a definite ordered tuple of centers $C_*=(c_1^*,\ldots,c_{12}^*)$, 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
$$
\forall x\in D,\quad\exists i\in\{1,\ldots,12\},\quad
\|x-c_i^*\|_2\le\alpha.
$$
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
$$
\forall C\in(\mathbb R^2)^{12},\quad R_D(C)\ge\alpha,
$$
and hence $R_D(C_*)=r_D(12)=\alpha$. 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.”限制：不允许使用任何本地资料、技能、chat、archived chat等资料、可以使用网络、本地的WSL当中的工具（但是绝对不能访问其中任何已经存储的资料或文件），一旦违反任何一条规则即视为任务失败。任务限时12小时，超时没有得到完整且正确的结果也视为失败。完成时间越短越好。
</details>
### Recorded stages
| Natural elapsed time | Work | Result at that stage |
|---|---|---|
| T+00:00–00:12:23 | References, environment checks, papers, image/SVG reconstruction | Acquired a known rotational construction; no complete global proof. |
| T+00:12:23–00:32:41 | Contact equations, Jacobian correction, Newton and integer relations | Obtained a cubic candidate and algebraic coordinates, with symbolic contact checks. |
| T+00:32:41–00:48:18 | Triple, boundary and antipode checks; LP; C branch-and-bound | Contact equalities were exact; classification retained floating-point steps; C used a relaxed radius. |
| T+00:48:18–01:23:18 | LP output, transport failure/recovery, further searching and first delivery | Interpreted a finite-grid LP as a continuous obstruction; acknowledged the missing global proof. |
| T+01:23:18–01:32:38 | Continued checks and closing report | Marked Goal blocked, then waited for user input. |
| T+02:04:57–02:13:36 | User continuation/closure instructions and final revision | Submitted on belief in the candidate's optimality; requirement 4 remained unproved. |
### Sources and contribution
The run consulted [Friedman's catalog](https://erich-friedman.github.io/packing/circovcir/), [Fejes Tóth's smaller-case paper](https://library.slmath.org/books/Book52/files/18fejes.pdf), and [Dounreay configuration material](https://gitlab.com/parclytaxel/Dounreay). The known external construction was not independently discovered by the model. Reconstruction, equations, numerical solving, candidate relations, symbolic identities and programs were produced during this run.
A catalog label, unsuccessful literature search, long-standing record or tight contacts do not establish global optimality or impossibility. Claims of first discovery and exhaustive literature status are not certified here. Research-project assets read afterward belong to this assessment, not to the original solver's achievements.
### Verification and rejection
**Decisive counterexample.** Twelve concrete rational centers were selected from a saved research asset, and a rational radius $\beta$ was chosen. An inspected exact consumer independently reconstructed every required critical candidate of the clipped Voronoi partition: interior triple circumcenters, all pair-bisector/unit-circle intersections, smooth-boundary maxima and relevant degeneracies. Python `Fraction` and guarded quadratic-surd comparisons checked 293 distinct candidates with 2,071 exact distance comparisons, returning `ACCEPT`. This is a complete continuous-cover reduction, not point sampling.
Exact arithmetic also gives $P(\beta)<0$. The submission correctly establishes that $P$ is strictly increasing and $P(\alpha)=0$. Therefore
$$r_D(12)\le\beta<\alpha.$$
This refutes the submitted optimum and its required universal lower bound. It does not rely on the research project's global classification or optimality claims, and does not certify the true optimum.
**Additional proof defects.** In §5.2, $m\,2\arcsin r\ge2\pi$ implies $\arcsin r\ge\pi/m$. For $m\ge9$ it does not imply the claimed stronger nine-arc bound. A proof that at most nine disks meet the boundary is missing. Rejection of this argument does not refute the bound's numerical value itself.
In §5.3, finite demand points and a finite set of allowed centers do not automatically give a fractional cover of the entire disk. A finite-grid objective below twelve does not establish the claimed continuous integrality gap or invalidate every LP/dual method.
In §4.2, high precision, tiny residuals and numerical margins are not certified enclosures. The C check in §4.3 uses a larger radius, so cannot prove the exact-radius statement. Moreover, uniformly rounding negative differences upward before taking absolute values can underestimate a claimed distance upper bound in `sup_sq`; decimal center endpoints also lack a complete exact enclosure argument. Its printed certificate is not accepted as a rigorous proof here.
**Review scope.** The original submission, directly supporting programs, model settings and usage were inspected; the better-cover certificate was actually checked. Original optimization, Newton, PSLQ, LP and C runs were not fully replayed. Original bytes, the rational counterexample, exact consumer, results and provenance are preserved privately. Prior research self-review is not promoted to independent peer review or an unconditional global theorem.
### Record files
[Machine record](record-n12-deepseek-v4-pro-max-20261002.json) · [English Markdown](record-n12-deepseek-v4-pro-max-20261002.en.md) · [中文](record-n12-deepseek-v4-pro-max-20261002.zh.html). The complete assessment ZIP contains private conversations, answers and counterexample data and remains local.