2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircaseentrance, staircase, five quantities · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

Entrance — the staircase, and the random walk, the ruler and the clock

What this chapter shows — Part 10 is the part that examines a road to the mass gap with Assumption H (the time over which fluctuations are smoothed out grows as a power of the square of the correlation length) as its axis. Assumption H is one sentence that restates a quantity of the problem statement (the gap in Euclidean time) as a quantity that can be computed on the lattice and that machine checking can reach (the relaxation time of Langevin dynamics). This part divides that sentence into a staircase of four steps, and lays out how far the claim has been built on the first step (the path measure of the double well in 0+1 dimensions). In this chapter, after looking at the picture of the staircase and the map of the part, we set up quantities such as the correlation length $\xi$ and the Poincaré constant $C_P$ one by one, with the metaphors of the random walk, the ruler and the clock.

Prerequisite chapters — 2-10 (relaxation time and the dynamical exponent), 4-11 (definition of the mass gap and the correlation length), 2-07 (the path integral and imaginary time).

Said first — This chapter is an entrance that lines up quantities through metaphors, and proves nothing. Assumption H is an unproved assumption. The precise definitions are placed in 10-01 (the side of the four-dimensional lattice) and 10-04 (the side of the double well).

The order of this chapter

A map of the part — the list of eighteen chapters, and how to read the labels
The four steps of the staircase — from the double well to the mass gap in the limit (Figure 1)
The ruler — the correlation length ξ — in how many steps correlations die out
The smearing time — the Poincaré constant C_P — L² steps to even out a length L (Table 3). Detour: why the random walk goes as a square
The scale and the clock — the remaining three quantities — the number of rescalings, the graduation of the clock, the clock that runs slow
Assumption H in one sentence — smearing time ≍ ruler². "Some f" and "every f"
How to walk through Part 10 — in which chapter each metaphor becomes a precise sentence

A map of the part — the list of eighteen chapters, and how to read the labels

This part is a study with Assumption H (the time over which fluctuations are smoothed out grows as a power of the square of the correlation length) as its axis. H is not the mass gap itself but a sentence that restates it as a quantity of a dynamics measurable on the lattice (10-01 §05), and this part builds that sentence on the first step of a four-step staircase — the path measure of the double well in 0+1 dimensions. We split the claim of the first step into four conditions, and for each condition lay out in one table the rows where a proof is given, rows citing original sources, computer-assisted rows, rows of numerical reading, and assumption rows.

The eighteen chapters of Part 10

10-00 Entrance — the staircase, and the random walk, the ruler and the clock [Y2] (this chapter)
10-01 The line of construction and Assumption H [Y2]–[Y3]
10-02 Roads tried on the four-dimensional lattice, and walls [Y3]
10-03 The four steps of the staircase — why we start from 0+1 dimensions [Y2]
10-04 The path measure of the double well, and kinks [Y2]–[Y3]
10-05 The statement of the first step — four conditions and Proposition 4.5 [Y3]
10-06 (H1) Local Poincaré within a phase [Res]
10-07 The map of (H2) — seven routes and the seam of the ring [Y3]
10-08 What computer-assisted means — the conditions on the chain inside the window [Y2]–[Y3]
10-09 The sector framework and the three strands of route 5 [Res]
10-10 The list of detour plans — types of breakage [Y3]–[Res]
10-11 (H3)(H4) and the constant of the core [Res]
10-12 The integrated table and the dependency graph [Y3]
10-13 Lean statements and the limits of machine checking [Y2]–[Y3]
10-14 Reconciliation — the numbers of the record and the known numbers [Y3]
10-15 What remains and its cost — the distance to a theorem [Y3]
10-16 What is not claimed [Y2]
10-17 Insights and small questions [Y2]–[Y3]

The chapters run: why this assumption (10-00 to 10-02), the statement of the first step (10-03 to 10-05), the four conditions (10-06 to 10-11), the overall tables (10-12, 10-13), and reconciliation and what remains (10-14 to 10-17). The marks [Y2], [Y3] and [Res] are reader levels: [Y2] is second-year, [Y3] third-year and above (using measures, operators and Poincaré inequalities), and [Res] is research level. This chapter is [Y2]. Viewpoints other than Assumption H, and new views that take H as a premise, are placed in Part 11 Further considerations.

The labels of this part

In this part the strength of a claim is divided by five labels (the labels of 0-01 §03, refined for the material of this part). Each sentence gets one label; a sentence needing two is split.

Table 1 The five labels of this part (Lean is a column outside the labels)
Label	Meaning	What it does not guarantee
Proved here	Elementary. The proof is placed in the text or at the end of the chapter	That it contains no error (it is placed in a form the reader can check)
Cited	A known theorem cited from the original source (or a secondary source whose statement was checked)	That it is a result of this book
Computer-assisted	Computer-assisted. Checked as an inequality by interval arithmetic (premises such as rounding conventions are written on the same row)	That it holds outside the range computed
Reading	A reading. Floating-point numbers, Monte Carlo, fits. The kind (a point value, a lower bound over a family, etc.) is added	That it is a proof of an upper or lower bound
Assumption	An assumption. A row put down without a proof. It is given a name	That it holds

Lean is a column separate from the five labels. A Lean theorem is the "algebra from assumptions to conclusion" checked by machine, and the theorem name and the assumptions of that theorem are written on the same row. Whether the assumptions are true is outside the check (10-13).

The four steps of the staircase — from the double well to the mass gap in the limit

Assumption H is the sentence "in some family of measures, the time over which fluctuations are smoothed out grows as a power of the square of the correlation length", and it can be stated independently of dimension. So we line up sentences of the same form in four steps, from easy places to hard ones.

Figure 1 The four steps of the staircase. This part deals with the inside of the first step; the arrows from the second step upwards stand for "a sentence of the same form", not for theorems of implication

Written out in words, Figure 1 is as follows. From the bottom: the first step = the double well in 0+1 dimensions (the path measure in imaginary time of a quantum-mechanical model); the second step = the non-abelian gauge theory on the four-dimensional lattice (the Wilson measure); the third step = the continuum limit with the lattice spacing taken to 0, and that limit satisfying the Osterwalder–Schrader axioms; the fourth step = the mass gap of the limiting theory.

Table 2 The objects of the four steps, and what can be called "closed" at each step
Step	Object	Form in which it can be called closed	State in this book
First step	The path measure of the double well in 0+1 dimensions (parameter $\hbar$ )	Show the upper side of the power form of H (an upper bound on $C_P$ ), with a table of finitely many assumption rows	Split into four conditions (H1)–(H4), and up to the algebra on top of them (Proposition 4.5). Every condition leaves Assumption rows
Second step	The Wilson measure on the four-dimensional lattice (parameter $\beta$ )	A sentence of the same form, as $\beta\to\infty$	Assumption H remains an assumption. Only estimates from below (10-02)
Third step	The limiting measure as the lattice spacing $a\to0$	The limit exists and satisfies the OS axioms	Not dealt with (for what is known to have been reached, 6-07, 6-10)
Fourth step	The reconstructed quantum field theory	The gap of the Hamiltonian is positive (4-11)	Not dealt with

Even if the first step is closed, there is no guarantee that it lifts to the second step and beyond. What has a prospect of lifting is "the form of the sentence" and "the form of the tools"; constants and ranges of parameters do not lift. The reasons are given from both sides in 10-03 §06.

The ruler — the correlation length ξ

Fluctuations are similar between nearby places and become unrelated when far apart. The length over which they become unrelated is the correlation length $\xi$ . As a metaphor, it is the one ruler the measure carries.

As a formula, the correlation of two points (the expectation of the product with the mean subtracted) decreases exponentially with distance $r$ , and the length of that decrease is $\xi$ .

\langle O(0)\,O(r)\rangle-\langle O\rangle^2\ \approx\ C\,e^{-r/\xi},\qquad \xi=\frac1m \tag{1}

$m$ is the gap of the transfer matrix (or transfer operator), that is, the difference between the lowest eigenvalue and the next, and $r$ is the distance in the imaginary-time direction. That the decay of correlations and the gap are the same quantity is in 4-11 §02, and the restatement as a correlation length is in 4-11 §07 Cited.

Example 1 (free field). For the free field on a one-dimensional lattice (a Gaussian chain with neighbours joined by springs, mass $m_0$ ), correlations decrease exactly as $e^{-r/\xi}$ , and $\xi$ is determined by $m_0^2=2(\cosh(1/\xi)-1)$ . For $m_0=1$ , $\xi\approx1.04$ ; for $m_0=0.1$ , $\xi\approx10.0$ (8-03 §05).

Example 2 (double well). In the quantum-mechanical double well (2-05 §07), the gap is the tunnelling splitting $\Delta$ , determined by passing through between the two valleys, and it becomes exponentially small the higher the hill. Seen through paths in imaginary time, the correlation length is $\xi=\hbar/\Delta$ , and as $\hbar$ is made smaller it grows at the rate $e^{S_0/\hbar}$ ( $S_0$ is the action of the path that moves from valley to valley). This property, that the ruler grows exponentially, is one of the reasons for choosing the model of the first step (10-03 §04, 10-04 §03).

The $\xi$ measurable on the lattice is a number in units of one graduation of the lattice (5-09 §01). The physical length is $\xi$ times the lattice spacing $a$ , $a\xi$ , and in the continuum limit $a\xi$ must remain finite while $a\to0$ . That is, $\xi$ in lattice units must grow to infinity (5-09 §05). The $\hbar\to0$ of the double well is the smallest specimen of this regime, "the ruler becomes infinitely long compared with the graduations".

The smearing time — the Poincaré constant C_P

Drop a drop of ink on paper, and the ink gradually smears and spreads, and in the end becomes uniformly pale. What the smearing really is, is a random walk of the ink particles. Each particle moves one step left or right at random at each tick. After $t$ steps the spread of the particles (the root-mean-square distance) is $\sqrt t$ , and evening out a strip of length $L$ takes on the order of $L^2$ steps.

The Poincaré constant $C_P$ is this "longest time until evening out is finished", defined in general for fluctuations. As a metaphor, it is the reading of the one clock the measure carries: the smearing time.

We write it as a formula. Given a probability measure $\mu$ and the "size of the change from place to place" $\mathcal E(f)$ of a function $f$ (the Dirichlet form),

\operatorname{Var}_\mu(f)\ \le\ C_P\,\mathcal E(f)\quad(\text{for every } f),\qquad C_P=\sup_f\frac{\operatorname{Var}_\mu(f)}{\mathcal E(f)} \tag{2}

the best constant is $C_P$ (1-10 §07). If there is a function with large variance but small change, that is, "a large, slowly varying fluctuation", $C_P$ becomes large. For the diffusion dynamics (Langevin dynamics) with $\mu$ as its equilibrium, $1/C_P$ is the gap of that dynamics, and the slowest fluctuation decreases by a factor $1/e$ every time $C_P$ Cited.

Example: the random walk on a single road

Consider a single road of $L$ cells in a row (reflecting at both ends). $\mu$ is the uniform distribution, and $\mathcal E(f)=\frac1L\sum_{i}(f_{i+1}-f_i)^2$ . Then the $C_P$ of (2) is the reciprocal of the eigenvalue next after 0 of the Laplacian matrix of the road, and can be written in closed form.

C_P(L)=\frac{1}{2\,(1-\cos(\pi/L))}\ \approx\ \frac{L^2}{\pi^2}\qquad(L\to\infty) \tag{3}

Proved here Proof: $f_i=\cos(\pi(i-\tfrac12)/L)$ ( $i=1,\dots,L$ ) is an eigenvector of the Laplacian matrix of the road reflecting at both ends, with eigenvalue $2(1-\cos(\pi/L))$ (use $\cos(A+B)+\cos(A-B)=2\cos A\cos B$ on the sum of the two neighbouring terms, and check that at both ends the mirror conditions $f_0=f_1$ , $f_{L+1}=f_L$ hold). The other eigenvectors are $\cos(\pi k(i-\tfrac12)/L)$ ( $k=0,\dots,L-1$ ), with eigenvalues increasing with $k$ . Since $\mu$ is uniform, $\operatorname{Var}$ and $\mathcal E$ are divided by the same $1/L$ , and the supremum of the ratio is the reciprocal of the eigenvalue next after 0. ∎

Proved hereTable 3 $C_P$ of the random walk on a single road (values of the closed formula (3)). $C_P/L^2$ approaches $1/\pi^2\approx0.10132$
$L$	Gap $2(1-\cos(\pi/L))$	$C_P$	$C_P/L^2$
16	0.0384294	26.02	0.10165
32	0.00963055	103.8	0.10140
64	0.00240909	415.1	0.10134
256	0.000150596	6640	0.10132

Doubling $L$ makes the time until evening out is finished four times as long.

The $C_P$ used in this part replaces this single road by "the space of configurations of whole paths". For the double well, the measure $e^{-S/\hbar}$ on the whole of the paths $x(\tau)$ in imaginary time is $\mu$ , $\mathcal E$ is the expected square of the change when a path is moved slightly, and Langevin dynamics diffuses whole paths along an "artificial time" $s$ (10-04 §01). On the four-dimensional lattice, $\mu$ is the Wilson measure and what moves are the angles of the links (10-01, 6-09).

For the free field, the slowest fluctuation is in the direction of shifting the whole path uniformly, and its rate is the square of the mass, $m_0^2$ . So $C_P=1/m_0^2$ , and from $m_0^2=2(\cosh(1/\xi)-1)$ of Example 1, $C_P/\xi^2$ at $\xi=1,4,16,256$ is 0.921, 0.995, 0.9997, 0.999999, approaching 1 Proved here (values of the closed formula). The smearing time is the square of the ruler: that is the free field's answer.

Precisely, the $C_P$ meant here is the Poincaré constant of "the distribution of configurations over the whole of spacetime (in 0+1 dimensions, imaginary time)" (the path measure). The Poincaré constant of the distribution at a single time (the $|\psi_0|^2$ of the ground state) is a different quantity, and it is tied to the gap of the Hamiltonian. The distinction among the three $C_P$ is placed in 10-01. Known examples in which the power $z$ (the dynamical exponent) with which $C_P$ grows in $\xi$ departs from 2 are in 2-10.

Detour: why the random walk goes as a square

Detour

A subsection you may skip. The position $X_t$ after adding $t$ independent steps $\pm1$ has mean 0, and its variance is $t$ times the variance 1 of one step. This is because for sums of independent things, variances add (1-08).

\mathbb E[X_t^2]=\sum_{i=1}^{t}\mathbb E[\varepsilon_i^2]+\sum_{i\ne j}\mathbb E[\varepsilon_i]\,\mathbb E[\varepsilon_j]=t \tag{4}

The spread is $\sqrt t$ , so reaching distance $L$ takes $t\approx L^2$ . A wave advances one step per tick, but a random walk goes back and forth, so reaching far is delayed by a square. The "square" of Assumption H is this square of the random walk.

The scale and the clock — the remaining three quantities

Besides the ruler $\xi$ and the smearing time $C_P$ , there are three quantities that have been used in the record for the four-dimensional lattice. They are not the main players on the first step (the double well), but the reason for putting Assumption H as a "power statement" lies here, so we only line up the metaphors.

① The rulerThe correlation length

\xi

the reciprocal of the gap

m

(§03)

② The smearing timeThe Poincaré constant

C_P

L^2

steps to even out a length

L

(§04)

③ The number of rescalingsThe bookkeeping length

\xi_{\rm RG}

the number of steps that coarsen the map

④ The graduation of the clockThe normalisation of

\mathcal E

in what unit the same fluctuation is measured

⑤ The clock that runs slowRenormalisation of stochastic timethe clock itself stretches with scale

③ The number of rescalings. In four-dimensional lattice Yang–Mills there is no way to obtain $\xi$ by formula as a function of the coupling $\beta$ . So we coarsen the scale of the map one step at a time, and make separately a length $\xi_{\rm RG}$ that counts how many times it takes to reach a picture of fixed coarseness. In the one-loop count, raising $\beta$ just a little makes $\xi_{\rm RG}$ grow exponentially (4-07). That $\xi_{\rm RG}$ and $\xi$ agree up to a constant factor is an assumption. The definitions and tables have been moved to 10-01 and 10-02.

④ The graduation of the clock. The $C_P$ of (2) changes value depending on the unit in which $\mathcal E$ is measured. Multiplying $\mathcal E$ by $c$ multiplies $C_P$ by $1/c$ . The same happens for the double well. Between the $\mathcal E$ that measures paths by the canonical field $x/\sqrt\hbar$ and the $\mathcal E_x=\mathcal E/\hbar$ that measures them by the position $x$ as it is, $C_P^{(x)}=\hbar\,C_P$ Proved here LeanratioR_x_normalisation (no assumptions; if $C_P^{(x)}=\hbar C_P$ , the ratio is also multiplied by $\hbar$ ). The difference of graduation shows in "the width of a constant factor", but the factor of a power of $\hbar$ is small compared with $\ln\xi$ , so it does not show in "the power".

⑤ The clock that runs slow. Langevin time itself stretches and shrinks when the scale is changed. In the perturbation theory of stochastic quantisation, the graduation of stochastic time also has to be renormalised (6-09)Cited (at the level of perturbation theory). On the four-dimensional lattice its effect is a factor of a fractional power of $\beta$ , and this too does not show in the power. How to compare the four clocks is placed in a detour of 10-01.

Precisely, ③–⑤ move sentences stated in "the width of a constant factor" (band statements), and do not move sentences stated in "a power". That is why Assumption H in this part is a power statement.

Assumption H in one sentence — smearing time ≍ ruler²

With the quantities in place, we put Assumption H into one sentence as a metaphor: "the smearing time grows with the same power as the square of the ruler". When a family of measures is lined up by one parameter, and moving the parameter makes the ruler grow without bound,

\text{H}:\quad \frac{\ln C_P}{\ln\xi}\ \longrightarrow\ 2\qquad(\xi\to\infty) \tag{5}

On the four-dimensional lattice the parameter is $\beta$ with $\beta\to\infty$ ; for the double well the parameter is $\hbar$ with $\hbar\to0$ . For the free field it holds with $C_P/\xi^2\to1$ , as in §04. The question is whether it holds on the four-dimensional lattice with interaction, and the first step is to build the upper side on the double well, which is easier than that.

For the double well, the "width of a constant factor" sentence stronger than (5) ( $C_P\le K\xi^2$ ) is read as being on the side that breaks. This is because the lower bound on the path $C_P$ grows from 1 times to 6.5 times $\xi^2$ as $\hbar$ is lowered Reading (lower bound over a family, step $\varepsilon=0.25$ , $\hbar\in[0.07,0.4]$ , 10-04 §04). In the count at the level of physics that counts kinks, the growth is first order in $S_0/\hbar\approx\ln\xi$ (the numbers in this range alone cannot distinguish it from exponential growth). If it is first order, the power statement (5) is kept. That is why the goal of the first step is not the band statement but the upper side of the power statement, $C_P\le K\xi^2S_0/\hbar$ (10-03 §05).

"Some f" and "every f"

The $C_P$ of (2) is a supremum, so bounding it from below and from above differ in difficulty. To bound it from below ( $C_P$ is large), it is enough to show one test function with large $\operatorname{Var}(f)/\mathcal E(f)$ . To bound it from above ( $C_P$ is not too large), the inequality must be shown for every $f$ .

C_P\ge c\,\xi^2\iff\exists f:\ \frac{\operatorname{Var}(f)}{\mathcal E(f)}\ge c\,\xi^2,\qquad C_P\le C\,\xi^2\iff\forall f:\ \frac{\operatorname{Var}(f)}{\mathcal E(f)}\le C\,\xi^2 \tag{6}

As a metaphor, that smearing is slow can be said by showing one stain, but that it is not too slow must be said for every stain. What lies on the side of the mass gap is the upper side ( $C_P$ not too large), and test functions and numerics reach only the lower side. The algebra that rereads the upper side as a lower bound on the mass, and the definitions of the upper and lower halves H↑ and H↓, are placed in 10-01.

Precisely, the lower bound $C_P\ge\xi^2$ for the double well is a proposition of the form that a family of test functions local in time (functions of a single time summed over time) reaches up to equality (Proposition L of 10-04 §04). On the upper side, even for the first step there are only conditional propositions.

How to walk through Part 10 — in which chapter each metaphor becomes a precise sentence

Each metaphor of this chapter is replaced once by a precise sentence somewhere in Part 10. Table 4 matches metaphors, symbols and the chapters that make them precise.

Table 4 Metaphors and the corresponding precise sentences
Metaphor	Symbol	Precise definition or sentence	State
The staircase	Four steps	10-03	The convention for how this part is built
The ruler	$\xi=1/m$ (for the double well $\xi=\hbar/\Delta$ )	4-11 §07, 10-04 §02	Cited Definition
The smearing time	$C_P$ (the path measure)	10-01, 10-04 §01	Cited Definition. The value on the four-dimensional lattice is unknown
The number of rescalings	$\xi_{\rm RG}$	10-01, 10-02	Convention (that it agrees with $\xi$ is Assumption)
The graduation of the clock, the clock that runs slow	Normalisation of $\mathcal E$ , stochastic time	10-01	Convention, level of perturbation theory
Smearing time ≍ ruler²	Assumption H (power statement)	10-01, 10-03 §05	Assumption
The upper side of the first step	$C_P\le K\xi^2S_0/\hbar$ (Proposition 4.5)	10-05	Algebra on top of four conditions. The conditions leave Assumption rows

To follow the line first: 10-03 → 10-04 → 10-05 → 10-12 (the integrated table) → 10-15 (the distance to a theorem). To see the relation with the four-dimensional lattice first: 10-01 → 10-02. To see what is not claimed first, 10-16 is the shortcut. The Lean statements and assumptions are in 10-13, and how to read Lean is in A guide to Lean.

What this chapter can say

Said	Not said
Cited That the correlation length and the gap are the same quantity ((1)); the definition of the Poincaré constant, and that the gap of the diffusion dynamics is its reciprocal ((2))	That $\xi$ diverges as $\beta\to\infty$ in four-dimensional lattice Yang–Mills; the value of $C_P$
Proved here The closed form (3) of $C_P$ for the random walk on a single road; $C_P/\xi^2\to1$ for the one-dimensional free field; the mean-square distance $=t$ of the random walk ((4))	That the same ratio agrees in power in a model with interaction (Assumption H)
Proved here That multiplying the normalisation by $\hbar$ also multiplies $C_P$ by $\hbar$ (④)Lean`ratioR_x_normalisation`	That the bookkeeping length $\xi_{\rm RG}$ of the four-dimensional lattice and the correlation length $\xi$ agree up to a constant factor
Reading That the lower bound on the path $C_P$ of the double well grows from 1 times to 6.5 times $\xi^2$ (lower bound over a family, $\varepsilon=0.25$ , $\hbar\in[0.07,0.4]$ )	An upper bound on $C_P$ for the double well. The upper side of the first step is conditional on the four conditions (10-05)
The way of building it as a four-step staircase (a convention)	Implication from the first step to the second step and beyond. About the fourth step (the mass gap in the limit), this chapter shows nothing

Sources and reproduction

Item	Kind	Source / tools
Correlation length and gap, correlations in imaginary time	Cited	The table of sources of 4-11.
Poincaré constant, gap of diffusion	Cited	The table of sources of 1-10.
$C_P$ of the random walk on a single road ((3), Table 3)	Proved here	The proof in §04 of the text. The values in the table are values of the closed formula (3).
Tunnelling splitting of the double well, and $S_0$	Cited	(9)(10) of 2-05 §07 and its table of sources.
Reading of the lower bound on the path $C_P$ (§06)	Reading	10-04 §04 and its table of sources.
Rereading the normalisation (④)	Proved here, Lean	Theorem `ratioR_x_normalisation` (no assumptions). The statement is in 10-13.
Bookkeeping length, renormalisation of stochastic time	Convention, Cited (perturbation theory)	The tables of sources of 4-06, 4-07 and 6-09. The detailed formulas are in 10-01.
Figure 1 (the staircase)	Convention	A schematic drawing of how this part is built.

Words used in this chapter — correlation length, Poincaré constant, Dirichlet form, path measure, staircase (four steps), Assumption H (power statement), the five labels (proved here, cited, computer-assisted, reading, assumption), tunnelling splitting. Their meanings are in A-02 Glossary.

Next chapter to read: 10-01 The line of construction and Assumption H — splits the problem into eight statements, and defines Assumption H and its upper and lower halves. To go straight to the first step, 10-03 The four steps of the staircase.

Revised 2026-10-09: rewritten to match the revised chapter structure of Part 10.