# AI provenance: Yang–Mills Mass Gap on the Lattice and One Cube from the Continuum

Authors:
- Deep Bhattacharjee, corresponding author. ORCID 0000-0003-0466-750X; d.bhattacharjee@erl-forschung.de, itsdeep@live.com; GitHub contact egspl@proton.me (github.com/creelie). Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India; Formerly, University of Calcutta (Heramba Chandra College), 23/49 Gariahat Road, Golpark, Kolkata 700029, West Bengal, India.
- Ushashi Bhattacharya. ORCID 0009-0002-2254-3914. Formerly, National Taiwan University, Taipei 106319, Taiwan.
- James Montgomery Lockwood. Independent Researcher, San Antonio, USA; spectralityworkinggroup@gmail.com (secondary Blaze61029dogQ@proton.me).
- Tristan Romanov. Independent Researcher, San Francisco, CA, USA; romanov360@gmail.com.

Version 5 of Hexagon preprint 2610.00341 (earlier title: Explicit Lattice Yang–Mills Mass Gap for Every Compact Simple Group Reducing the Continuum Gap to One Cube).

- Paper: https://github.com/creelie/4Y-M/blob/main/preprintYangMillsForestGauge/YangMillsForestGauge.pdf
- Source, programs and Lean certificate: https://github.com/creelie/4Y-M, release v4.31.0, https://doi.org/10.5281/zenodo.23895262 (all versions: https://doi.org/10.5281/zenodo.23444338).

## What the paper proves

The gauge of four-dimensional lattice Yang–Mills theory is fixed on a period-two field of straight lines that meets every plaquette in exactly one link, so that every free link has twelve Dobrushin partners instead of eighteen.

- **Fixed lattice spacing.** A unique free-boundary state, reflection positivity, exponential clustering and a transfer-matrix gap, for SU(2) up to beta_W = 0.351 and for SU(3) up to beta_W = 0.77 through plaquette blocks whose constants are certified in ball arithmetic, and for every compact simple group G when N|beta| < 2 kappa_G/29, with kappa_G = h^vee/(2 l_rho) the least Ricci curvature and the explicit gap m_G(beta) = arccosh(1 + (kappa_G/(N|beta|) - 29/2)/12).
- **The continuum step.** A Dobrushin–Shlosman condition on one cube of fixed physical size, uniform along the continuum tuning, gives a mass gap in physical units for every continuum limit; the same holds for observables smoothed by the spatial gradient flow, whose renormalised continuum limits exist along subsequences and are nontrivial exactly when one ratio of two-point functions stays away from zero.
- **What the cube estimate needs.** It holds at strong coupling, fails at every fixed cube size at weak coupling, cannot come from bounds on single lines of links, needs for SU(N) a cube of side of order (beta_W/log beta_W)^{1/4}, and fails for the abelian Villain theory and for the quadratic approximation of the line gauge.
- **Block averages.** A constrained quadratic gap 5000/537; a continuous, covariant and coercive cut-off of Balaban's averaging that composes on fields of small curvature and, under the SU(2) Wilson measure, carries the field through the large-field clusters up to a coarse spacing of the order the cube needs; effective forms Q_k <= S_k <= 256 Q_k local uniformly in the scale; log-concave charts of the fibres of the average with decay constants explicit for every compact group; Peierls bounds on the mass off the chart on every fibre; calm completions of every exterior along the tuning.
- **Fixed large-field sets and coarse defects (new in this version).** Under every exterior, excess action in a box is rare and a link conditioned on a fixed large-field set keeps a bounded Dobrushin response. The curvature of the block averages costs action independent of the scale. The Haar tilt, a right translation of the Haar measures, bounds the tails of the force along a fixed direction by the second variation of the action at every scale and under every exterior; for U(1) this makes coarse defects rare at the rate beta_W x^2 at every scale, while for SU(2) the second variation grows like 4^k on the gauge orbit of the vacuum. Transporting the direction along a forest keeps the Haar measures and gives the same bound with the supremum over the gauge-fixed section, but for the block forest that supremum is again of order 4^k, reached by pure gauge fields constant on the blocks; so the one estimate that remains open is a second-variation bound on a smooth section of the gauge orbits, with its Jacobian.

## What is not claimed

The cube estimate along the continuum tuning and the existence of a nontrivial Osterwalder–Schrader continuum limit are not proved. The Clay Millennium problem is not solved, and the paper says so. The proofs are written in full and have not yet been refereed.

## Who did what

Text of the contribution: a mix of human-written and AI-generated text. Human understanding: all parts; the four authors read every proof and approved the final text.

Deep Bhattacharjee chose the problem, directed the work and ran the computations. The proofs, the text, the figures, the Lean file, the Python and shell checks and the reference list were produced with AI assistance under his direction (Claude Opus 5.5 in Claude Code, Claude Fable 5.1, GPT-6.1 Sol Ultra and GPT-6 Astra Ultra). The programs were written by Deep Bhattacharjee, James Montgomery Lockwood and Tristan Romanov, and T. Romanov constructed the certificates of Proposition 88 and Theorem 191. Every reference was checked against the publisher's, Crossref's or DataCite's record.

The Anaxia engine of the BLLM system first assembled the combination of the forest gauge with the sharp quarter behind the SU(2) theorem (release v3.0.5, doi:10.5281/zenodo.23469389; run records in release v3.1.0, doi:10.5281/zenodo.23484160); it supplied no step of the proofs. The block-constrained programme follows J. M. Lockwood's report, and the paper proves it in full.

## How the result is checked

- **Proofs by hand.** Every theorem is proved in the text, with its constants derived in the appendices. The only computations the proofs use are evaluations with rigorous error bounds, in ball arithmetic (python-flint, FLINT and Arb): the SU(3) covers (2341 boxes at r = 3/2, 1063 at r = 38/25 and 2825 at r = 77/50) and the plaquette-block constants.
- **Lean 4 (v4.33.0) with Mathlib.** The combinatorics of the line field and of the block gauge, the arithmetic joining the analytic steps on perturbations of the action, and the chain of rational bounds of the SU(2) plaquette blocks (`strong-coupling/gauge-fixed/blocks/lean/`, which also shows that the bound exceeds one at beta_W = 0.352); every theorem on the standard axioms, with no `sorry`. The Lean files cover finite data and arithmetic only; the analytic proofs are checked by hand.
- **Independent checks** (in `anc/` of the source zip): exact rational checks of the finite counts of the continuum and block sections, the curvature table from the root systems, and numerical checks of the path-block bounds of the slice windings for SU(2) and SU(3), of the cut-off average, of the combinatorics, constants and conditional laws of the large-field section, of the propagation of defects across scales, and of the failure of the quadratic cube condition in the forest and axial gauges. The Monte Carlo estimates in the archive are not used by any proof.
- **Certificate bound for the Hessian.** Appendix L of the paper derives the bound h_F ≤ 14.420001 (Theorem 191) from a certificate of T. Romanov, one of the authors (github.com/romanov360/yang-mills-4d-forest-certificate). It extends the curvature range to |β| < 1/28.840002. The derivation is written out by hand: the corner and site identities, the certificate lemma and the reduction from U(N) to U(2). What it leaves to the computer is the positivity of the 96 plaquette matrices at every phase and the arithmetic of the rational data. In Romanov's repository the Lean 4 kernel checks the finite data and the positivity, through exact sums of squares with positive definite integer Gram matrices, using only propext and Quot.sound. The written steps of the appendix are not formalised. Our own check (`romanov_certificate.py` in `anc/`, with his data `cert4d_h14.42.json` and its licence notice) covers three things. His fixed links are ours up to a lattice symmetry. The finite data hold in exact arithmetic. The 96 plaquette matrices are positive definite at every phase (adaptive grid with a Lipschitz bound, in floating point). J. M. Lockwood checked the certificate independently in exact rational arithmetic with outward-rounded interval elimination. These checks concern the fixed lattice, not the continuum limit. The theorems keep the constant 29/2, proved by hand.
- **Block gap 5000/537.** Proposition 88 improves the constant 52 of Theorem 87 to 5004/537 through a certificate of T. Romanov on a window of five blocks. `blockgap_certificate.py` in `anc/` (with `certificate_c537_5000.json`) was written independently from the definitions of the paper. In exact arithmetic it checks the identity of quadratic forms, the positivity of all 245 leading minors and the vanishing of the multiplier terms on the constrained fields (4 of 4 checks).
- **Figure of the certificate.** `make_certificate.py` in `anc/` draws Figure 23 from the certificate data. Its eigenvalues are in floating point and no proof uses them.
- **Box extension.** `box_extension.py` in `anc/` checks the lift and extension of Lemma 147 (the facet-by-facet axial lift with McShane extensions, its bound 13118 l rho_s, the path metric of the earlier facets, the gauge on the boundary, and the radial Whitney extension by quadrature, on random boxes of side 2 to 4) and the constants of Proposition 149. The proofs in the paper are by hand.
- **Nonabelian filling.** The gluing of facet gauges through principal logarithms in Lemma 148 was first drafted in a note supplied to the authors that was prepared with AI assistance; the authors checked every step, proved the transport estimate that the note only sketched, and rewrote the lemma. `nonabelian_filling.py` in `anc/` checks the facet gauges, the gluing and its bounds C_k l rho_s, the curl bound, and the full filling with every plaquette at most K rho_s on random SU(2) boundary fields of boxes of side 2 and 3, in 60-digit arithmetic; these are numerical checks, not interval certificates.
- **Small fields on the fibres.** `small_field_fibre.py` in `anc/` checks Lemma 117 and Theorem 118 on SU(2) backgrounds in the tree gauge of a periodic lattice of side 4 with plaquettes at most kappa (block-gauge backgrounds at three sizes, 8 and 16 constrained blocks): the frame errors, the linearised constraint, the tangent gap Q_U >= |xi|^2/52, the Hessian formula and its lower bound, the Gram matrix of the constraint and the Hessian along curves in the fibre, with 12 exact checks of the constants; these are numerical checks in double precision, not interval certificates. The proofs in the paper are by hand.
- **Chart of the fibres.** `fibre_chart.py` in `anc/` (which imports `small_field_fibre.py`) checks Theorem 120 on the same SU(2) backgrounds with two to sixteen constrained blocks: it solves for the chart of the fibre over an l-infinity ball of the tangent space, checks that the chart lies on the fibre and contracts as in the proof, computes by second differences the Hessian of the action in the chart and of the Haar, coarea and area factors, and checks the arithmetic of the constants exactly; these are numerical checks in double precision, not interval certificates. The proofs in the paper are by hand.
- **Decay on a product chart.** `chart_decay.py` in `anc/` (which imports `fibre_chart.py`) checks the product chart of Theorem 124 on SU(2) backgrounds on the periodic lattice of side 12 with a row of five constrained blocks: that each path link enters only its own constraint, that the section S is injective with B S = 0 and path-link row sums at most 14 (the proof uses 15.1), that the chart factor of the projection is at least the identity, the decay with distance of the blocks of the inverse Gram matrix, of that factor and of the chart Hessian of the action (from 15 at distance zero to 1e-8 at distance nine), and the arithmetic of the constants exactly; these are numerical checks in double precision, not interval certificates. The proofs in the paper are by hand.
- **Mass off the chart.** `off_chart.py` in `anc/` (which imports `small_field_fibre.py`) checks the first step of Theorem 127 on SU(2) backgrounds with two and eight constrained blocks: from random starts it finds points of the fibre by solving the constraints for the path links and descending the action, and at every such point checks that each link of the constrained set moves by at most 92.3 times the largest plaquette distance (the largest ratio observed is 1.5); it checks the covariance comparison of part (iii) in exact arithmetic on random finite laws and the arithmetic of the constants exactly. `make_offchart.py` draws Figure 18. The search is in double precision and is not a certificate; the proof in the paper is by hand.
- **Mass off the chart on calm fibres.** `private_submersion.py` in `anc/` checks Lemma 128: the constants α, Λ < 1.0927, 1 − 7Λ/8 > 0.0439 and σ_0 > 0.031, and, on random SU(2) and SU(3) fields with Haar inner links and every value of the cut-off, the test variation of the proof and the least singular value of the derivative along the private links (least observed 0.188 against the proved 0.031). `calm_fibre.py` in `anc/` checks Lemma 130 and Proposition 131 for SU(2): the constants (M < 2.4·10^4, 1 + 7Λ/8 < 1.957, the Haar measure of small balls), the determinant bound and the uniqueness of the solution on the regions of the density formula, and on calm backgrounds the contraction that solves each constraint for its path link inside the ball of radius 44ρ. Both are numerical checks in double precision; the proofs in the paper are by hand.
- **Peierls bound on every fibre.** `every_fibre.py` in `anc/` checks Lemma 133 and Proposition 134 for SU(2): it builds the aligned field on runs of blocks in each of the four directions, with random averages and random fixed links, and confirms that every site of every average equals I and that the averages take their prescribed values; and on regular backgrounds with Haar random links inside the blocks it checks the contraction that solves each constraint for its path link inside the ball of radius 44ρ, the cut-off and the determinant bound, together with the constants of the proof. A numerical check in double precision; the proofs in the paper are by hand.
- **Explicit constants of the fibre chart.** `explicit_constants.py` in `anc/` checks Proposition 121: it evaluates the chain of bounds of the proof at the radii used (the third derivative bound 14810 at radius 0.07), the bounds for log j and the sum that gives c_6, the SU(2) thresholds λ ≥ 5.6·10^16, κ < 4.9·10^{-18} and β_W > 3.8·10^38, and, on random SU(2) backgrounds, the first three derivatives of the averages by finite differences (largest 0.13, 0.025 and 0.017 against c_3 = 21213); 8 of 8 checks. `make_offchart.py` redraws Figure 18 at the new threshold. A numerical check in double precision; the proof in the paper is by hand (Cauchy estimates).
- **Explicit rate and amplitude of the chart decay.** `explicit_decay.py` in `anc/` checks Proposition 126: it evaluates the chain of bounds of the proof (the Gram operator, the section, the projection and the Hessian, K ≤ 2068λ at weight 0.197) and every stated number, α_G = 2.5·10^{-4} and c_9 ≤ 1.2·10^4 for every compact group; and, on random banded Hermitian matrices, the Schwarz bound for the Hermitian part of a weighted matrix, the sinh bound for S*S and the decay of inverses in the weighted norm (largest ratios 0.119, 0.567 and 0.975 against the proved bounds); 12 of 12 checks. A numerical check in double precision; the proof in the paper is by hand.
- **Comparison of fibres with the same exterior.** `fibre_compare.py` in `anc/` checks Proposition 136 and Corollary 137: in exact rational arithmetic the constants of the proofs (n_P ≥ 74, ρ < 1.5·10^{-4}, 16√2ρ + 16ρ² < 22.63ρ, 14·90.6³ < 1.05·10^7, the arithmetic of the corollary) and, in double precision, the sufficient condition β_W ≥ 6.4·10^{43}|W|²(1 + ln|W|) for |W| up to 10^6; on random SU(2) links it checks δ ≤ d, the growth of the squared plaquette deviation when four links move by at most ρ (largest ratio 7.7 against 22.63) and the Haar mass of small balls; 15 of 15 checks. The proofs in the paper are by hand.
- **Bad cubes under every exterior.** `any_exterior.py` in `anc/` checks Proposition 151 and Corollary 152: it enumerates boxes of side 2 to 6 for the counts of links, plaquettes, boundary plaquettes and interior unit cubes and the bound of four unit cubes per plaquette, and checks the constants (24π³ < e^7, the mass of the Haar caps, 6x + 31 ≤ e^{x/4}/128 for x ≥ 44 and the final bracket) in exact arithmetic (16 of 16 checks). The proofs in the paper are by hand.
- **Calm windows and the large-field cases.** `calm_windows.py` in `anc/` checks Corollary 153 and Proposition 154: it enumerates the blocks of W⁺ met by the plaquettes of a unit cube (at most 256), checks the constants of both proofs in exact arithmetic (the bound 9.72·1.371 < 14, n_P ≥ 73.5|W|, 14·10^{-24}/(512·73.5³) < 7·10^{-32}, the logarithm above 162.8 and β_W > 10^{24}|W|²), and compares the cap probability of an SU(2) plaquette under the conditional Wilson density, by quadrature, with the bound 14(β_W ρ'²)^{3/2} (largest ratio 0.106); 14 of 14 checks. The proofs in the paper are by hand.