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CS-647-A1 · revision r1 · on CS-647

gpt 6 astra ultra attempt on CS-647

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. Symmetry-Aware Rank-One Convex Interpolation Algorithms
  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. What symmetry averaging preserves, and what polar reduction loses
  13. Verification and remaining gaps
  14. References and source attribution
  15. Integration review.
  16. Completion Estimate
  17. Final status and attribution

Symmetry-Aware Rank-One Convex Interpolation Algorithms

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.

An energy $W$ on deformation-gradient matrices is rank-one convex if $t\mapsto W(F+t ab^T)$ is convex wherever defined. Frame indifference means $W(RF)=W(F)$ for permitted rotations $R$.

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 interpolation constraints and what suitable rank-one-convexity means are not fully specified in the supplied question.

Source-record limitation: The required interpolation property is only called suitable and its input/accuracy/efficiency target is not specified.

Statement recorded in the supplied source.
Develop an efficient algorithm that exploits frame indifference or material symmetry to compute an interpolation with suitable rank-one-convexity properties from values of an energy density on a mesh of deformation gradients. [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. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Efficiently interpolate mesh energy values while preserving the stated material symmetries and rank-one-convexity properties.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30000359 (OWR-1113-003). 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 Mini-Workshop: Analysis and Computation of Microstructures in Finite Plasticity $2006$, Oberwolfach Reports, DOI 10.4171/owr/2005/52, pp. 2979--3018 https://doi.org/10.4171/owr/2005/52.
  • S3 later finite-plasticity algorithm survey https://ems.press/content/serial-article-files/46025?nt=1.
  • Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
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 30000359 and code OWR-1113-003 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 full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

What symmetry averaging preserves, and what polar reduction loses

The source asks for interpolation respecting rank-one convexity and frame indifference. A function $W$ on real matrices is rank-one convex if its restriction to every affine line with rank-one direction is convex. We test two possible routes: average an already valid interpolant over a finite symmetry group, or interpolate directly in polar coordinates.
Proposition: Let a finite group $\Gamma$ act linearly by maps $T(F)=QFR$, with $Q,R$ invertible. If $W$ is rank-one convex, then $\overline W(F)=|\Gamma|^{-1}\sum_{T\in\Gamma}W(T(F))$ is rank-one convex and invariant under $\Gamma$. If a sample set is closed under $\Gamma$, its specified values are constant on each orbit, and $W$ interpolates them, then so does $\overline W$.
Proof: If $H$ has rank one, $QHR$ also has rank one because multiplication by invertible matrices preserves rank. For any $F,H$ with this property, each summand $t\mapsto W(T(F)+tT(H))$ is convex; their average is convex. For $T_0\in\Gamma$, composition $T\mapsto T\circ T_0$ permutes the group, proving $\overline W(T_0(F))=\overline W(F)$. At a sample point $F$, every $T(F)$ is another sample with the same specified value. Each summand therefore equals that value, proving interpolation. ∎
This route presupposes existence and construction of $W$; averaging does not produce one from inconsistent or otherwise difficult data. Finite-group invariance also does not by itself imply invariance under all rotations. For the second route consider the rank-one affine line \[ F(t)=\begin{pmatrix}1&t\\0&1\end{pmatrix},\qquad U(t)=(F(t)^TF(t))^{1/2}. \] Its polar stretch values satisfy \[ U(0)=I,\qquad U(\pm1)=\frac1{\sqrt5}\begin{pmatrix}2&\pm1\\\pm1&3\end{pmatrix}. \] Squaring the last matrices gives $F(\pm1)^TF(\pm1)$, and they are positive definite because their leading entry and determinant are positive. Hence they are the positive square roots. But their midpoint is $\operatorname{diag}(2,3)/\sqrt5\ne I$. The polar map therefore does not carry this rank-one line to the corresponding straight line in stretch space.

Verification and remaining gaps

The two matrix squares, determinants and midpoint inequality were checked exactly up to the displayed symbolic square root. The attempted inference that ordinary straight-line interpolation of stretch matrices automatically checks rank-one convexity fails at this geometric step. This does not prove that all methods using polar coordinates fail; they could track the resulting curved paths or impose stronger sufficient conditions. No existence theorem or algorithm for a general frame-indifferent rank-one-convex interpolant is obtained, and the report's stated approximation results are not claimed as new.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2005/52. Relevant passages read: 2005/52 pp. 3007--3009, rank-one-convex envelopes, interpolation and frame indifference.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

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 batch016/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.