2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasestaircase, power form of H, form of Proposition 4.5 · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

The four steps of the staircase — why we start from 0+1 dimensions

What this chapter shows — Assumption H is a sentence of a dimension-independent form: "the relaxation time of a path measure grows as a power of the square of the correlation length of that measure". So we divide H into four steps (the double well in 0+1 dimensions / the four-dimensional lattice, non-abelian / the continuum limit and the OS axioms / the mass gap in the limit), and give the reasons for choosing the double well for the first step. The statement of the first step is "the power form of H", $\ln C_P/\ln\xi\to2$ , and putting its upper side in the form $C_P\le K\xi^2S_0/\hbar$ (Proposition 4.5) is the centre of the assembly of the first step. Proposition 4.5 is a conditional proposition on top of four conditions, and is not proved. We give both the prospects of lifting from the first step to the second and beyond, and the reasons it does not lift.

Prerequisite chapters — 10-00 (the entrance, the five labels), 2-07 (the path integral and imaginary time), 2-05 §07 (the tunnelling splitting of the double well). Definitions on the side of the four-dimensional lattice are in 10-01.

Said first — Assumption H is an assumption both on the four-dimensional lattice and for the double well. Proposition 4.5 is algebra on top of conditions (H1)–(H4), and every one of the four conditions leaves assumption rows. This chapter fixes the form of the statement and proves nothing new.

The order of this chapter

H is a sentence of a dimension-independent form — a family of measures, the correlation length, the relaxation time
Definition of the four steps — the form in which each step can be called "closed" (Table 1)
Why we do not attack it on the four-dimensional lattice as it is — there are no tools for the upper side
Conditions on the model chosen for the first step — a ruler that grows exponentially, two phases, quantities that can be compared (Table 2)
The power form of H and the form of Proposition 4.5 — the lower bound ξ² and the target for the upper bound
What lifts and what does not (Table 3)
A map inside the first step — a guide to 10-04 to 10-11

H is a sentence of a dimension-independent form — a family of measures, the correlation length, the relaxation time

Three things appear in Assumption H: a family of probability measures $\mu_s$ lined up by one parameter, the correlation length $\xi(s)$ of each $\mu_s$ , and the Poincaré constant $C_P(s)$ of each $\mu_s$ (the relaxation time of the Langevin dynamics having that measure as equilibrium). When moving the parameter gives $\xi(s)\to\infty$ , Assumption H is the one sentence

\text{H}:\quad \lim\frac{\ln C_P(s)}{\ln\xi(s)}=2 \tag{1}

This sentence does not say on what space the measure lives. On the four-dimensional lattice $\mu_s$ is the Wilson measure with $s=\beta\to\infty$ (10-01); for the double well in 0+1 dimensions $\mu_s$ is the measure on paths in imaginary time $e^{-S/\hbar}$ with $s=1/\hbar\to\infty$ (10-04). Sentences of the same form can be lined up with the object changed.

Example: for the free field (a Gaussian chain of mass $m_0$ ), $C_P=1/m_0^2$ and $\xi\approx1/m_0$ , and (1) holds as $m_0\to0$ (10-00 §04)Proved here. With interaction, new mechanisms that slow relaxation enter (crossing barriers, moving between configurations in different phases), and (1) is no longer trivial.

Precisely, "Assumption H" in this part refers only to the sentence (1) at the power level. A sentence stated within the width of a constant factor (a band statement) is called by another name, not H (10-01). Also, the four conditions of the first step are written "conditions (H1)–(H4)", always with the word "conditions" so as not to be confused with Assumption H. The names of the conditions are in Lean theorem names, so they are not changed.

Definition of the four steps — the form in which each step can be called "closed"

We place four steps, from easy places to hard ones.

The first step: the path measure of the double well in 0+1 dimensions. Parameter $\hbar\to0$ .
The second step: non-abelian gauge theory on the four-dimensional lattice (the Wilson measure of SU(2) or SU(3)). Parameter $\beta\to\infty$ .
The third step: the continuum limit as the lattice spacing $a\to0$ . That the limiting measure exists and satisfies the Osterwalder–Schrader axioms (4-10).
The fourth step: the mass gap of the limiting theory (4-11).

Table 1 Object, target, tools and state in this book, step by step
Step	Object	Form in which it can be called "closed"	Main tools	State in this book
First step	The path measure of the double well	State the upper side of the power form of H with a table of finitely many named assumption rows, and fill each row with a proof, an original source, or computer assistance	Splitting of the variance, local Poincaré, reduction to the box, interval arithmetic	Up to Proposition 4.5 (algebra on top of conditions (H1)–(H4)). Every one of the four conditions leaves Assumption rows (10-05, 10-12)
Second step	The Wilson measure on the four-dimensional lattice	The sentence (1) of the same form, as $\beta\to\infty$	Prospect of transferring the types made on the first step (§06)	Assumption H is Assumption. Only estimates from below have been reached (10-02)
Third step	The measure of the continuum limit	Existence of the limit and the OS axioms	Sequences of renormalisation-group steps, constructive field theory (Part 6)	Not dealt with
Fourth step	The reconstructed quantum field theory	The gap of the Hamiltonian is positive	The OS reconstruction (4-10)	Not dealt with

"Closed" for the first step means putting the proposition of the upper side into a form where "which rows are proofs and which are assumptions" can be read on one table. Only when every row is filled with a proof, an original source or computer assistance, and the stage of Proposition 4.5 goes through as a single proof, does the first step become a theorem. At present it is not so (10-12).

To go up from the second step, one more bridge is needed between each pair of steps. Even if the sentence (1) of the second step holds, that alone does not give the existence of the continuum limit (the third step), and the gap in the limit (the fourth step) is a form that can be reread only after adding to the upper side of (1) the assumptions joining the correlation length to physical length (the chains of 10-01). This part has no theorem of implication for any bridge between steps.

Why we do not attack it on the four-dimensional lattice as it is — there are no tools for the upper side

In the record for the four-dimensional lattice (10-02), every road through test functions, gauge fixing and spectroscopy ended in lower bounds on $C_P$ or upper bounds on the mass, and no tool reaching the upper side ( $C_P$ not too large, over every $f$ ) has been found. There are estimates on the strong-coupling side ( $C_P$ bounded) and the Bakry–Émery type that works for convex measures, but neither reaches the regime $\xi\to\infty$ . In an example applying a known volume-uniform criterion to the path lattice of the double well, the upper bound obtained is of the type $e^{c\xi}$ (doubly exponential in $\hbar$ ), read as far from a power statement Cited (criterion: Theorem 2.1 of Bauerschmidt–Dagallier) Reading ( $c\approx1.4$ to $1.8$ ; numbers for $\varepsilon=0.25$ , $\hbar\in[0.1,0.4]$ ). So we first make an "assembly from above" on the smallest test bed where H is expected to hold. The detailed roads and walls are in 10-02.

Conditions on the model chosen for the first step — a ruler that grows exponentially, two phases, quantities that can be compared

There are three conditions we ask of the model of the first step.

The ruler grows exponentially in the parameter. The correlation length of the four-dimensional lattice grows exponentially in $\beta$ in the one-loop count (4-07). To make a test bed that does not confuse powers with exponentials, a ruler growing at the same rate is needed.
The slow motion is the motion of "changing phase". What slows relaxation is a rare event of moving between two (or more) phases. There are two scales: fluctuations inside the two phases are fast, and only the motion of changing phase is slow.
The quantities compared are known exactly. The correlation length can be found to high precision by an independent method. If what it is compared with is blurred, the growth of $C_P/\xi^2$ will be misread.

The path measure of the double well in 0+1 dimensions, $V(x)=(x^2-1)^2/8$ , satisfies all three. The ruler grows exponentially as $\xi=\hbar/\Delta\propto\hbar^{1/2}e^{S_0/\hbar}$ ( $\Delta$ is the tunnelling splitting, $S_0=2/3$ the action of the path moving from valley to valley); the two phases are "the intervals where the path is in the right valley" and "those where it is in the left valley", and the event of changing phase is a kink (a short transition with action $S_0$ ). $\Delta$ is found by two routes, the semiclassical formula ((10) of 2-05 §07) and numerics of the transfer operator (10-04).

The action $S_0$ of one kink, and the correlation length $\xi\propto e^{S_0/\hbar}$ . This pair has, in the smallest dimension, the structure in which "the action of an instanton" and "a correlation length growing exponentially" sit side by side on the four-dimensional lattice. This is an analogy, however, not a theorem of correspondence.

Table 2 How the double well and the four-dimensional lattice line up (the right column is an analogy, not a theorem of correspondence)
Item	Double well (first step)	Four-dimensional lattice (second step)
Measure	The path measure $e^{-S/\hbar}$ (the line of imaginary time)	The Wilson measure (four-dimensional lattice)
Parameter and limit	$\hbar\to0$	$\beta\to\infty$
Ruler	$\xi\propto\hbar^{1/2}e^{S_0/\hbar}$ Cited (leading semiclassical term)	$\propto e^{3\pi^2\beta/11}$ from one-loop bookkeeping (SU(2); that it agrees with $\xi$ is Assumption)
Slow degrees of freedom	The positions of kinks (action $S_0=2/3$ )	The positions, sizes and orientations of instantons, the density of topological charge (analogy)
Lower side	$C_P\ge\xi^2$ (Proposition L, 10-04 §04)	Lower bounds from test functions (10-02)
Upper side	Conditional on four conditions (Proposition 4.5)	None

The first step is an easy model, but not too easy. No known theorem giving the upper side of the power form of H was found in what we searched. There are theorems of Eyring–Kramers type that hold the length of the interval fixed and take $\hbar\to0$ , but the regime that makes the length of the paths $T$ infinite first ( $T\gg\xi$ ) is outside the scope of those theorems Cited (Brooks–Di Gesù 2026 state explicitly in the text that the regime of infinite length and the family of translations are outside their scope).

The power form of H and the form of Proposition 4.5 — the lower bound ξ² and the target for the upper bound

The statement of the first step, for the path measure of the double well (taking $T\to\infty$ first), is

\text{power form of H}:\quad \lim_{\hbar\to0}\frac{\ln C_P(\hbar)}{\ln\xi(\hbar)}=2 \tag{2}

Setting $R:=C_P/\xi^2$ , $\ln C_P/\ln\xi=2+\ln R/\ln\xi$ , so (2) is the same as " $\ln R$ is small compared with $\ln\xi$ ".

The lower side. $R\ge1$ , that is $C_P\ge\xi^2$ , holds for every well in 0+1 dimensions (Proposition L; the proof is in 10-04 §04).

The target for the upper side. The position of a kink moves with diffusion coefficient $D=\hbar/S_0$ , and the time to cross a length $\xi$ by diffusion is $\xi^2/D=\xi^2S_0/\hbar$ (10-04 §03). So we put the target for the upper side in the form

C_P\ \le\ K\,\xi^2\,\frac{S_0}{\hbar}\qquad(K\ \text{does not depend on}\ \hbar\ \text{}) \tag{3}

In the count at the level of physics (counting kinks as a dilute gas) it is read as $C_P\approx\xi^2S_0/\hbar$ , and (3) is that with a margin of a constant factor Assumption (the count is heuristic, not a proof).

From (3) to (2). From (3) and $R\ge1$ , $1\le R\le KS_0/\hbar$ . When $\hbar$ is small, $\ln\xi\ge s/\hbar$ (for some $s\gt0$ ), so

0\ \le\ \frac{\ln R}{\ln\xi}\ \le\ \frac{\hbar\,\bigl(\ln(KS_0)+\ln(1/\hbar)\bigr)}{s}\ \longrightarrow\ 0\qquad(\hbar\to0) \tag{4}

and (2) follows Proved here Leanpower_H_of_kink_ratio (assumptions: $0\lt K$ , $0\lt s$ , $0\lt\delta$ , and for $0\lt\hbar\lt\delta$ , $1\le R\le K/\hbar$ and $s/\hbar\le\ln\xi$ . The $K$ of Lean is the $KS_0$ of the text). $\ln\xi\ge s/\hbar$ is a row coming from the leading semiclassical term $\ln\xi\approx S_0/\hbar$ , and in the lattice model it is Reading. The values are in 10-04 §07.

The form of Proposition 4.5

(3) is not proved. The assembly of the first step splits (3) into the following four conditions, and puts it in the form of a conditional proposition: on top of them, (3) follows.

Proposition 4.5 (conditional; statement only)

For the path measure of the double well in 0+1 dimensions, assuming conditions (H1)–(H4), taking $T\to\infty$ first, as $\hbar\to0$ , $C_P\le K\xi^2S_0/\hbar$ ( $K$ depends only on the constants of the conditions). Hence the power form of H (2) follows.

Condition (H1) Local Poincaré within a phase
Condition (H2) Transport along the family of flat saddles
Condition (H3) Reduction to the box dynamics
Condition (H4) The gap inside the box

The content of the four conditions, the algebra of the splitting of the variance for Proposition 4.5, and how what has been shown is separated from the assumption rows are in 10-05. The current status of each condition is in 10-06 to 10-11, and the table putting all rows on one sheet is 10-12.

Precisely, every one of the four conditions leaves assumption rows at present. And there is not yet a single $\hbar$ at which all rows of all conditions hold at once (10-12). Proposition 4.5 is a proposition naming "what must be shown for the first step to close", not a proposition saying that the first step has closed.

What lifts and what does not

Even if (3) closes on the first step, the sentence of the second step (the four-dimensional lattice) does not follow. This section states, with reasons, what has a prospect of lifting and what does not. Both are descriptions of prospects, not theorems of implication.

Table 3 From the first step to the second (prospects)
	Prospect of lifting	Reason it does not lift
Form	The sentences (1) and (2) do not depend on dimension. The two-scale split "fast inside a phase / slow when changing phase" can also be written in the same form for any measure with slow degrees of freedom	The slow degrees of freedom of the four-dimensional lattice (collective coordinates of instantons, the density of topological charge) increase in number in proportion to the volume. Even on the first step, when the path is much longer than $\xi$ , a split taking only one slow variable cannot be used (the band of 10-04 §05). In four dimensions this situation is expected to be stronger still
Tools	The splitting of the variance, the lemma of two projections, the Efron–Stein inequality and the stage of sums over sectors are algebraic stages that do not use the dimension of the space of the measure, and can be written as they are	Many stages of the first step depend on properties peculiar to one-dimensional nearest-neighbour measures (the transfer operator, the boundary between phases being a point, kinks being countable as a dilute gas). Gauge symmetry and the geometry of the orbit space (3-04) are absent from the first step
Constants	—	The constants of the first step were fixed by computer assistance and numerical readings at particular nodes and windows of $\hbar$ , and give no information about the $\beta$ of the four-dimensional lattice. The correspondence between $\hbar$ and $\beta$ is also an analogy

We still start from the first step because the procedure itself of assembling the proposition of the upper side in a form where "which rows are assumptions" is visible is also needed on the second step. The procedure of naming the assumption rows, checking the direction of dependence (10-12), and separating what machine checking guarantees (10-13) does not depend on the model. Taking the procedure through on the first step means that when a proposition of the same form is set up on the second step, we can name which rows are newly needed.

Precisely, setting up "a sentence of the same form" on the second step is not evidence that the sentence of the second step holds. Even if H is read as holding for the model of the first step (counts at the level of physics and readings of lower bounds), whether it holds on the second step is a separate question (10-15).

A map inside the first step — a guide to 10-04 to 10-11

Chapters of the first step

10-04 The path measure of the double well, and kinks — definition of the setting, kinks, the lower bound $C_P\ge\xi^2$ [Y2]–[Y3]
10-05 The statement of the first step — the splitting of the variance and the four conditions, the assembly of Proposition 4.5 [Y3]
10-06 Condition (H1) — local Poincaré in a phase without kinks [Res]
10-07 The map of condition (H2) — seven routes and the seam of the ring [Y3]
10-08 What computer-assisted means — checking inequalities by interval arithmetic [Y2]–[Y3]
10-09 The sector framework and the three strands — the common wall of the plateau at the top [Res]
10-10 The list of detour plans — types of breakage [Y3]–[Res]
10-11 Conditions (H3)(H4) and the constant of the core — reduction to the box and the gap inside the box [Res]

The [Res] chapters place at the start a summary section that can be read at the [Y3] level, so one can read only the summaries and go on to 10-12.

What this chapter can say

Said	Not said
The way of building it by writing Assumption H in the dimension-independent form (1) and dividing it into four steps (a convention)	Any implication between steps. Assumption H on the second step, the continuum limit, the mass gap in the limit
Proved here The algebra (4) by which the power form of H (2) follows from (3), the lower bound $R\ge1$ and $\ln\xi\ge s/\hbar$ Lean`power_H_of_kink_ratio` (assumptions are the rows of §05)	(3) itself. Proposition 4.5 is conditional on conditions (H1)–(H4), and every condition leaves assumption rows
Cited That Eyring–Kramers-type theorems with the interval held fixed place the regime taking $T\to\infty$ first outside their scope	That there is no known theorem giving the upper side in this regime (only as far as: none was found in what we searched)
Prospects of lifting and reasons it does not lift (Table 3, prospects)	That a proposition of the same form can be set up, or holds, on the second step

Sources and reproduction

Item	Kind	Source / tools
Definitions of the Poincaré constant and the correlation length	Cited	The table of sources of 10-00.
Tunnelling splitting of the double well, and $S_0$	Cited	(9)(10) of 2-05 §07 and its table of sources.
Eyring–Kramers-type theorems, and their scope	Cited	G. Di Gesù, D. Le Peutrec, arXiv:1506.04434 (Theorem 1.1); B. Brooks, G. Di Gesù, arXiv:2606.16083 (the statement of scope on p.4 of the text). Both checked in the text.
A volume-uniform criterion, and its values on the path lattice of the double well	Cited, Reading	R. Bauerschmidt, B. Dagallier, arXiv:2202.02295 (Theorem 2.1, checked in the text). Values from numerics of the transfer operator (step $\varepsilon=0.25$ , $T=\infty$ ).
The algebra (4) of the power form of H	Proved here, Lean	§05 of the text. Theorem `power_H_of_kink_ratio` (assumptions in §05). The statement is in 10-13.
The count $C_P\approx\xi^2S_0/\hbar$	Assumption	A count at the level of physics using a dilute gas of kinks and the diffusion coefficient of the projection (10-04 §03).
Statement of Proposition 4.5	With a table of assumptions	The content is in 10-05.

Words used in this chapter — staircase (four steps), the power form of H, conditions (H1)–(H4), Proposition 4.5, kink, tunnelling splitting, phase (phase without kinks), local Poincaré, box dynamics (reduction to the box). Their meanings are in A-02 Glossary.

Next chapter to read: 10-04 The path measure of the double well, and kinks — defines the setting of the first step, and derives the lower bound $C_P\ge\xi^2$ and the form of the target for the upper bound.

Revised 2026-10-09: first version.