Mathematical Conjecture Generation with Machine Intelligence

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 conjecture is a proposed mathematical assertion. A surrogate is an efficiently computed quantity used to estimate a harder quantity. Formal proof assistants mechanically check derivations in their stated formal foundations.

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 specific bounds, free-group action, and required mathematical guarantees are not supplied; workflow goals remain source-described research directions.

Source-record limitation: The hard quantities, proposed bounds, and free-group action are unspecified; the remaining items are workflow design.

Statement recorded in the supplied source.

Develop machine-guided methods that discover useful bounds between hard matrix quantities and efficiently computable surrogates; characterize invariants of free-group actions on spaces of conjectures; and integrate conjecture-generation pipelines with formal proof assistants and natural-language representations. [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

Use automated methods to generate useful conjectures and connect them to formal checking.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30005554 (OWR-13750338-001). 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 Machine Learning for Science: Mathematics at the Interface of Data-driven and Mechanistic Modelling $2023$, Oberwolfach Reports, DOI 10.4171/owr/2023/26, pp. 1453--1484 https://doi.org/10.4171/owr/2023/26.
• S3 OWR 2023 26 https://ems.press/content/serial-article-files/47019.
• S4 Mishra Moulik Sarkar https://arxiv.org/abs/2306.07277.
• 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 30005554 and code OWR-13750338-001 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. 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

I read the primary report's conjecture-generation discussion on pages 1462--1464. It proposes matrix-quantity bounds, invariants for free-group actions and interaction with proof assistants, without specifying one matrix access model, group action or formal proof target. The following three auxiliary calculations make those missing choices concrete. They are not claims of a complete machine conjecture-generation system or a new invariant theory for the source space.

Research attempt: Surrogate indistinguishability, finite orbit invariants, and exact certificates

A surrogate has an information limit even before learning.

Consider real $n\times n$ matrices with Frobenius norm one, $n\geq1$, and try to estimate their Euclidean operator norm from that single surrogate. The inequality \[ n^{-1/2}\leq\|A\|_{\rm op}\leq1 \] follows as follows. Each column has norm at most $\|A\|_{\rm op}$, so $1=\sum_j\|Ae_j\|^2\leq n\|A\|_{\rm op}^2$. Conversely for unit $x$, applying Cauchy--Schwarz to every row gives $\|Ax\|^2\leq\sum_{ij}a_{ij}^2=1$. The endpoints are attained by $A=I/\sqrt n$ and $A=e_1e_1^T$. A rule observing only this surrogate must output the same positive value $c$ on both matrices. Its worst multiplicative error is at least $\max\{c\sqrt n,1/c\}$, minimized at $c=n^{-1/4}$ with value $n^{1/4}$. That constant output also attains this worst factor for the entire displayed interval. Thus the best possible approximation factor using only this feature is exactly $n^{1/4}$; more samples of the same feature cannot improve it. This is a deliberately elementary quantity, not a claim that spectral norm of an explicit matrix is computationally hard.

Trace features can also miss nonnormal size altogether: for $A_t=t e_1e_2^T$ in dimension at least two, $A_t^2=0$, so every positive-power trace is zero, exactly as for the zero matrix, whereas $\|A_t\|_{\rm op}=|t|$. Any bound inferred solely from those traces needs additional assumptions such as a restricted matrix class; this example is not Hermitian and is not a counterexample for Hermitian matrices.

What invariants mean for a finite encoded conjecture space.

Suppose a finite set $\mathcal C$ of $M$ candidate statements is supplied with explicit permutations $s_1,\ldots,s_k$ and their inverses. These define an action of the free group on $k$ generators (the action itself need not be free). Build the graph joining $c$ to $s_jc$ and $s_j^{-1}c$. Breadth-first exploration gives its components using at most $2kM$ directed edge examinations. These components are exactly the group orbits: a graph path is a generator word, and every generator word gives a graph path. A function on $\mathcal C$ is invariant exactly when it is constant on those components. Assigning each component its least encoded element therefore gives a complete orbit invariant for this finite model. The proof gives no finite search bound for an infinite syntactic or geometric conjecture space, where an orbit can be infinite and equality of represented statements adds another problem.

A concrete certificate separates testing from proving.

For a row vector $a\in\mathbb R^d$ and $x\in\mathbb R^d$, \[ \left(\sum_i a_i^2\right)\left(\sum_i x_i^2\right) -(\sum_i a_ix_i)^2 =\sum_{i<j}(a_ix_j-a_jx_i)^2. \] Expand the right side: each $a_i^2x_j^2$ for $i\ne j$ occurs once and each cross term is $-2a_ia_jx_ix_j$, exactly the difference on the left. Summing over matrix rows is an exact sum-of-squares certificate for $\|Ax\|^2\leq\|A\|_F^2\|x\|^2$. A polynomial-identity checker can verify this by comparing coefficients; its input includes the proposed squares and all quantified variable assumptions. This is an explicit certificate interface, not an implemented proof-assistant integration or a proof that a learned generator will find useful identities.

Finite testing by itself cannot substitute for that certificate. Given finitely many real test points $t_1,\ldots,t_N$ and a different point $t_0$, the polynomial \[ p(t)=1-M\prod_{i=1}^N(t-t_i)^2 \] equals one at every test point, but is negative at $t_0$ when $M>\prod_i(t_0-t_i)^{-2}$. Repeated test points cause no difficulty. The example permits a degree growing with the number of tests; no contradiction to interpolation with an a priori bounded-degree model is intended.

Verification and adversarial audit

The surrogate factor is checked on both endpoint matrices and is attained on the whole admissible norm interval. The trace obstruction states its nonnormal scope. The orbit algorithm assumes explicit finite permutations, and free group is not confused with a free action. The sum-of-squares identity is proved by expansion, while the testing counterexample states its unrestricted degree.

Audit corrections.

Specified the information available to the surrogate rule and the finite action model. Did not classify explicit matrix spectral norm as a hard problem, and kept the nonnormal trace example out of the Hermitian setting.

External computation.

None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The results specify obstacles and exact procedures in three finite auxiliary models. No discovery algorithm for hard implicit matrix quantities, invariant classification for the original unspecified conjecture space, or implemented natural-language/proof-assistant pipeline has been produced. The source agenda remains broad and unresolved.

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/47019 (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. 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.