Large Language Models (LLMs) have made notable progress in automated theorem proving, yet existing formal benchmarks remain limited in both mathematical coverage and difficulty. Most are concentrated in areas that are easier to formalize, such as algebra and elementary number theory, and provide limited coverage of subfields that require deeper reasoning, including mathematical analysis. To address this gap, we introduce MA-ProofBench, to the best of our knowledge, the first formal theorem-proving benchmark dedicated to Mathematical Analysis. The benchmark contains 200 formalized theorems covering 6 core topics and 27 subcategories, including measure and integration theory, complex analysis, and functional analysis. The problems are divided into two difficulty levels, an undergraduate level (Level I, 100 problems) and a Ph.D. qualifying level (Level II, 100 problems), to evaluate how well LLMs perform formal reasoning at different mathematical depths. Each problem is constructed through a human-led, LLM-assisted formalization pipeline followed by independent expert review, ensuring that the formal statements remain faithful to the original mathematics. We evaluate a range of recent general-purpose reasoning models and formal theorem provers on MA-ProofBench. However, most models perform poorly: even the best-performing model, GPT-5.5, achieves only 16% Pass@8 on Level I and 5% on Level II, while most models stay close to 0% on Level II. Further analysis identifies Mathlib hallucinations and incomplete proofs as the two dominant failure modes, while an evaluation on the natural-language version of the benchmark exposes a clear gap between informal and formal reasoning. MA-ProofBench is intended to serve as a reliable reference for tracking progress in formal mathematical reasoning in advanced domains.
Two major periods of reduced funding and confidence in artificial intelligence research, commonly called the first and second AI winters, are usually explained through engineering failure, commercial disappointment, and inflated expectations. This article develops a complementary thesis: that the dominant paradigms of those periods also met genuine formal barriers, including limitations of representation, optimisation, computational complexity, statistical learnability, and high-dimensional approximation. The contribution is synthetic rather than archival. We do not claim that particular theorems mechanically caused the winters; rather, we show that several central disappointments of early AI were aligned with mathematically precise bottlenecks. We analyse these bottlenecks through the perceptron impossibility results of Minsky and Papert, the complexity-theoretic hardness of exact neural-network training established by Blum and Rivest, minimax rates for nonparametric estimation in high dimension due to Stone, vanishing-gradient analyses by Hochreiter and by Bengio and collaborators, and classical statistical learning theory in the tradition of Vapnik and Chervonenkis, Valiant, and Blumer and collaborators. We then relate these barriers to the later breakthroughs that mitigated, rather than eliminated, them.
Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have \[ \Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p}, \] where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound \[ \Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p}, \] where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.
Sun Woo Park, Yun Young Choi, U Jin Choi +1cs.LG math.AT
We provide a mathematical interpretation of convolutional (or message passing) neural networks by using presheaves and copresheaves of the set of continuous functions over a topological space. Based on this interpretation, we formulate a theoretical heuristic which elaborates a number of empirical limitations of these neural networks by using obstructions on such sets of continuous functions over a topological space to be sheaves or copresheaves.