2026-10-15 · chapter Yang–Mills and the mass gapAppendiceslist of papers, labels and conditions of the main theorems · Kiichi, Shiori and Rin

Papers that came out of this book

The "stage" in Table 1 is either published (the PDF and LaTeX source are on this site's papers page) or draft (material for the study in this book, not published). Pages are pages of the PDF. The authorship follows the form of publication.

The "window" in the original title of Paper 2 is, in this book's vocabulary, "the crossover band". It is a different thing from the windows of Parts 10 and 11 (an interval of sites, a range of $\hbar$, an interval of times), and in the text we write "band" (13-00 §02). The "rigorous" in the original title of Paper 3 is the original's word, and is independent of the book's label.

Abstract. For $H=-(\hbar^2/2)\Delta_M+V$ on a finite-dimensional configuration space (a compact connected Riemannian manifold, or $\mathbb R^n$ with a constant metric and a smooth confining potential), assume a discrete spectrum, a simple positive ground state $\psi_0$ and a gap $\Delta=E_1-E_0\gt0$. For the Euclidean path measure stationary over all time, the Poincaré constant of the Dirichlet form of stochastic quantisation (Langevin) on smooth time-averaged functionals satisfies $C_P\ge\kappa(\hbar/\Delta)^2$. The test function is the time integral of $F=\psi_1/\psi_0$, and among functionals local in time it is the best. The converse is false in general: for a free particle on the circle, $C_P=\infty$ while $\Delta\gt0$. It touches neither four dimensions nor gauge fields.

Scope. Wilson's lattice with discretised time, gauge fields, infinite volume and the continuum limit are outside the statements of the paper. To use it for four-dimensional lattice gauge theory, assumption (D) on the lattice with discretised time remains (11-04 §08).

Abstract. For the $SU(2)$ Wilson action on $(\mathbb Z/L)^4$, if the links of a "complete family", containing exactly one link of every plaquette, are integrated first, the Haar integral splits star by star, and the marginal density of the remaining links takes the form $e^{-\Phi}$. The second derivative of $\Phi$ along geodesics is at least $-27\beta^2\sum_e\|X_e\|^2$ for every $\beta$ and every configuration. Together with the Ricci curvature 2 of the unit $S^3$ this gives Bakry–Émery curvature $2-27\beta^2$ and the threshold $\beta\lt\sqrt{2/27}=0.2722$. A complete family exists in $\mathbb Z^d$ only when $d\le4$. The mass-gap step is not a Lean theorem; it is on paper, and moreover a sketch.

Scope. All that can be said is that "the constant improves within the Bakry–Émery method"; it does not say that the range of strong coupling has been widened. It says nothing about the continuum limit (5-11).

Abstract (draft). The $SU(2)$ Wilson lattice gauge theory is written in the twist-summed ensemble of Mack–Petkova, and a family of identities without approximation is shown for local centre-vortex insertions. For an involutive insertion $\tau$ aligned with the sector (change of action $\Delta S$) and an observable $H$ odd under $\tau$: $\mathbb E[H]=\mathbb E[H\tanh(\Delta S/2)]$ and $\mathbb E[H\sqrt\rho]=0$ ($\rho=e^{-\Delta S}$). Combining insertions in separated cells and averaging over the position of the insertion gives $|\langle W(C)\rangle|\le\mathbb E\prod_c|\bar t_c|$ (each factor in $[-1,1]$). For insertions driven by reversible heat-bath dynamics, a Crooks-type identity $\mathbb E[H]=\mathbb E[H\tanh(W/2)]$ holds for the work $W$. The slab-confinement theorem for every $\beta$ is recovered as the special case of a thin insertion. The draft places these as rewritings within the framework of Tomboulis–Yaffe, Mack–Petkova and Kovács–Tomboulis, and says nothing about the continuum limit or about the problem itself.

Novelty. The draft lists things such as "local identities in the twist-summed ensemble" as its own contributions, but the book has not checked them against the earlier literature and does not claim novelty (12-05 §08 (d)).

Abstract (draft). Heads towards lattice confinement for four-dimensional $SU(2)$ by the road of the slab $T^2_L\times\mathbb Z^2$ through a small twisted torus $T^2_L$. It states four points. (i) The naive condition "$\theta(L,L)\le\theta_0\lt1$ uniformly in $L$" is also satisfied by four-dimensional $U(1)$, so it does not imply confinement; the correct target is $\ln(1/\theta(L,L))\ge cL^2$ (a local form of the 't Hooft criterion). (ii) Combined with nesting in the planar directions, the target reduces to a single lateral-widening inequality. This inequality fails for $U(1)$ and $\mathbb Z_2$, and for $SU(2)$ in the dilute regime its sign is set by $11/3-2=5/3\gt0$. (iii) Reflection positivity, the identities of centre and duality, and positivity of the transfer matrix hold just as well for $\mathbb Z_2$ and $U(1)$, so structural inputs alone cannot show the lateral-widening inequality. (iv) Monte Carlo on the slab shows the predicted contrast. At one loop, the comparison from maps isometric in the fluctuations can give only the $U(1)$ direction $\beta'\ge\beta+0.023$, while maps that reshape the fluctuations give $\beta-\beta'=0.258$. It shows neither confinement in the continuum limit nor the mass gap.

A planned addition in the record. Adding to Paper 2 the scan with step functions and the $U(1)$ control (the sign flip at $\beta_c\approx1.01$) is listed as one row in the order of work (14-03 §06).

Abstract (draft). In four-dimensional $SU(2)$ Wilson lattice gauge theory on $T^2_N\times T^2_M$, with the 't Hooft twist $n_{12}=1$ put on the small torus $T^2_N$, it considers the bias $\theta(N,M,\beta)$ of the flux $n_{34}$ (the quantity that bounds large planar Wilson loops through the Tomboulis–Yaffe inequality). At fixed $N$, an area law with rate $\approx e^{-2\beta N^2}$ follows elementarily from the cost of the surface of a thin $\mathbb Z_2$ vortex. The draft shows a lower bound that replaces the rate by the minimal action $S_*(N,\ell)$ of a fractional instanton inside a frame frozen to the vacuum. At fixed $M$ it also shows the bound in the opposite direction, sandwiching the exponential rate between $S_{\min}(N,M)$ and $S_*(N,M/2)$. It is not uniform in $N$ and says nothing about the continuum limit.

What is open. The converse bound uniform in $M$ (an upper bound by a dilute gas), and the agreement $S_*=S_{\min}$. The draft reduces the former to a single positivity lemma (14-03 Questions 12a and 12b). The direction of increasing $N$ is the same problem as the middle of the band.

Table 7 lists the machine-checked part of each paper by bundle (a set of Lean files). Every bundle guarantees only the algebra of finite sums and real numbers from the list of assumptions to the conclusion; the Haar measure, path measures, the topology of sectors, that $\theta$ is the bias of a sector, and positivity of the transfer matrix are outside Lean. The location and naming of the bundles follow appendix A-04.

The bundles of Parts 12 and 13 come to 11 files and 32 theorems and lemmas in all. In addition there is a separate bundle (150 theorems) containing the slab-confinement theorem and the container lemmas (SlabConfine, SlabAverage, FrameLemma, CommutingFlips and others), which 12-03 and 12-04 refer to. It is not a bundle of the papers, and it has passed the same re-check. The names and location of the bundles will be decided when they are added to the table of bundles in appendix A-04.

What this chapter can say

Sources and reproduction

Terms used in this chapter — published, draft, the main theorem, label, bundle, the twist-summed ensemble, the crossover band, the lateral-widening inequality, fractional instanton, container. Their meanings are in the A-02 Glossary.

Next page to read: Contents. Which condition rows of the book the statements of the papers bear on is in 14-02 The ledger of conditions.

← A-06 Index of assumptionsContents