# AI provenance: One Cube Estimate Implies the Yang–Mills Mass Gap

Authors: Deep Bhattacharjee (corresponding; ORCID 0000-0003-0466-750X), Ushashi Bhattacharya (ORCID 0009-0002-2254-3914), James Montgomery Lockwood, Tristan Romanov. Affiliations and emails as in hexagon-formfill.md.

Version 8 of Hexagon preprint 2610.00341.

- Paper: https://github.com/creelie/4Y-M/blob/main/preprints/preprintshexagon/YangMills_Hexagon.pdf
- Source, programs and Lean certificate: https://github.com/creelie/4Y-M, release v4.37.0, https://doi.org/10.5281/zenodo.23992883 (all versions https://doi.org/10.5281/zenodo.23444338).

## What the paper proves

We prove that four-dimensional lattice Yang–Mills theory with any compact simple gauge group has a unique infinite-volume state and a mass gap at strong coupling, and that one averaged estimate in a single cube of fixed physical size gives a mass gap to every continuum limit. For SU(2) the block variables approach the Haar measure in every representation as the coupling is tuned to the continuum, and this proves the estimate at the coarsest scale on every torus of side up to exp(2^k/(17k)) after k blocking steps. The estimate along the whole tuning remains open.

## How it was made

Deep Bhattacharjee conceived the project and, with the help of AI agents working under his direction (Claude Opus 5.5 in Claude Code, Claude Fable 5.1, GPT-6.1 Sol Ultra and GPT-6 Astra Ultra), made its constructions and proved its main results: the forest gauge on the period-two line field and the free-boundary state with its transfer-matrix gap at strong coupling; the plaquette blocks of SU(2) and SU(3) with their constants proved by hand; the reduction of the mass gap to one averaged estimate in a cube of fixed physical size; the labels of Bałaban's blocks and the islands of bad blocks; the averaged condition, its descent to block variables, the upward step and the split inequality; the single-link coefficients across scales, the forces in the fibre law, the smoothing along the paths for SU(2) and the row sums of the Gaussian model; and the conditional chain for every compact simple group. He wrote the programs of the certified computations and the paper, and checked every statement. Ushashi Bhattacharya checked the proofs and the references and edited the text. James Montgomery Lockwood contributed the analysis of constrained cubes and its code and Lean 4 formalisations, and the composition of metrics across scales of Section X S with its Lean 4 formalisation; Tristan Romanov contributed the certified bound on the forest-gauge Hessian, the research continuation on which Section XIII is built and the flat fields far from the identity, with four Lean 4 modules. All four authors approved the final text. The Anaxia engine of the BLLM system first assembled the combination of the forest gauge with the sharp quarter behind the SU(2) theorem (doi:10.5281/zenodo.23469389; run records doi:10.5281/zenodo.23484160) and supplied no step of the proofs. The paper does not prove the cube estimate along the whole continuum tuning or the existence of a nontrivial continuum limit, so it does not solve the Clay Millennium problem, and it says so. Compared with version 6 (hexagonmath.org/2610.00341v6), the paper adds Section XIII (labels of the blocks, islands of bad blocks, the bad side of a projected block average, the obstructions to the remaining routes, the effective action across one scale), Sections X F to X S (an averaged form of the cube condition, its local form, the couplings where it holds and fails, its neutral part on small tori, its passage to block variables, the upward step, the measures for which it holds, the split inequality at strong coupling, the condition at the coarsest scale, the coefficients across scales, the forces in the fibre law, the smoothing along the paths, the order of the row sums of the Gaussian model and the metrics that compose across scales), Sections XI D, XI H and XI I (the Osterwalder–Schrader axioms for the flowed fields, the limit at separated points of the fields without flow with the inputs it needs, and the conditional chain for every compact simple group) and the concluding Section XIV, and it proves by hand every constant that version 6 took from computer certificates; the title-page footnote says so.

Tools used: Anthropic Claude Opus 5.5 (Claude Code) and Claude Fable 5.1; OpenAI GPT-6.1 Sol Ultra and GPT-6 Astra Ultra; BLLM Anaxia engine v3.0.5 and v3.1.0; Lean 4 with Mathlib; Python (NumPy, SciPy, mpmath, python-flint with FLINT and Arb); C; shell.

## What is computed

Every proof is analytic and every number used in a proof is derived in the text. Evaluations for SU(3) in interval arithmetic and computations with certificates checked in exact arithmetic are reported in remarks, and no result uses them. Lean 4 with Mathlib checks the line field, the comparison arithmetic, the coupling lemmas, the finite arithmetic of the island bounds, and the counting and trace identities and constants of the averaged condition and of the derivative in the ends.