Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory

Abstract: We investigate the capacity of current LLMs to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of LLMs for mathematical discovery and the continuing importance of expert verification.

• The paper documents five AI-generated proof candidates in Banach space theory, including results on toroidal separation, Calkin algebras, operator ideals, basis factorization, and primarity, with expert verification and repair establishing their mathematical value.
• The authors demonstrate that lightweight workflows using ChatGPT and Codex can contribute to live research problems, while emphasizing that human mathematicians remain responsible for selecting, checking, formalizing, and publishing proofs.
• An automated pipeline extends the approach by extracting open-problem signals from arXiv sources, but the paper highlights unresolved challenges in formal verification, reproducibility, peer review, and responsible disclosure of AI assistance.

Overview

This paper, by Antonio Acuaviva and Pablo Acuaviva (2607.17388), documents an experiment in which LLMs were used to generate proofs of research-level open problems in Banach space theory. The work has two components. The first is a set of five human-selected problems—proposed by Tomasz Kania, Kevin Beanland, and the first-named author—for which the model produced proof candidates that survived human verification and editing. The second is an automated pipeline that extracts open-problem signals from arXiv sources and attempts solutions at scale with minimal human guidance. The authors' central claim is deliberately measured but substantive: current LLMs, placed in suitable workflows and checked by experts, can already do useful work on live research mathematics, not merely polished exercises or exposition.

The tooling is intentionally lightweight: direct interaction with ChatGPT 5.5 Pro and Codex agents operating over a filesystem, rather than a bespoke formal-verification architecture. The authors are explicit that this is not claimed to be optimal; the point is that a relatively ordinary setup can produce serious proof candidates.

The five selected problems

The five problems in Part II were chosen by specialists as substantial targets, not discovered by the pipeline. Their provenance matters: for Problems 1–4, the raw model outputs were "essentially correct, up to verification, small modifications, and editing," with most required work being presentational rather than mathematical. Problem 5 was different: the raw output contained the right proof architecture and most technical content, but could not be regarded as a complete proof; human intervention involved reorganizing the argument and making indicated steps precise.

P1 — Toroidal Elton–Odell theorem. The classical Elton–Odell theorem gives a uniform margin $\varepsilon>0$ in Kottman separation on the unit sphere of any infinite-dimensional normed space. The toroidal variant, raised in Hájek–Kania–Russo, asks whether the same holds when vectors are identified up to unimodular scalar multiplication, i.e., whether every infinite-dimensional complex normed space admits unit vectors $(x_n)$ with $(x_n,x_m)\ge 1+\varepsilon$. Prior results gave the strict non-uniform analogue in full generality and uniform versions under extra hypotheses (finite cotype, lower $q$-estimates, asymptotic uniform convexity). The paper proves the uniform statement unconditionally. The proof proceeds by case analysis: explicit constructions in $c_0$ and $\ell_1$ transferred via James' distortion theorem; a boundedly complete case; and a strongly summing case reached via Rosenthal's $\ell_1$ and $c_0$ theorems. The combinatorial core is a "rapid flat-block alternative": inside the span of an asymptotically monotone basic sequence, failure of toroidal separation forces successive almost-flat blocks $d_1<d_2<\cdots$ and phases $\alpha_j$ with uniformly bounded partial sums—an obstruction that contradicts bounded completeness or strong summability. Notably, the model produced two genuinely distinct proofs; only the first has been manually verified to inclusion standard, the second being released unverified on the project website.

P2 — Non-Calkin unital Banach algebras. The realisation problem asks which unital Banach algebras occur as Calkin algebras $(x_n)$0. Horváth and Kania had constructed simple unital AF $(x_n)$1-algebras that are not Calkin algebras of any separable Banach space, leaving the general question open. The paper answers it affirmatively with two independent constructions. The first uses a Leavitt-type quotient $(x_n)$2 built from one vertex with $(x_n)$3 loops: density rules out separable $(x_n)$4, while topological simplicity plus a countable matrix-unit system with uniformly bounded rows forces, via a separable-subspace absorption argument, a noncompact operator with separable range—a contradiction. The second uses a shift algebra on $(x_n)$5 with $(x_n)$6: the strong-limit property traps any realizing space $(x_n)$7 at density at least $(x_n)$8, compact errors are simultaneously killed on a unit vector, the shift relations then force a copy of $(x_n)$9 inside $(x_n,x_m)\ge 1+\varepsilon$0, and Josefson–Nissenzweig yields a noncompact separable-range operator, destroying simplicity of the Calkin algebra.

P3 — Strict cosingularity and adjoints. Pełczyński showed that weakly compact strictly cosingular operators have strictly singular adjoints, and that weak compactness cannot be dropped ($(x_n,x_m)\ge 1+\varepsilon$1). The question, arising from Androulakis–Beanland's transfinite Schreier hierarchy and posted by Beanland on MathOverflow, was whether separability of the range suffices. The paper proves: if $(x_n,x_m)\ge 1+\varepsilon$2 is separable, $(x_n,x_m)\ge 1+\varepsilon$3 is strictly cosingular if and only if $(x_n,x_m)\ge 1+\varepsilon$4 is strictly singular. The key technical device is a preadjoint extraction lemma: using a countable dense set in $(x_n,x_m)\ge 1+\varepsilon$5 and finite-dimensional interpolation, an arbitrary operator $(x_n,x_m)\ge 1+\varepsilon$6 becomes, after passing to an infinite-dimensional subspace $(x_n,x_m)\ge 1+\varepsilon$7, the restriction of an adjoint $(x_n,x_m)\ge 1+\varepsilon$8. This yields a quotient factorization $(x_n,x_m)\ge 1+\varepsilon$9 whenever $q$0 is bounded below on some infinite-dimensional subspace, which directly obstructs strict cosingularity. Separability enters only here; no assumption on $q$1 is needed, which also explains why the $q$2 counterexample escapes the theorem.

P4 — Weakly compact basis factorization. Davis–Figiel–Johnson–Pełczyński knew their interpolation space admits a Schauder basis when the range has a shrinking basis; Figiel–Johnson–Tzafriri handled unconditional bases, and Ghoussoub–Maurey–Schachermayer the $q$3 case. Whether a basis could always be arranged when the range merely has a Schauder basis remained open (with the $q$4 case settled by Bill Johnson in 2016). The paper proves the general result: if $q$5 is separable with the bounded approximation property, every weakly compact $q$6 factors through a reflexive Banach space with a Schauder basis. The delicate step is producing finite-rank approximants whose adjoints converge pointwise in norm on both $q$7 and, after double adjoints, $q$8; this combines Baire-one regularity of $q$9 on the weak-star ball (via Stegall's theorem) with a tail-convexification/Mazur argument. A bridge theorem then encodes $c_0$0 into $c_0$1, applies DFJP interpolation to a carefully constructed weakly compact set stabilized by freezing projections, obtains an FDD, and invokes Johnson–Rosenthal–Zippin basisification. This removes the principal obstacle to extending Dodos–Ferenczi-style strong boundedness from reflexive spaces to classes of weakly compact operators via uniform factorization.

P5 — Primariness of $c_0$2. Lechner–Motakis–Müller–Schlumprecht proved $c_0$3 primary and flagged $c_0$4 as a prominent open case. The paper proves that $c_0$5 has the uniform primary factorization property, hence is primary, for all $c_0$6. The strategy follows LMMS: reduce arbitrary operators on the doubly cancellative product-Haar part $c_0$7 to product Haar multipliers, then apply their scalar-compression theorem. The new machinery includes a local $c_0$8 trace theorem—extracting from any operator on $c_0$9 a limiting averaged Haar diagonal represented by an $\ell_1$0 density, via Reynolds averaging over finite dyadic symmetry groups and the Semenov–Uksusov branch-variation criterion—and a probabilistic alternating outer/inner tree construction of faithful Haar systems, using randomized splittings, Khintchine-based variance estimates, and ultrafilter limits justified by uniform integrability. Off-diagonal coefficients of three types (same outer strip, same inner strip, mixed) are driven below summable budgets, so the compressed operator is within $\ell_1$1 of a bounded multiplier while retaining a left inverse. Pełczyński decomposition then yields primarity. This is the longest and most structured problem, and the one where the gap between raw model output and finished proof was largest.

Methodological positioning

The paper distinguishes itself carefully from adjacent work. Unlike FrontierMath, RealMath, LemmaBench, IMProofBench, or First Proof, it does not compare systems under fixed protocols or produce leaderboard results; it asks what models contribute under repeated exploration of live problems. Unlike FunSearch, AlphaEvolve, and related evaluator-guided search—which require computationally scoreable candidate objects—the problems here admit only natural-language proof candidates checked by reading. Unlike agentic systems such as Kosmos, Arbor, Iteris, Aletheia, Rethlas/Archon, or Aristotle, the setup involves no Lean formalization, theorem-retrieval infrastructure, or hypothesis trees. A mathematician did not supply intermediate lemmas; the mathematical content of promoted proofs originated in model attempts, with humans selecting, verifying, repairing, and writing up.

Limitations and open questions

The authors state several limitations plainly. The proofs are not formally verified; they rest on human mathematical reading, and the authors take responsibility for correctness. The second proof of P1 remains unverified. The field restriction to Banach space theory reflects a practical constraint—generated mathematics requires verifiers who know the local literature—not a claim of domain specificity. All exploratory work used ChatGPT 5.5 Pro prior to newer model releases, so the results should be read as a snapshot that may understate current capability. Beyond these, the paper raises structural questions it does not resolve: how disclosure norms should be designed so that honest reporting of AI assistance is not penalized relative to nondisclosure, particularly for early-career researchers; how peer review should absorb an increased volume of plausible-but-unverified arguments without referees becoming de facto cleanup labor; and whether AI-generated proofs can be translated into formally verified mathematics cheaply enough to relieve what the authors identify as the principal bottleneck—verification rather than generation. The automatic-search component of Part III is documented but its outputs are not held to the same inclusion standard as Part II.

Conclusion

The paper's contribution is twofold: five genuine results in Banach space theory—the toroidal Elton–Odell theorem, the existence of non-Calkin unital Banach algebras, the separable-range duality between strict cosingularity and strict singularity, the general weakly compact basis factorization theorem, and the primarity of $\ell_1$2—and a documented demonstration that language-model proof search, combined with expert selection, verification, and repair, can settle questions that specialists regard as serious. The authors' own framing is the appropriate one: the results evidence a workflow, not autonomous reliability, and the burden of correctness, understanding, and stewardship remains entirely with the mathematicians who publish them.