2026-10-15 · chapter Yang–Mills and the mass gapPart 9 Where things stand, and open taskstable of seven stages, two roads, index · Kiichi, Shiori and Rin

The stages a proof must contain — the table of seven stages

We split the way from the lattice to the theory on $\mathbb R^4$ and its mass gap into seven stages. In order: (1) the container (the measure on the lattice and the machinery that builds a quantum theory from it), (2) the ultraviolet (uniform control at all scales that does not break down as the lattice spacing $a$ is sent to 0), (3) the gap on a small T², (4) confinement on a small T², (5) the crossover band (the band for short), (6) the theory on R⁴, (7) general groups.

The "small T²" of stages 3 and 4 is the case where we write four dimensions as $\mathbb R^2\times T^2$ and take the side $L$ of the T² much smaller than the physical length $1/\Lambda$ ($\Lambda$ is the mass scale the theory generates by itself, 4-07). If we put 't Hooft's twist on the T² (a boundary condition that glues opposite sides shifted by an element of the centre of the group), the zero modes (directions in which one can move without raising the action) disappear, and there is a prospect of keeping the coupling weak at scales up to $L$. It is a waypoint for splitting the problem into the stages up to there (3 and 4) and the stage of widening the T² (5). The content is treated in Part 12 (12-00).

The band of stage 5 is the intermediate scale $L\sim1/\Lambda$ at which the coupling passes from weak to strong. In four dimensions no tool has crossed this scale non-perturbatively (13-00). In the language of the space road, the statement that crosses the band takes the following form.

$\theta(L,L)$ is the bias of the sector when one vortex winding the T² is put into a container of side $L$ (a quantity that moves away from 1 the "cheaper" the vortex is). (1) is the area law in the form of 't Hooft's criterion; that "$\theta$ moves away from 1" alone is not enough is treated in 13-01.

What to read from Table 1. (a) The "□" stages are two (the ultraviolet and the band), plus stage 7, which changes the group. The ultraviolet and the band are each, on their own, long-standing open problems of the field. (b) The "◐" of stages 3 and 4 is in both cases a "◐" under the assumption of stage 2. As long as stage 2 is open, the marks of stages 3 and 4 do not go up. (c) Of the claims written "Proved here" in this book, most of the chapters that place the paper proofs in the book are chapters still to be written (Parts 12 and 13). Until they are placed, the rows of the ledger (14-02) do not disappear.

The marks of Table 1 mark how the record of this book's work treats each stage. ■ means closed in the record (there is a paper proof, and where there is a Lean entry its arithmetic is closed too), ◐ means part is closed and the rest stays a strand or a numerical sign, and □ means open.

These marks are not a graduation of how much is proved. There are three reasons. First, ■ says only that something is closed within the conditions written in the row, not the whole stage. Second, a ◐ stage does not necessarily have "less remaining" than a □ stage. The ◐ of stage 3 says that the twisted part is closed, but the rest depends on stage 2 (□) itself. Third, the book's labels (proved here, cited, computer-assisted, reading, assumption) attach to each claim, while the marks attach to each stage. It is normal for "Proved here" and "Assumption" to stand side by side in the same stage. We do not add the marks up and read them as "so many of the seven stages".

Example. The reflection positivity of z in stage 1 is closed in the record (a claim corresponding to ■). Even so, the mark of stage 1 is ◐. This is because writing out the whole construction of the container (from block observables to the gap of the transfer matrix) remains a strand.

What the problem statement asks for is the mass gap (that the spectrum of the Hamiltonian $H$ of the quantum theory is separated by $\Delta\gt0$ above the vacuum, 4-11); it does not ask for confinement (the area law of Wilson loops). As in 9-05 §06, confinement and the gap are different claims, and in the four-dimensional continuum theory neither implication is a theorem Cited. Physics sees the two as two faces of one scale (the ratio $m/\sqrt\sigma$ approaches a finite positive number), but that is a statement at the level of expectation Reading (physics).

The two roads of this book arrive at different statements.

• The road to the gap (stage 3 + the transfer matrix): carry the gap of the transfer matrix on a small T² (stage 3) through the band to $\mathbb R^4$ with the same constant. If it can be carried, the gap survives in the limit (rule R3 of 9-04 §03, with premises (i)–(iii)). The road of Part 10 (from the direction of time) also heads to the same statement of the gap, through the diffusive-relaxation-time assumption (formerly Assumption H) and the one-sided bridge.
• The road to confinement (stages 4 and 5 + stage 6): make confinement on a small T² (stage 4) uniform in the size of the T² through the band (stage 5), and turn it into an area law in the theory on $\mathbb R^4$ (stage 6).

There is no known arrow joining the two roads, that is, no known road "from the area law to the mass gap". Even the statement of what must be added to get a gap from the area law is not yet settled (the last arrow of 14-01, the separate box of 14-02). So even if the road to confinement were closed entirely, that alone would not reach the requirement of the problem statement. This chapter sets the two roads side by side, but does not count the arrow from confinement to the gap as a "road that joins them".

Figure 1 is the flow of the logic from Part 9 to Part 14. The first road comes from the direction of time (Parts 10 and 11), the second from the direction of space (Part 12), and at the level of the four-dimensional goal both arrive at a statement of one type: "the same constant at all scales". That statement is the statement that crosses the band, and Part 13 treats it.

What to read from Figure 1. (a) "What remains of the first step" on the first road (a uniform logarithmic Sobolev inequality for a measure that contains non-convex parts, the path measure of the 0+1-dimensional double well, 10-15 Table 1) is a wall of the same type as the band (there are different tools at the two ends, and the bound on the worst configuration stops in the middle), but it is not the band. The length of the crossover in the first step is fixed by a classical quantity, the first conjugate point of the Jacobi equation at the top; it does not depend on $\hbar$ and is finite in lattice units. The length of the crossover in four dimensions, $1/\Lambda$, is generated quantum mechanically (4-07) and diverges in lattice units. This difference is treated in 13-07 together with the proof of the computation of the conjugate point on the first-step side (until the proof is placed, Assumption). (b) What flows into the band is, on both roads, the "four-dimensional goal" row. On the first road it is the upper side ξ²↑ of the diffusive-relaxation-time assumption uniform in $\beta\to\infty$ and the one-sided bridge (level (T2) of 10-15 §07); on the second road it is the slab tension uniform in $L$. (c) Stage 2 is a separate wall outside the band, and both roads proceed by placing it as an assumption.

The meeting in Figure 1 means "arriving at a statement of the same type at the level of the goal"; it does not mean that the present walls of the two roads are in the same place. The "band" of the band part and the "windows" of Parts 10 and 11 (the site window, the window in $\hbar$, the time window) are different things (the note on words in 13-00).

Table 2 is an index of which cell of which stage of Table 1 the chapters from Part 10 on supply material to. A "cell" is a column of Table 1 (known, this book, unknown). Parts 12–14 have chapters still being written, and the links are made by chapter number.

K1–K6 of 9-04 Table 1 split the tasks needed for the continuum theory according to the requirements of the problem statement. The seven stages of Table 1 split the stages of one road that starts from the lattice. Since the axes of the splitting differ, they do not correspond one to one. Table 3 is the correspondence, and it is this book's arrangement.

What to read from Table 3. In the counting of 9-04 (9-04 §06), if the arrows of the roadmap are admitted, the minimal set needed was one, {K1, K3, K4, K5, K6}. In the language of the seven stages, it is stage 2 (K1, K5, K6), stage 5 (K3), and the axioms of stages 1 and 6 (K4). In either way of splitting, two remain: the ultraviolet and the band.

What this chapter can say

Sources and reproduction

Words used in this chapter — the seven stages (container, ultraviolet, gap on a small T², confinement on a small T², band, R⁴, general groups), the crossover band (the band), 't Hooft's twist, the bias of the sector $\theta$, the marks of the state of the record (■◐□), the road to the gap and the road to confinement, the four-dimensional goal, a miniature of the band. Their meanings are in A-02 Glossary.

Next chapter to read: 10-00 Entrance — the first road (from the direction of time). If you read from the list of conditions, 14-02 The ledger of conditions.

← 9-05 Common misconceptionsContents10-00 Entrance — the staircase, and the random walk, the ruler and the clock →