AI‑Tractable Hard Mathematics: Structural Difficulty Patterns and the CVRT Framework

This repository presents a preliminary but unified structural framework—Categorical Resonance and Convergence Theory (CVRT)—for analyzing why deep mathematical problems are hard, and which of them are likely to be solved by AI theorem provers, LLM‑based systems, and search‑based mathematical engines.

CVRT characterizes hard problems using five structural indicators: d_cat (category‑crossing distance), d_sub (subcategory mismatch), Gap II (consistency–existence gap), R (circularity depth), and B (bottleneck concentration). These indicators form a semi‑quantitative difficulty index that explains why some problems (e.g., combinatorics, local analytic problems) are AI‑tractable, while others (e.g., abc, BSD, Langlands) remain structurally inaccessible to current AI systems.

The note includes: • A unified definition of CVRT and its category system • A structural analysis of recent AI breakthroughs (AlphaTensor, AlphaGeometry, FunSearch, AlphaFold, AlphaProof Nexus) • A predicted list of AI‑solvable open problems and AI‑intractable problems • A 5‑indicator scoring rubric for 100 major problems (50 solved + 50 unsolved) • A systems‑engineering perspective on the consistency→existence barrier in mathematics

This work aims to support researchers in AI theorem proving, automated reasoning, mathematical search, and symbolic‑neural hybrid systems, providing a structural map of where AI is strong, where it is weak, and where breakthroughs are most likely to occur.

Full CVRT definitions and the complete scoring dataset will be released in future versions.

Keywords: AI theorem proving, LLM mathematics, automated reasoning, AlphaProof, symbolic AI, mathematical problem difficulty, CVRT, category theory, structural complexity, hard problems, Gap II, circularity, bottleneck, mathematical AI benchmarks.