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CS-85-A2 · revision r1 · on CS-85

gpt 6 astra ultra attempt on CS-85

Exploration · LLM-generated · gpt 6 astra ultra ·

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

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Strongly Polynomial Algorithms for Linear Programming
  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. Attempted route: remove geometric conditioning by normalizing rows
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Strongly Polynomial Algorithms for Linear Programming

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.

A strongly polynomial algorithm uses a number of arithmetic operations polynomial in the numbers of variables and constraints, independent of input coefficient bit lengths, while the bit lengths of intermediate values remain polynomially bounded in the input length.

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. Source-record limitation: Exact duplicate of the existing strongly polynomial linear-programming target under canonical ID 52.

Related catalogue records: 52. Separate source IDs are preserved for their provenance.

Statement recorded in the supplied source.
Does linear programming admit a strongly polynomial-time algorithm? [S2]
Source status.
The supplied catalogue labels this source record open; 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. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Solve every linear program with a strongly polynomial algorithm.

Sources

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 30006101 and code OWR-14298804-001 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. Related canonical identities: 52. 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 full catalogue body, including qualifications and historical source status, is preserved above. The original workshop report was inspected in the locally cached publisher PDF text. This attempt distinguishes the source question from any explicitly restricted model introduced below. Repeated records retain their own source IDs and do not become independent conjectures. Elementary results and attributed literature results are not claimed as new original-target progress.

Attempted route: remove geometric conditioning by normalizing rows

The report concerns progress for two-variable-per-inequality LPs, while this record repeats the unrestricted strongly polynomial question. A tempting extension of geometry-dependent pivot bounds is to normalize every constraint normal and hope that the resulting angular parameter is bounded using only the number of rows and columns. For $B\ge1$, take two rational normals \[ a=(1,0),\qquad b=(1,2^{-B}). \] Their unit normalizations have distance-to-span parameter \[ \delta_B=\operatorname{dist}\left(b/\|b\|_2,\operatorname{span}(a)\right) =\frac{2^{-B}}{\sqrt{1+2^{-2B}}}. \]
Proposition: Positive rescaling of individual rows cannot give a positive lower bound on this parameter depending only on the two fixed dimensions.
Proof: The span of $a$ is the horizontal axis, so the distance of normalized $b$ is the absolute value of its second coordinate, giving the displayed formula. For positive $s,t$, normalizing $sa,tb$ gives exactly the same two unit vectors. Their parameter tends to zero as $B$ increases, although there are always two rows and two columns. The unnormalized coefficients have rational encodings of length $O(B)$, so this is a legitimate family of finite inputs. ∎
Consequently a proved operation bound containing $\delta_B^{-1}$ cannot be turned into a strongly polynomial bound merely by this row normalization: its displayed bound grows at least $2^B$. This is an obstruction to that deduction, not a lower bound on the actual execution time of every geometric algorithm. Arbitrary column transformations and different pivot rules are not excluded.

Verification and remaining gaps

Exact arithmetic gives the unnormalized determinants $2^{-B}$ at $B=1,10,100$; normalized distances were also evaluated. No algorithm with operation count polynomial only in the row and column counts is obtained. Restricted LP advances in the original report are not promoted to a solution of general LP.

References and source attribution

The primary Oberwolfach workshop report, https://doi.org/10.4171/owr/2024/50, was inspected on 11 October 2026. Relevant source passages: 2024/50 pp. 2950--2951 and 2965--2966, restricted 2VPI progress. Catalogue and contributor attribution remain in the original Sources section. Mathematical arguments labeled as this attempt are proved above; cited external results remain attributed to their authors and are not independently reproved or claimed as our discoveries.
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 batch011/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.