When flattening and symmetry-aware collapse are informative

Research-attempt provenance

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 table flattening regroups a multiway array into a matrix. Category collapsing is a linear summation map. A group action on the table's coordinates can decompose a representation into invariant or irreducible components, providing alternative coordinate descriptions.

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 data model and meaning of more useful are not defined. No universal superiority of a transform or preservation theorem is asserted. The supplied catalogue marks this record partially solved; its present research status was not independently checked. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim.

Statement recorded in the supplied source.

Question 44 (Landsberg). Why is flattening used so frequently in statistics? Might other linear transformations on categorical data be more useful? Why not project onto the irre-ducible representations of the symmetric group? What about other forms of collapsing data that takes into account group actions? [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

The source asks why flattening is useful and whether symmetry-based transformations could be preferable.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001220 (AIM-COMPUTATION-0058). 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-COMPUTATION-0058. https://aimath.org/WWN/compalgstat/compalgstat.pdf.

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 20001220 and code AIM-COMPUTATION-0058 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.

Literature and context check

The original AIM Computational Algebraic Statistics workshop PDF was consulted. The preserved extract does not specify every model, parameterization or performance criterion needed by its broad question. The following declared model is an actual test of one route, not a replacement target or a claim of novelty for elementary statistical or algebraic facts.

Research attempt: Symmetry averaging can lose even symmetry-invariant information

Reshaping a tensor into a matrix is a bijection of entries, so a flattening itself loses no information. Information is lost only when one retains a coarser statistic such as rank or selected minors. For example many different strictly positive two-by-two probability tables have rank two; their common rank does not determine the table or its likelihood.

Consider instead the group generated by transposition, acting on two-by-two tables, and average over that group: $R(P)=(P+P^T)/2$. This is an idempotent linear projection, because applying transposition twice is the identity and a symmetric matrix is fixed. But it is not an orbit identifier. Let \[ P=\begin{pmatrix}3/10&1/10\\3/10&3/10\end{pmatrix},\quad Q=\begin{pmatrix}3/10&3/20\\1/4&3/10\end{pmatrix}. \] Both have positive entries summing to one and both averages equal $\left(\begin{smallmatrix}3/10&1/5\\1/5&3/10\end{smallmatrix}\right)$. Yet $Q$ is neither $P$ nor $P^T$. Moreover the group-invariant quantity $|p_{12}-p_{21}|$ is $1/5$ for $P$ and $1/10$ for $Q$. Thus averaging loses information even relevant to a symmetry-invariant target; the assertion "the target is invariant, so the trivial component suffices" is false.

For this group the missing antisymmetric component is $A(P)=(P-P^T)/2$, and $P=R(P)+A(P)$ reconstructs the table exactly. Keeping both components is just an invertible coordinate decomposition. Discarding $A$ is justified only for a task that depends on $R$ alone, a stronger condition than invariance under transposition. Similar distinctions must be checked before replacing flattening by group-aware collapse in a statistical procedure.

Verification and adversarial audit

Normalization, averages, transposes and absolute antisymmetric differences were checked exactly. The example refutes orbit recovery as well as recovery of a specified invariant. No representation-theoretic completeness theorem is used without proof.

Audit corrections.

Invariant targets were distinguished from dependence solely on the group average; full coordinate transforms were not confused with lossy summaries.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The source selects no group, statistical loss or preserved quantity. The first gap is a task-specific sufficiency argument for the proposed retained components; the existence of a group action does not supply one.

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% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

• https://aimath.org/WWN/compalgstat/compalgstat.pdf (consulted 2026-10-11).

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.