The continuum limit of three-dimensional SU(2) lattice Yang–Mills theory on a torus V: Short-distance and weak-coupling asymptotics

WARNING!

We have extensively audited this manuscript using large language models, including adversarial reviews by multiple models. However, the proof is too complex for Lean formalization at the present time, and it has not yet undergone verification by a human expert.

Image: detail from Abraham Ortelius, Theatrum orbis terrarum (1570). Library of Congress, Geography and Map Division. https://www.loc.gov/resource/g3200m.gct00126/?sp=106

Authors’ Note

With the exception of this note, the entire document is machine-written. We have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited.

The manuscript began as an attempt to pursue the program of Magnen, Rivasseau, and Sénéor [132] using large language models. Our hope was that understanding the problem on the torus might offer some insight into the general case. The construction that resulted from this starting point is three-dimensional and lattice-based and draws extensively on Bałaban’s renormalization group methods.

As we were verifying these results, we learned that Alpöge and Sussman announced that they appear to have a YM4 existence argument, following Bałaban’s program (https://github.com/ethanwsussman/Yang--Mills-LLM-experiment). We are thrilled and look forward to learning more about these exciting developments. We hope that this manuscript nonetheless contributes to understanding constructive approaches to three-dimensional Yang–Mills theory. The first version of these results was posted to the first author’s GitHub on October 17, 2026 (https://github.com/pmaronow/3d-ym-torus). Prior to posting the first version of this work, we did not inspect the ongoing work of Alpöge and Sussman. Further, to the best of our knowledge, the LLMs we used did not access their work during the mathematical development and drafting of this manuscript. This version contains additional clarifications relative to the first version; it has also been partitioned into five volumes.

We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.

P. L. was partially supported by NSF grant DMS-2450004.

Gaussian short-distance limits

Reconstruction estimates and divergence-free currents

This section and § 2.2 prove the short-distance theorem, Theorem 2.8. We study the heat-smoothed circulations defined in Definition 31.1: circulations, around small lattice loops of size $\varepsilon$, of the linear heat evolution of the reconstructed fine field, divided by $\sqrt{g_3^2\varepsilon}$. They are linear in the link logarithms of one fixed gauge representative; flowed Wilson loops are compared with them in Lemma 32.11.

The main result of this section is an exact decomposition of the heat-smoothed circulation $I^{K,s}$ under the conditioned Wilson measure, proved in Lemma 31.8:

Here $T_s$ is the contribution of the coarse field on the lattice of side $n_s$; the $j$-th summand is linear in the centered normalized fluctuation variables of the $j$-th RG step, so that the sum is a martingale; and $N_{K,s}$ collects conditional means and nonlinear terms. For $a\leq\varepsilon\leq\varepsilon_0(s)$ and $\eta=g_3^2\varepsilon$, we show that

uniformly in the number $K$ of RG steps, where $H_s=L/n_s$ is the coarse mesh and $D_s$ is bounded uniformly in $s$. Theorem 31.6 identifies the covariance of the martingale increments at the zero field:

In the linearized Gaussian theory, $C^{a,\varepsilon,L}\otimes I_3$ is the covariance of the circulation of the lattice Maxwell Gaussian field, and $C^{\mathrm{ret}}_{s,a,\varepsilon}$ is the covariance of its conditional expectation given the linearized coarse field. As $a\to0$ and then $\varepsilon\to0$, with $s$ fixed, the right side converges to $\Sigma\otimes I_3$, where $\Sigma$ is the covariance in (2.6). § 2.2 uses these facts to show that the martingale is asymptotically Gaussian.

The normalization by $\sqrt{g_3^2\varepsilon}$ comes from the Gaussian Maxwell field. If $A$ has covariance $g_3^2(-\Delta)^{-1}$ in three dimensions, then $e^{\sigma\varepsilon^2\Delta}A$ has typical size $g_3\varepsilon^{-1/2}$, and its circulation around a loop of length of order $\varepsilon$ has typical size $g_3\sqrt{\varepsilon}=\sqrt{\eta}$.

Two ideas enter the proof of the decomposition. The first concerns the zero-fluctuation reconstruction, in which every fluctuation variable is set to zero. Let $\widehat{L}_i$ be the linearization at the zero field of the $i$-th zero-fluctuation reconstruction map; it takes link logarithms on the lattice of mesh $a_{i-1}$ to link logarithms on the lattice of mesh $a_i$. Each $\widehat{L}_i$ is a constrained minimization of the Maxwell quadratic form. By Theorem 31.3, a product of them agrees, up to a periodic gradient, with a single constrained minimization over all the averaging steps at once:

The circulation around a closed lattice loop annihilates the gradient exactly, and $E_j^a$ obeys bounds that do not depend on $k-j$. The nonlinear Taylor remainders $\widehat{R}_i$ of the reconstruction maps are inserted at their own scale and then propagated to the fine lattice by the linearizations:

Converting link logarithms at mesh $a_i$ into a potential at the scale $a_j$ introduces the factor $a_j/a_i=B^{i-j}$. Since $\widehat{R}_i$ vanishes to second order, every monomial contains at least two inputs, each carrying a contraction factor $\alpha_*$ for every preceding step. For $i-1-j$ preceding steps, the physical rescaling is therefore controlled by

The propagated remainders are therefore summable. The result is an analytic field $\widehat{E}_j^a$ on the fine lattice with the same circulations as the reconstruction, whose coefficients are bounded uniformly in the number of steps; we need no such bound for the composed gauge representative itself.

The second idea concerns conditional means. The conditional law of the fluctuation variables is interacting. Using its density, the local covariance estimates of Section 13, and global color-rotation symmetry, we show that the conditional means of the nonlinear terms are of order $t_j^2$ times the size of the coarse field. Their conditional variances are bounded independently of the number of fluctuation variables.

We keep four kinds of variables apart. The link logarithms $\widehat{\theta}_j$ on the lattice of mesh $a_j$ are dimensionless; the potentials $\widehat{\theta}_j/a_j$ are in physical units; the normalized fluctuation variables $w_j$ are distributed according to the conditional fluctuation kernel $k_j$ of Theorem 31.5, which is supported on a product of balls; and independent standard Gaussian coordinates occur only in the linearized calculation of Theorem 31.6. All circulations are computed for the same gauge representative and with the componentwise heat semigroup.

The inputs come from Sections 4, 8, 10 to 13, 15, 16, and 23. § 2.1.1 fixes notation. § 2.1.2 defines the circulations and computes their covariance under the lattice Maxwell Gaussian field. § 2.1.3 compares linearized reconstructions and controls the nonlinear remainders. § 2.1.4 estimates the coefficients and the conditional means. § 2.1.5 proves the covariance identity, § 2.1.6 proves the decomposition and sums over scales, and § 2.1.7 records that all estimates hold at every fixed positive coupling. All results are used in § 2.2.

Setting and conventions

Fix $L > 0$, an odd integer $M \ge3$, and an upper bound $\Gamma> 0$ on the dimensionless coupling. Let $J$, $B = M^J$, the parameter tuple $\Theta$, the reconstruction maps, and the minimal coarse side $n_{\Gamma}$ be as in Theorem 23.5; they are chosen before the coupling. Fix $0 < \gamma\le\Gamma$ and set $g_3^2 = \gamma/L$. Fix $n_* \in4\mathbb{N}$ and the sequence of lattice sides $N_K = n_*B^K$. For a coarse index $s \ge0$ with $n_s \ge n_{\Gamma}$ and for $K \ge s$, set