2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe road of Part 10, the wall of the first step, conjugate points · Kiichi, Shiori and Rin

The band seen from the side of time

Part 10 split the goal of the mass gap on the four-dimensional lattice into three levels (10-15 Table 5): (T1) the correlation length is finite at each $\beta$ on the lattice; (T2) the lattice mass measured in physical units, $m_{\rm lat}(\beta)/a(\beta)$, tends to a positive finite value $\Delta_\infty$; (T3) the gap is positive in the continuum theory. On the road from the side of time, (T2) is what corresponds to the band.

The lower half of (T2) (that the mass does not collapse to 0) follows by one line of algebra from two statements: the upper side $\xi^2\!\uparrow$ of the diffusive-relaxation-time assumption (formerly H↑, (14) of 10-01 §05) and the one-sided bridge (in the form of Proposition B of 10-01 §06).

Proved here Proof. From the two parts of (1), $\kappa\,\xi^2a^2\le a^2C_P\le C$; taking the reciprocal of the square root of both sides gives (2). ∎ (The same algebra as (16) of 10-01 §06. $C_P$ is the Langevin Poincaré constant of the Wilson measure, $\xi=1/m_{\rm lat}$ the correlation length in lattice units, and $a(\beta)$ the lattice spacing.)

The content of (1) lies in the order of the two quantifiers: $C$ and $\kappa$ do not depend on $\beta$, that is, it holds with the same constant on all lattices as $\beta\to\infty$. The one-sided bridge can be shown, for continuous time and a finite spatial lattice, as a proposition with named assumptions (Proposition B), but its lifting to the four-dimensional lattice is only a prospect of form Assumption (the row of the fourth step in 11-07 Table 2). $\xi^2\!\uparrow$ is an assumption in four dimensions Assumption.

Why this statement crosses the band Reading (physical estimate). As $\beta$ grows, the lattice spacing $a(\beta)$ goes to 0, and the correlation length in lattice units grows like $\xi(\beta)\propto e^{3\pi^2\beta/11}$ (the one-loop bookkeeping, 10-03 Table 2). The Wilson measure at one fixed $\beta$ contains all scales from one lattice step up to $\xi(\beta)$, and the effective coupling at the scale of length $L$ is small for $L\ll1/\Lambda$, becomes $O(1)$ at $L\sim1/\Lambda$, and beyond that enters strong-coupling behaviour. $C_P(\beta)$ is a supremum over all test functions, so it includes the slowest relaxation, that is, the behaviour at the longest scales. So showing (1) with a $C$ independent of $\beta$ includes passing, with the same constant, at every $\beta$, through the scales at which the effective coupling becomes $O(1)$ — the crossover band. The goal of the road from the side of space (the uniform slab tension: a tension positive uniformly in $L$, 13-00) is a statement of the same type. That the two roads arrive at a statement of one type, "the same constant at every scale", is what the confluence of Part 13 means.

This uniformity takes up again something Part 10 already said. The end of 10-15 §07 wrote that what is missing is not the number of steps but uniformity, holding at every scale with the same constant, and that it appears at the first step in a form uniform as $\hbar\to0$ and on the four-dimensional lattice in a form uniform as $\beta\to\infty$. The question of this chapter is whether that first-step version is the same thing as the four-dimensional band.

First, the place. The first step is the path measure $\mu\propto e^{-S/\hbar}$ of the double well in 0+1 dimensions, with

(10-04). The wall appears in the conditional measure on the interval (the core) $Q_k=[-x_c,0]^k$ ($x_c=0.89$, Neumann faces) in which the path stays between the top $z=0$ and the well, and the goal is that its Poincaré constant is uniform in the length $k$ of the core, the end values and $\hbar$ (row (T0′_P), 10-15 §02).

The Hessian at the top configuration $z\equiv0$ is the tridiagonal $\mathrm{tridiag}(-2,\tfrac{15}4,-2)$, and the diagonal $\tfrac{15}4$ of one row falls short of the sum of couplings $4$ by $\tfrac14$ (10-09 §06). We reread this shortfall in the language of the calculus of variations.

The Jacobi equation at the top. Linearising the Euler–Lagrange equation of $S$ at the top $z\equiv0$ gives, for the displacement $u_n$,

For general $\varepsilon$ the diagonal is $2/\varepsilon+\varepsilon V''(0)=2/\varepsilon-\varepsilon/2$ and the coupling to neighbours is $1/\varepsilon$, and (4) can be written $u_{n-1}+u_{n+1}=(2-\varepsilon^2/2)u_n$. Substituting $u_n=e^{\pm in\theta}$, the characteristic equation is $2\cos\theta=2-\varepsilon^2/2$, that is,

and since $|\cos\theta|\lt1$, the solution is a rotation $u_n=\sin(n\theta)$ ($u_0=0$). Seen through the linearisation $\ddot u=V''(0)u=-\tfrac12u$ of the Euclidean equation of motion $\ddot z=V'(z)$ too, the top is not a point of exponential departure but a point of rotation.

The first conjugate point. The position $n=\pi/\theta$ at which the solution starting from $u_0=0$ next becomes 0 is the first conjugate point. At $\varepsilon=\tfrac12$,

Proved here (the numerical value of $\theta=\arccos(15/16)$. As an independent check, running the recursion (4) from $u_0=0,u_1=1$ gives $u_8=0.844\gt0$, $u_9=-0.164\lt0$, and the zero of the linear interpolation is $8.837$).

The threshold of convexity = the conjugate point. The smallest eigenvalue of the Hessian $\mathrm{tridiag}(-2,\tfrac{15}4,-2)$ at the top of a lump of length $k$ with both ends fixed is

(the closed formula for the eigenvalues of a tridiagonal matrix), and it is positive only when $\pi/(k+1)\gt\theta$, that is, $k+1\lt\pi/\theta$. From (6), it is positive for $k\le7$ and negative for $k\ge8$ Proved here ($\kappa_{\rm C7}(7)=0.0545$, $\kappa_{\rm C7}(8)=-0.0088$. The difference between numerical diagonalisation and (7) is of order $10^{-15}$ for $k=5$–$10$). "Convex up to length 7, broken from 8" (Lemma BV of 10-11 §05 Stage 1) is exactly the discrete version of the Jacobi criterion of the calculus of variations: whether a conjugate point falls in the interval $[0,k+1]$.

This length is classical and does not depend on $\hbar$. $\hbar$ never appears in (4)–(7). What appears is only the form of the action $S$ (the coefficient $1/\varepsilon$ of the kinetic term and the curvature at the top $V''(0)=-\tfrac12$). The limit taken at the first step is $\hbar\to0$ with $\varepsilon=\tfrac12$ fixed, so the crossover length stays at about 7.84 sites throughout the limit and is finite in lattice units. In units of time it is $\varepsilon\cdot\pi/\theta\approx4.42$, close to the value $\pi/\sqrt{|V''(0)|}=\pi\sqrt2\approx4.44$ (the half period of rotation at the top) in the limit where time is made continuous. Sending $\varepsilon$ to 0 makes the number of sites grow as $1/\varepsilon$, but that is not the limit the problem of the first step takes.

Example. For $\varepsilon=\tfrac14$ it is $\pi/\theta-1\approx16.75$ sites, which in units of time is $\approx4.44$. As a number of sites the length changes with the step; as a physical time it is fixed by the number $V''(0)$ in the action.

A numerical agreement with the mixing length (the mechanism is different). The number of sites over which the influence of the worst end value (the top) drops below $10^{-2}$ in total variation hardly moves for $\hbar=10^{-2}$–$10^{-5}$ and is about 9; changing the step to $\varepsilon=0.35,0.5,0.7,0.9$ gives $12,9,7,5$, numerically close to $\pi/\theta=12.7,8.8,6.3,4.9$ Reading. But the mechanism is not "staying near the top for half a period". The optimal path as $\hbar\to0$ starts going down from the top at once and reaches the corner $-x_c$ of the box in a finite number of steps (7 at $\varepsilon=\tfrac12$), and the mixing length is roughly that number of steps plus 2 Reading. Why the two values are close is not known. What this chapter claims about the conjugate point is only the convexity threshold of (7).

Setting the two walls side by side shows the same type Reading (physical estimate; a comparison of types, implying no statement on either side).

There are different tools at the two ends. At the first step, short lumps ($k\le7$) are bounded uniformly in the end values and $\hbar$ by the convexity of (7) (Lemma BV Proved here), and inside the wells local Poincaré within the phase (condition (H1), 10-06) works. In the four-dimensional band, the expansion works at the strong-coupling end, and the one-loop comparison at the weak-coupling end (13-06 §03; on the strong-coupling side the strand $u\le u_0\approx10^{-80}$, on the weak-coupling side the one-loop level).

In the middle, the estimate of the worst configuration stops. At the first step, every tool that takes the worst configuration as its entrance comes back to the same one-line shortfall $\tfrac{15}4\lt4$ Proved here (each row is a two-line computation).

• The one-site Dobrushin influence coefficient: the conditional one-site measure is $\tfrac{15}4$-convex, and with the coupling $1/\varepsilon=2$ to neighbours the influence from one neighbour is $\le2/\tfrac{15}4=\tfrac8{15}$, and from two $\tfrac{16}{15}\gt1$.
• The influence of a block of length 7: by Brascamp–Lieb and an entrywise comparison of M-matrices, the influence from the left end to the right end is $2\,(M_7^{-1})_{71}=2\cdot2^6/\det M_7\approx1.18\gt1$ ($7\times7$ for $M_7=\mathrm{tridiag}(-2,\tfrac{15}4,-2)$, $\det M_7=1770735/16384\approx108.08$).
• The Otto–Reznikoff criterion: it needs the positivity of $\mathrm{tridiag}(-2,\tfrac{15}4,-2)$, which fails on the infinite chain with lower end $-\tfrac14$.

In the four-dimensional band, the road that estimates with the worst frame plays the same role. The tension that comes out of the slab-confinement estimate $\sup_\Phi\theta\le\tanh(\beta L^2)^{\ell^2}$ (12-04) is $\sigma\ge-\ln\tanh(\beta L^2)\approx2e^{-2\beta L^2}$, which gets worse superexponentially in $L$ ($2e^{-80}$ at $\beta=2.5$, $L=4$). Structural inputs alone (reflection positivity, duality, the transfer matrix) hold in the same form for $\mathbb Z_2$ and $U(1)$, and cannot cross the band (13-04). Each side has a "worst single point": at the first step the face $x\equiv0$ where every site is at the top, in four dimensions the place where the plaquettes are at the centre element $-1$.

Moving to averaging tools, there is a prospect that uniformity comes out. At the first step, the correlation length of the transfer operator gets shorter as $\hbar$ is made smaller (from $1.68$ to $0.48$ as $\hbar=0.1$ goes to $3\times10^{-4}$) Reading, and the influence of the worst end value also vanishes over a length almost independent of $\hbar$ (§02). This is because the rarity of configurations near the top ($P(z\gt-0.5)\approx10^{-252}$ at $\hbar=3\times10^{-4}$) Reading is paid automatically inside the transfer operator ("paying with rarity" of 11-06 §06). On the four-dimensional side, if $L$ is fixed, frame decorrelation is replaced by the single chessboard density $\kappa_{\rm cb}$ (an averaged quantity) (13-05). Estimates across scales are entrusted to one of the four faces (13-06 §07) Assumption.

A first-step version of the averaging tool has been proposed down to a concrete form (§06). We set out, with proof, the uniqueness proposition at its entrance.

Proposition K′ (hidden convexity). For any $\varepsilon\gt0$, $c\gt0$, $l\ge1$ and end values $a,b\in[-c,0]$, the action $S$ (with the $V$ of (3) and both ends fixed at $a,b$) has exactly one local minimum on the box $[-c,0]^l$, and it is the global minimum.

Proved here Proof. Put $z_i=-\sqrt{\rho_i}$ ($\rho_i\in[0,c^2]$). (i) $(z_{i+1}-z_i)^2=\rho_{i+1}+\rho_i-2\sqrt{\rho_i\rho_{i+1}}$, and the geometric mean $\sqrt{xy}$ is concave on $[0,\infty)^2$, so $-\sqrt{\rho_i\rho_{i+1}}$ is convex. (ii) The end term $(z_1-a)^2=\rho_1+a^2-2|a|\sqrt{\rho_1}$ is also convex, since $-\sqrt{\rho_1}$ is convex (likewise at the right end). (iii) $V(z)=(z^2-1)^2/8=(\rho-1)^2/8$ is strictly convex in $\rho$ with second derivative $\tfrac14$. Hence $\tilde S(\rho):=S(-\sqrt\rho)$ is strictly convex on the convex set $[0,c^2]^l$ with $\nabla^2\tilde S\ge(\varepsilon/4)I$, and its local minimum is unique and global. $\rho\mapsto-\sqrt\rho$ is a homeomorphism from $[0,c^2]^l$ to $[-c,0]^l$, so the local minima correspond to each other. ∎

What Proposition K′ says is the uniqueness of the minimum, not a Poincaré inequality. The Dirichlet form is written in the flat metric of $z$, so convexity in the coordinates $\rho$ does not give Bakry–Émery as it stands. Writing the measure in $\rho$ attaches $\prod_i(2\sqrt{\rho_i})^{-1}$ to the density, and the concavity of its logarithm breaks convexity only for $\rho_i\lesssim\sqrt{2\hbar/\varepsilon}$ (a thin layer very close to the top) (comparing $\tfrac{\varepsilon}{4\hbar}$ with $\tfrac1{2\rho^2}$). One can read the non-convexity of "15/4 against 4" as moving from what looked, in the coordinate $z$, like lumps at the top of length 8 or more, to rare thin layers site by site in the coordinate $\rho$ Assumption (the argument that pays for this layer with rarity has not been built).

Even with the same type, three points that decide the weight of the wall differ Reading (physical estimate).

(i) The origin of the crossover length. The crossover length of the first step is the first conjugate point (about 7.84 sites), a classical length fixed only by numbers written in the action $S$ (the coefficient of the kinetic term and the curvature at the top). Throughout the limit $\hbar\to0$ it stays finite in lattice units (§02). The length of the four-dimensional band is $1/\Lambda$, and the action has no ruler of length (in four dimensions the coupling is dimensionless, 11-07 §01). The length is born by quantisation, in the one-loop bookkeeping in the form $\Lambda=\mu\exp(-1/(2b_0g^2(\mu)))$ (dimensional transmutation, 4-07) Cited, and in lattice units $1/(a\Lambda)$ diverges like $e^{3\pi^2\beta/11}$ as $\beta\to\infty$. All derivatives of $\exp(-1/g^2)$ vanish at $g=0$, so this length appears at no order of the expansion in the coupling.

(ii) Whether bad configurations are rare or typical. At the first step, configurations near the top are rare (the $10^{-252}$ of §03), and the currency of "paying with rarity" of 11-06 works. In the four-dimensional band, the bad configurations (thick centre vortices, fractional instantons) are the mechanism of confinement itself, and are typical. The rate of rarity runs out in the band. This is the same point as the row of root (R2) in 11-07 §05 (in the infrared, configurations for which the Gaussian approximation fails become typical, and there is no currency of rarity).

(iii) Whether a phase transition is possible. The core of the first step is a one-dimensional nearest-neighbour chain over a compact one-site space. The transfer operator is a positive integral kernel, so there is no phase transition that would prevent estimates uniform in the length of the chain Assumption (a fact about the principal eigenvalue of positive integral kernels; the same bibliography as [001] of 10-15 Table 1, the original still to be obtained). At fixed $\beta$ in four dimensions, an estimate uniform in the volume has the same weight as "there is no phase transition", and $U(1)$ actually has a transition inside the band ($\beta\approx1.01$). An argument that closes the band must distinguish itself from $U(1)$: this is criterion (m) (13-06 §08).

One decisive difference follows from (i). At the first step, if one chooses a finite block (length 12) longer than the crossover length, a two-stage road stands: analysis around a single minimum inside the block, and strong mixing between blocks (§06). In four dimensions the crossover length $1/\Lambda$ diverges in lattice units, so there is no finite block longer than it. So we cannot expect the tools of the first step to cross the band as they are. The "dimensionless coupling = no classical length" of 11-07 §01 is the reason for this difference.

A supplement. Through the reduction lemma of 13-06 §06, the inside of the band becomes "an estimate uniform in the volume at fixed $\beta$", a statement of the same form as the wall of the first step (an estimate uniform in the length $k$ at fixed $\hbar$). Three things are common: a fixed parameter, uniformity in the volume, and passing through a non-convex region. That the weights still differ is because of (ii) and (iii). And after the reduction, the difficulty of crossing scales gathers at the seam between the upper end of the band and ultraviolet control (ibid. §06). At the first step there is no flow of the coupling (no root (R1)), so there is nothing corresponding to this seam.

Table 1 gathers §02–§04 into six columns. The label of each cell is written inside the cell. Comparisons of type (readings across rows) are all estimates Reading.

How to read Table 1. Reading columns (3)–(5) across, the type is the same (different tools at the two ends, the worst-case estimate stops in the middle, then averaging tools); comparing columns (2) and (6) down, the content differs (whether the origin of the length is classical or quantum, whether a phase transition is possible). The answer to the question "can 15/4 against 4 be written in the same words as the band?" is: the type can be written in the same words, and the content (the origin of the crossover length) differs.

The following road has been proposed as a candidate for supporting row (T0′_P) without going through the LSI constant $A_W$ of the site window (10-09). As the third branch, "road C", of the fork of Part 10 (10-15 §02), its table of rows and dependency graph are in the subsection on road C in 10-15. Here we write only the skeleton needed for the relation to the band.

1. Block LSI Assumption: on blocks of length $\ell_B\ge12$ (avoiding the conjugate point $\approx7.84$ and the length 8 at which convexity breaks), uniform in the end values and $\hbar$. The entrance is Proposition K′ (§03) Proved here; the curvature at the minimiser for the worst end values $\lambda_*(12)\ge0.255$ Reading (interval arithmetic for the minimisers at the four corners remains); and Witten's Morse theory together with the Menz–Schlichting decomposition, which derive from a single non-degenerate minimum that there are no exponentially small eigenvalues (the form at the corners of a Neumann box not yet checked).
2. Strong mixing Assumption: in three stages — the optimal path as $\hbar\to0$ reaches the corner of the box within 7 steps from every starting point (the record is a 50-digit KKT certificate, Proposition K′ and Topkis monotonicity; Reading until it is checked whether this is interval arithmetic with outward rounding), concentrates within a width $\approx0.8\sqrt\hbar$, and has a Doeblin overlap in the boundary layer at the corner — one gets $\mathrm{TV}(n)\le(1-\delta)^{\lfloor n/10\rfloor}$ with $\delta$ independent of $\hbar$. The second and third stages are along the lines of the constrained Laplace method.
3. The standard theorem for one-dimensional chains Assumption: from block LSI and strong mixing, an LSI uniform in the length of the chain (of the Zegarlinski and Martinelli–Olivieri type; the version for compact continuous spins not yet checked). Poincaré follows from LSI and reaches (T0′_P).

Where it stands. Even if road C closes, the first step does not close. The rows common to the two roads (the tube, the reduced kink gas, the cross term, the extension of (H1) and others: the 13 common rows of 10-15 Table 1) remain on road C too. If a mild dependence on $\ln(1/\hbar)$ remains in the mixing length, the power statement ($\ln C_P/\ln\xi\to2$) is expected to be unaffected, but rows that need the constant to be uniform in $\hbar$ are affected (details in the subsection on road C in 10-15).

The relation to the band. Road C is the first-step version of the stage "move to averaging tools" in the type of §03. And as in §04 (i), its key was that there is "a finite block longer than the crossover length". The four-dimensional band has no such finite block. Even if road C succeeds at the first step, we cannot say it gives a tool that crosses the band. In the form carried over to four dimensions (a box criterion with fixed boundary + strong mixing, the face $\delta(L_0)$), that "strong mixing" is precisely where $U(1)$ breaks Reading (physical estimate).

(a) Neither half of the statement (1) that crosses the band has been shown in four dimensions. (2) is one line of algebra on top of (1).

(b) The comparison of the types of the two walls (§03, §04, readings across the rows of Table 1) is an estimate, and we do not say that results at the first step imply anything in four dimensions. Only the form of the statement lifts (10-03 §06).

(c) All we claim about the conjugate point is that the convexity threshold of (7) is fixed by the first conjugate point. The reason for the numerical agreement with the mixing length is not known.

(d) The rows of road C, apart from Proposition K′, are "Assumption" and "Reading". We do not say that road C closes the first step.

(e) That a one-dimensional chain has no phase transition is a statement resting on bibliography whose original has not been obtained.

(f) That four dimensions "has no finite block" is not a claim that the road of blocks is impossible in four dimensions. It means that the form of the first-step tools does not carry over as it is.

The dependency graph (the numbers of premises are smaller than the number of the conclusion):

[01] PRV ALG    (1) ⟹ m_lat/a ≥ √(κ/C)
[02] PRV BRG    one-sided bridge (Proposition B)                  continuous time, finite spatial lattice
[03] ASM BRG4   lifting the bridge to four dimensions ⟸ [02]
[04] ASM XIUP   ξ²↑ (C independent of β)                          four dimensions
[05] RDG CROSS  showing (1) with one constant crosses the band ⟸ [01][03][04]
[06] PRV JAC    Jacobi equation at the top, cos θ = 1 − ε²/4
[07] PRV CONJ   first conjugate point π/θ − 1 ≈ 7.84 ⟸ [06]       ε = ½
[08] PRV THR    convexity threshold k ≤ 7 = conjugate point ⟸ [06][07]
[09] PRV CLS    crossover length independent of ħ, finite ⟸ [06][07]  ħ → 0 with ε fixed
[10] RDG MIX    numerical closeness of mixing length ≈ 9 and π/θ
[11] PRV WORST  worst-configuration tools come back to 15/4 vs 4 ⟸ [08]
[12] PRV SUP    worst frame gets superexponentially worse in L (12-04)
[13] PRV KP     Proposition K′
[14] ASM RHO    non-convexity moves to thin layers in ρ ⟸ [13]
[15] CIT DT     1/Λ from dimensional transmutation, diverges in lattice units
[16] ASM NOPT   the one-dimensional chain of the core has no phase transition
[17] RDG TYPE   same type, different content ⟸ [09][11][12][15][16]
[18] ASM WAYC   road C ⟹ (T0′_P) ⟸ [13]
[19] RDG NOBLK  no finite block in four dimensions ⟸ [09][15][18]

The check is to see by eye that every number to the right of a "⟸" is smaller than the number on the left. There are no cycles. The "Proved here" nodes of this chapter are the one line of algebra [01], the linear computations at the top [06]–[09] and [11], and Proposition K′ [13]; the statements that cross the band, [04] and [05], are "Assumption" and "Reading". There is no node for the mass gap.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the crossover band, the diffusive-relaxation-time assumption ($\xi^2\!\uparrow$, formerly H↑), the one-sided bridge, the Jacobi equation, conjugate point, hidden convexity, block LSI, strong mixing, road C, criterion (m). Their meanings are in A-02 Glossary.

Next chapter: 13-08 A map of the roads tried and stopped — a list of the roads tried and stopped, and why they stopped. The two points that work in four dimensions are in 11-07 Because it is four dimensions.

← 13-06 Renormalisation comparison — a comparison that distinguishes groupsContents13-08 A map of the roads tried and stopped →