gpt 6 astra ultra attempt on CS-792

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 representation maps observations into features. Its quality may concern information preservation, prediction loss, invariance, or computational utility, each requiring a specified task.

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 supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Good representation is not given a mathematical criterion in the extracted text.

Source-record limitation: Source flags the statement as unrecoverable: The official report poses this as a philosophical organizing question in an expository discussion, not as a standalone mathematical problem. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.

What makes a good representation? The report passage is retained verbatim as the relevant context for this extracted problem. [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

The fragment asks what makes a representation useful.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006179 (OWR-14299083-005). 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 Overparametrization, Regularization, Identifiability and Uncertainty in Machine Learning $2025$, Oberwolfach Reports, DOI 10.4171/owr/2025/4, pp. 175--208 https://doi.org/10.4171/owr/2025/4.
• S3 ICLR 2024 Representational Alignment workshop https://representational-alignment.github.io/.
• S4 Symmetry-Based Representations https://arxiv.org/abs/2203.09250.
• 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 30006179 and code OWR-14299083-005 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 primary report, printed pp. 199--202, discusses instrumental causal representations and already supplies several possible desiderata. The phrase "good representation" has no unique objective until tasks and allowed interventions are fixed. The formal model below makes those choices explicit and does not replace the broader source agenda.

Research attempt: A coarsest task-preserving finite intervention model

Fix a finite nonempty state set $X$ of size $N$, deterministic interventions $A_1,\ldots,A_k$, and a specified observation/task map $o:X\to Y$. Include the empty word among finite intervention words. Define \[ x\sim y\quad\Longleftrightarrow\quad o(w x)=o(w y) \quad\text{for every finite word }w. \] This is an equivalence relation. It refines equality of $o$ by the empty word. If $x\sim y$, then for each $i$ and each $w$, the word applying $A_i$ first and $w$ next gives $o(w A_i x)=o(w A_i y)$. Hence the relation is forward stable. Conversely, any forward-stable equivalence relation refining equality of $o$ has equal observations after every word, by induction on word length. It is therefore contained in $\sim$. The quotient by $\sim$ is the coarsest quotient that retains this task after every intervention sequence, and induced interventions are well-defined by forward stability.

There is a finite algorithm. Start with partition $P_0$ by equal observations. Given $P_t$, put two states in the same $P_{t+1}$ block exactly when the tuples \[ \left(P_t(x),P_t(A_1x),\ldots,P_t(A_kx)\right) \] agree, where $P_t(x)$ denotes a canonical block identifier. Induction shows $P_t$ agrees exactly on observations of all words of length at most $t$. Each strict refinement increases the number of blocks, which never exceeds $N$, so after at most $N-1$ strict refinements the partition is stable. Stability makes it forward stable and task preserving; the preceding maximality proof identifies it with $\sim$. A naive implementation comparing all state pairs at every round uses at most $O((k+1)N^3)$ elementary comparisons when the maps are given by tables and task equality can be tested.

Dependence on the task.

On $X=\{0,1\}^2$ with independent bit-flip interventions, task $o(x)=x_1$ produces exactly the two blocks of equal first bit: the second bit never changes any future first-bit observation. Task $o(x)=(x_1,x_2)$ gives the discrete partition already at $P_0$. A constant task gives the one-block quotient. Thus a universal task-independent compression criterion cannot be inferred from intervention commutation alone. Changing the task changes the provably optimal quotient even with identical state space and interventions.

Verification and adversarial audit

The refinement proof checks words in the correct composition order and includes the empty word. Stability, not a fixed number of successful rounds, certifies all word lengths. The case $N=1$ stops immediately; $k=0$ gives exactly the task partition. The complexity counts a table model and is not a statistical learning guarantee.

Audit corrections.

Added a specified observation map and intervention tables; the optimality statement is only among quotients preserving those tasks exactly.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The finite criterion solves the explicitly stated auxiliary compression problem, not the original unrestricted notion of a good learned representation. No noisy/stochastic, continuous-state, sample-efficient, or task-selection result is claimed.

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://ems.press/content/serial-article-files/51348 (consulted 2026-10-11).

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking.

Material model collaboration.

Root Codex GPT-6 supplied the coarsest intervention-compatible task quotient, partition-refinement algorithm and its proof of maximality. The gpt-6-astra author inspected and reworked these arguments; this is model-assisted mathematical work, not independent human verification. 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 and repository-format review; 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.