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

This note presents CVRT (Categorical Resonance and Convergence Theory), a preliminary engineering framework for describing and comparing the structural difficulty of deep mathematical problems. It does not prove or solve any open problem; its aim is to organize the structural barriers in problem-solving — in particular the consistency→existence gap (Gap II) — using five semi-quantitative indicators: d_cat (category-crossing distance), d_sub (subcategory distance), Gap II, R (circularity depth), and B (bottleneck concentration).

Version 24 incorporates the results of an independent blind re-scoring of the 358 Erdős problem statements attempted by AlphaProof Nexus (May 2026), archived as a pre-registered dataset at doi:10.5281/zenodo.20672502. Based on that data, this version: (i) restates the CVRT-AI Hypothesis in two parts — proximity (problems with low Gap II and low B appear structurally closer to resolution by any solver) and solver type (d_cat is hypothesized to index the kind of breakthrough required: C0–C1 search-type methods vs. C2–C3 cross-category conceptual bridging) — and withdraws the earlier "high B" component, which the blind data contradicted; (ii) discloses the measured reliability of the five indicators (Gap II κw = 0.65, B = 0.53, d_sub = 0.62, R = 0.48, d_cat = 0.36, the last below the pre-specified threshold, with d_cat-dependent claims downgraded accordingly); (iii) unifies the previously conflicting Class definitions (Class = d_cat-based; point bands renamed S-band; totalling convention fixed as T-score v2) and recomputes all classification tables.

The note includes the CVRT definitions and fixed 9-category system, a scoring rubric with calibration anchors, a structural reading of the AlphaProof Nexus results presented alongside the contradicting blind scores, classification tables for representative solved and unsolved problems, correspondences with reverse mathematics, proof mining, and categorical logic, and an explicit scope-and-limitations section (undecidability, ZFC-independence, and P vs NP are outside or at the boundary of CVRT's scope).

All hypotheses are stated at the Heuristic or Conjecture level with falsification conditions; the companion dataset provides the frozen baseline against which future AI and human resolution reports can test them. CVRT's "categories" are subject-area labels for proof languages, not categories in the sense of category theory.

Keywords: AI theorem proving, automated reasoning, mathematical problem difficulty, structural classification, CVRT, Gap II, proof dependency structure, bottleneck, Erdős problems, AlphaProof, blind scoring, pre-registered baseline, hypothesis revision