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CS-610-A1 · revision r1 · on CS-610

gpt 6 astra ultra attempt on CS-610

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Certified degree windows for truncated algebra commands
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Record scope and qualifications.
  5. Statement recorded in the supplied source.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Research attempt
  11. Literature, source context and scope
  12. Testing same-window truncation of a kernel computation
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Certified degree windows for truncated algebra commands

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

For an $R$-linear map $f:M\to N$, $\ker f=\{m\in M:f(m)=0\}$. The module $\operatorname{Hom}_R(M,N)$ consists of $R$-linear maps. A graded calculation can organize generators and relations by degree. A degree limit requests bounded-degree work or output according to the semantics of the particular method; it need not constitute a complete calculation of the unbounded module.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The desired stopping semantics, supported gradings, and completeness guarantees are not given. The statement about current options is historical and not independently verified for current Macaulay2.

Catalogue screening limitation: API and documentation requests for degree limits, without identifying one mathematical computation and its semantics.

The source snapshot labels this record partially solved; this is catalogue metadata rather than independent verification of a mathematical result or the current completion of a project.

Statement recorded in the supplied source.
Currently, "kernel", "Hom", and several other commands have no options for "DegreeLimit".

What commands should we add "DegreeLimit" to, and how can we better document "DegreeLimit" and its expected behavior in these functions? [S2]

Source status.
The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

Determine where to provide and document degree-limited kernel, Hom, and related calculations.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20002186 (AIM-INFRASTRUCTURE-0086). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 AIM Workshop Problem Lists, source linked by catalogue record AIM-INFRASTRUCTURE-0086. http://aimpl.org/macaulay2efie/1/.
Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20002186 and code AIM-INFRASTRUCTURE-0086 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The complete catalogue body above retains the original agenda, source status, and historical preparation. The AIMPL problem page could not be retrieved. The supplied complete source transcription was read, and the accessible primary AIM workshop announcement was checked for context. The announcement is not a substitute for verification of the inaccessible problem page. The catalogue's earlier proposed frameworks and research summaries are context, not established solutions credited to this attempt. The following is an explicitly limited investigation of one route; it neither substitutes a formalized toy model for the entire source agenda nor claims empirical or deployed-software results. Elementary auxiliary facts are not new original-target progress.

Testing same-window truncation of a kernel computation

The historical source says several commands lack DegreeLimit options. Current official documentation lists DegreeLimit for Hom and for a kernel entry with its own stated semantics. Thus the old sentence is preserved as history, not asserted as a verified present-day absence. We test a mathematical truncation hazard independently of those implementations. Let $V_D$ be the polynomials of degree at most $D$ in $K[x]$ over a field, with $D\ge0$. Multiplication by $x$ maps $V_D$ to $V_{D+1}$. A naive same-window computation instead composes it with the projection $\pi_D:V_{D+1}\to V_D$ that discards the coefficient of $x^{D+1}$.
Proposition: Multiplication by $x$ on $V_D$ has zero kernel, but $\pi_D\circ(x\cdot)$ has kernel $Kx^D$. More generally, if a homogeneous linear map has degree $r\ge0$, retaining all output degrees from zero through $D+r$ suffices to preserve its restriction to inputs of degrees at most $D$ without truncation loss.
Proof: Write $f=\sum_{i=0}^D a_ix^i$. Then $xf=\sum_{i=0}^D a_ix^{i+1}$, zero exactly when every $a_i=0$, proving injectivity. Its projection is $\sum_{i=0}^{D-1}a_ix^{i+1}$, zero exactly when $a_0,\ldots,a_{D-1}$ vanish, while $a_D$ is arbitrary. This gives the one-dimensional kernel, including $D=0$, when the projected map is zero on constants. For the general claim, a homogeneous degree-$r$ map sends an input component of degree $d$ into output degree $d+r$. For $0\le d\le D$ these outputs are all at most $D+r$; retaining that full range keeps every component and hence the restricted linear map and its kernel unchanged. This assumes nonnegative input grading and no omitted lower degrees. ∎
For $D=2$, the correct matrix in monomial bases has three columns $x,x^2,x^3$ in a four-dimensional target and rank 3. Removing its last row produces rank 2 and falsely adds $x^2$ to the kernel. The failed inference is that truncating the target at the same displayed degree as the source necessarily preserves a bounded-degree kernel.

Verification and remaining gaps

Both matrices were row-reduced over rational numbers, obtaining ranks 3 and 2. No behavior of a current M2 command was inferred from this example, and no command options or documentation were modified. General shifted modules, multigradings, partially computed Gröbner bases and the exact meaning of each command's DegreeLimit remain to be specified and implemented.

References and source attribution

Primary source: http://aimpl.org/macaulay2efie/1/. Attempted primary AIMPL page retrieval; failed with timeout or gateway error. The supplied full source transcription was read; the inaccessible live page was not independently inspected.

https://aimath.org/pastworkshops/macaulay2efie.html. Read the primary September 2023 workshop announcement.

https://aimath.org/pastworkshops/macaulay2efierep.pdf. Read the complete three-page primary workshop report. Existing threads/tasks, MPI work and parallelApply are acknowledged; no absence of all existing parallelism is inferred.

https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/___Degree__Limit.html. Read current official DegreeLimit option index. It lists Hom, gb, syz and a kernel entry; the latter describes a subring-related stopping behavior. This listing is not treated as proof that all kernel overloads have identical semantics.

All retrievals were checked on 11 October 2026. Original contributor and catalogue attribution remains in the preserved Sources section. The elementary arguments below are self-contained; their exposition is not a claim of first discovery.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch013/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.