# AI provenance: Explicit Lattice Yang–Mills Mass Gap for Every Compact Simple Group Reducing the Continuum Gap to One Cube

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.

- Paper: https://github.com/creelie/4Y-M/blob/main/preprintYangMillsForestGauge/YangMillsForestGauge.pdf
- Source, programs and Lean certificate: https://github.com/creelie/4Y-M, release v4.29.0, https://doi.org/10.5281/zenodo.23841532 (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 β_W = 0.345, for SU(3) up to β_W = 0.76, and for every compact simple group G when N|β| < 2κ_G/29, with κ_G = h^∨/(2ℓ_ρ) the least Ricci curvature and the explicit gap m_G(β) = arccosh(1 + (κ_G/(N|β|) − 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. Exponential decay with constants of the renormalised correlations of a spanning set of observables also gives the gap.
- **The open inputs, measured.** The cube condition holds at strong coupling, fails at every fixed cube size at weak coupling, cannot come from bounds on single lines of links, and for SU(N) needs cubes of side at least of order (β_W/log β_W)^{1/4}; it fails for the abelian Villain theory and for the quadratic approximation of the line gauge.
- **Block averages.** In a gauge whose fixed links are trees inside blocks of side two, the quadratic approximation of the cube condition holds at every coupling for large cubes, and one step of averaging reduces it to the averages. For the group-valued links the straight path between block roots gives a coercivity bound D_v² ≤ 224 Q for every compact group and every field, but it does not compose across scales, and for every compact connected gauge group no continuous covariant block map treats the positions of a block alike. A cut-off of Bałaban's averaging operation is continuous, covariant and coercive for every field, with D² ≤ 1431 Q and coarse energy at most 2956 Q for one step, and agrees with the block averages to first order; it breaks the transverse symmetry only at large fields, and its constants for one step do not compose in general. On fields with values in a maximal torus whose curvature stays small at every scale it composes: the coarse curvature is the mean of the sixteen fine 2×2 fluxes, the energy never increases, and the thin paths are contracted. For every compact group the commutators of the cut-off average are bounded by the square of the local curvature, so it composes on fields whose curvatures at the successive scales have a small sum, with the energy growing at most by a factor exp(12060 σ). Under the SU(2) Wilson measure, reflection positivity and the chessboard estimate show that the plaquettes farther than ρ from the identity form clusters whose physical size tends to zero in probability, and the cut-off average differs from the path average only next to them; such plaquettes are present along the one-loop tuning, and a frustrated exterior makes a plaquette bad with probability tending to one. A surface-cost bound with Haar caps holds under every exterior, so along the logarithmic tuning the fraction of bad unit cubes in a cube of fixed physical size is small with probability tending to one, uniformly over all exteriors. The composition is local and the defects spread by at most three spacings of the current scale per step, so the cut-off average carries the field through the clusters up to a coarse spacing of order (β_W/log β_W)^{1/4} a, the order of the cube side that the cube condition needs. In the quadratic approximation the effective forms after k averaging steps are equivalent to the coarse forms, Q_k ≤ S_k ≤ 256 Q_k uniformly in k, through a Whitney extension with nodes at the block centres whose averages are the fixed filter (1/8, 3/4, 1/8); in Feynman gauge they are local uniformly in k, sums over the six planes of convolution forms in the coarse curvature whose kernels decay exponentially with constants independent of k. With these forms the constrained cube condition holds at every scale with one cube side, and the composition step holds at every scale. On the fibres of the cut-off average at small fields the constrained law is log-concave on a chart of each fibre, with a Brascamp–Lieb constant 60/λ, and on a chart whose domain is a product of balls its covariances of local observables decay exponentially at a rate that depends only on the group. That chart contains every point of the fibre whose plaquettes are small, so under the SU(2) Wilson measure the mass off it is small on most fibres, and on those fibres the covariances decay on the whole fibre. Along its eight private links each cut-off average is a submersion with a constant that does not depend on the field, and for SU(2) its law under Haar measure on them has density at most 2.4·10^4, so a Peierls argument inside the fibre bounds the mass off the chart on every fibre through a field whose plaquettes near the blocks are small; fibres without such a field are open. Composition across the remaining scales is open. Within the quadratic approximation the cube condition fails in every gauge fixed on a periodic forest with infinite trees: a constant field corrected by a periodic gauge transformation costs no action and moves the whole cube.

## 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 87 and Theorem 180. 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. 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 and 1063 at r = 38/25) and the plaquette-block constants.
- **Lean 4 (v4.33.0) with Mathlib.** The combinatorics of the line field and of the block gauge, and the arithmetic joining the analytic steps on perturbations of the action; every theorem on the standard axioms, with no `sorry`.
- **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 180) 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 87 improves the constant 52 of Theorem 86 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 20 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 146 (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 148. The proofs in the paper are by hand.
- **Nonabelian filling.** The gluing of facet gauges through principal logarithms in Lemma 147 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 116 and Theorem 117 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 119 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 123 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 126 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 127: 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 129 and Proposition 130 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 132 and Proposition 133 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 120: 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 125: 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 135 and Corollary 136: 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 150 and Corollary 151: 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 152 and Proposition 153: 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.