2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe lateral-widening inequality, the doubling iteration, the conditional reduction · Kiichi, Shiori and Rin

Reduction to a single inequality

The notation is as in 13-01 §01. $\theta(R;T^2_L)$ is the sector bias at a typical frame of the container $D_R\times T^2_L$ of size $R$ in the directions of the plane, and we write $\theta(R,R):=\theta(R;T^2_R)$. The goal is, for some $c\gt0$ uniform in physical units,

There are two moves that change $\theta(R;T^2_L)$.

Nesting in the directions of the plane. Keeping $T^2_L$ fixed, split a large container $R$ by an interior frame $\Phi_{\rm int}$ into small containers $R_1,\dots,R_k$; conditioned on the frame, the small containers are independent, so

holds (the nested form of the Mack–Petkova container identity Cited, 12-03). If the container is enlarged $s$ times, the right-hand side is a product of $s^2$ factors. If the factors are close to their typical value and nearly independent of each other, $\theta(sR;T^2_L)\approx\theta(R;T^2_L)^{s^2}$, and the disorder propagates to large scales. But this is a matter inside a slab with $T^2$ fixed, and it has the same content as slab confinement (12-04).

Widening the transverse $T^2$. (1) asks that the container and the transverse torus be enlarged together. Nesting does not change $T^2$, so it alone does not reach (1). Replacing $T^2_L$ by $T^2_{L'}$ changes the very surface that the vortex sheet wraps, and a relation connecting containers at the two scales does not come out of structure (13-04). Setting this relation as an assumption is the lateral-widening inequality of the next section.

We call the assumptions by name (the correspondence with the old symbols is in A-06). All lengths are in physical units and at least $L_{\min}(a)$, where lattice artefacts can be neglected (§06).

(A) The small-T² base point Assumption For some physical $L_0\ge L_{\min}(a)$ and $\varepsilon_0\in(0,1)$,

In the estimate of a dilute gas of fractional instantons, $\varepsilon_0\propto(L_0\Lambda)^{11/3}$ (12-02). A line of showing it under ultraviolet control and uniform acceptance of insertions has been considered, but it has not been shown.

(B) Frame decorrelation Assumption There is a constant $C_{\rm D}\ge1$ such that for all $R$, $L$ and integers $s\ge1$,

The right-hand side of (2) is the average of a product of factors that share a frame, and Hölder's inequality alone loses the power (only the form $\mathbb E\theta_{2R}^4\le\mathbb E\theta_R^4$ comes out). (4) is the assumption that this product splits, up to a constant, into the product of typical values. If $L$ is fixed, it can be replaced, under reflection positivity in z, by the chessboard density $\kappa_{\rm cb}$ (13-05). The form uniform in $L$ is open.

(C) The lateral-widening inequality (strong form) Assumption There is a constant $C$ such that for $L_{\min}(a)\le L\le L'\le R$,

It is the statement that, even if the container is fixed and the transverse $T^2$ is widened by a factor $s=L'/L$, the bias grows by at most a factor $Cs^2$. The factor $s^2$ is the entropy of the transverse position of the vortex sheet. Setting $L'=L$, we need $C\ge1$ (if $\theta\gt0$).

(C′) The lateral-widening inequality (weak form) Assumption For the dyadic scales $R_k=2^kR_0$, there are $0\le\varepsilon_k\lt1$ with $\sum_k\varepsilon_k\lt\infty$ and a constant $C'\ge0$ such that

It is the statement that at each step of doubling the transverse directions one may lose a fraction $\varepsilon_k$ of $\ln(1/\theta)$ and a constant $C'$, provided the sum of the fractions lost is finite. The strong form (5) at $s=2$ loses only the constant $\ln(4C)$ from $\ln(1/\theta)$, so it is the case $\varepsilon_k=0$ of (6).

Theorem 1 (conditional reduction) Proved here Assume (A) the small-T² base point, (B) frame decorrelation and (C) the lateral-widening inequality (strong form). Set $K:=4C\,C_{\rm D}$ ($\ge1$), and choose an integer $s\ge1$ so that

(this is possible because $s^2(1-\varepsilon_0)^{s^2}\to0$). Setting $R_0:=sL_0$ and $R_k:=2^kR_0$, for all $k\ge0$

Moreover, under (B), by Proposition 2 of 13-01, for square loops on the slab $T^2_{R_k}\times\mathbb R^2$ with side a multiple of $R_k$, $|\langle W(C)\rangle|\le C_{\rm D}e^{-\gamma A}$. The lower bound $\gamma$ on the tension does not depend on $k$.

Proof In three steps.

Step 1 (jump). Using (B) with $R=L=L_0$ and $s$, and putting in (A), gives $\theta(sL_0;T^2_{L_0})\le C_{\rm D}(1-\varepsilon_0)^{s^2}$. Next, using (C) with $L=L_0$ and $L'=R=sL_0$,

Step 2 (one doubling). Set $t_k:=\theta(R_k,R_k)$. Using (B) with $R=L=R_k$ and $s=2$ gives $\theta(2R_k;T^2_{R_k})\le C_{\rm D}t_k^4$. Using (C) with $L=R_k$ and $L'=R=2R_k$,

Step 3 (iteration). Setting $c:=K^{1/3}\ge1$, (9) becomes $t_{k+1}\le c^3t_k^4$. Multiplying both sides by $c$ gives $ct_{k+1}\le(ct_k)^4$, so by induction $ct_k\le(ct_0')^{4^k}\le(ct_0)^{4^k}$ (using $0\le ct_0'\le ct_0$). Hence $t_k\le c^{-1}(ct_0)^{4^k}$. From $4^k=R_k^2/R_0^2$, $ct_0\lt1$ and $c\ge1$,

For the last claim, take $L=R_k$ in Proposition 2 and replace $e^{-cL^2}$ by $e^{-\gamma R_k^2}$. ∎

The algebra of the iteration in step 3 — from $t_{k+1}\le c^3t_k^4$ to $ct_k\le(ct_0)^{4^k}$ and $t_k\le(ct_0)^{4^k}/c$ — is checked in Lean (§07).

What Theorem 1 does not say.

• The conclusion is a statement on the dyadic sequence $R_k$. Extending it to the scales in between, $R_k\lt R\lt R_{k+1}$, needs an additional property such as monotonicity of $\theta(R,R)$ in $R$, which we do not assume here.
• The constant $\gamma$ is uniform in the lattice spacing $a$ only when the constants $\varepsilon_0,C_{\rm D},C$ of (A)(B)(C) and $L_0$ are uniform in $a$.
• (B) is used in two places — the power in the nesting (steps 1 and 2) and the bridge from the bias to the Wilson loop (Proposition 2).
• The meaning of "typical frame" is put inside (B). We have not used a form that treats the tail of the distribution of frames (the rarity of bad frames) separately.

Note (the global version). If the same chain is written with the ratio of twisted partition functions on the torus $T^2_L\times T^2_M$ ((8) of 13-01) instead of $\theta$ conditioned on the frame, the bridge to the area law is carried by the Tomboulis–Yaffe inequality (Corollary 3 of 13-01), and (B) is not needed. In exchange, the base point (A) must also be shown for the global partition functions — as a lower bound saying that vortices are cheap. This is also open.

Even with the weak form (6), the iteration keeps the area exponent. But the jump step cannot be made with the weak form, so the base point has to be set sufficiently small at one scale.

Theorem 2 Proved here Assume (B), and (C′) on the dyadic sequence $R_k=2^kR_0$, and set $P:=\prod_k(1-\varepsilon_k)\gt0$, $A:=\ln C_{\rm D}+C'$, $u_k:=\ln(1/\theta(R_k,R_k))$. If $u_0\gt A/(3P)$, then for all $k$

Proof Using (B) with $s=2$ gives $\ln(1/\theta(R_{k+1};T^2_{R_k}))\ge4u_k-\ln C_{\rm D}$. Putting this into (6) and using $0\le1-\varepsilon_k\le1$ and $\ln C_{\rm D}\ge0$ gives $u_{k+1}\ge(1-\varepsilon_k)(4u_k-\ln C_{\rm D})-C'\ge4(1-\varepsilon_k)u_k-A$. Setting $v_k:=u_k/4^k$, $v_{k+1}\ge(1-\varepsilon_k)v_k-A\,4^{-(k+1)}$. For $P_k:=\prod_{i\lt k}(1-\varepsilon_i)$ and $S_k:=\sum_{j=1}^k4^{-j}$ we show $v_k\ge P_kv_0-AS_k$ by induction: for $k=0$ it is an equality. If it holds for $k$, then from $AS_k\ge0$ and $1-\varepsilon_k\le1$, $v_{k+1}\ge(1-\varepsilon_k)P_kv_0-AS_k-A4^{-(k+1)}=P_{k+1}v_0-AS_{k+1}$. From $v_0=u_0\gt0$, $P_k\ge P$ and $S_k\lt1/3$, $v_k\ge Pu_0-A/3$. ∎

The strong form is the case $\varepsilon_k=0$ ($P=1$), $C'=\ln(4C)$, and the condition there, $u_0\gt(\ln C_{\rm D}+\ln4C)/3=\ln K/3$, is the same condition as $t_0\lt K^{-1/3}$ in Theorem 1. The condition on $u_0$ in Theorem 2 moves the role that the jump step played in Theorem 1 onto the assumption of the base point.

We define the string tension of the two-dimensional theory obtained by shrinking the transverse directions to $T^2_L$ (the slab tension) as

The uniform slab tension ((1) of 13-00) is $\inf_{L}\sigma_{\rm eff}(L)\gt0$.

Proposition 3 Proved here If the strong form (5) holds for all $R\ge L'$, then $\sigma_{\rm eff}(L')\ge\sigma_{\rm eff}(L)$ for $L_{\min}(a)\le L\le L'$.

Proof For $M\ge L'$, taking the logarithm of (5) gives $-\ln\theta(M;T^2_{L'})\ge-\ln\theta(M;T^2_L)-\ln\bigl(C(L'/L)^2\bigr)$. Dividing by $M^2$ and taking the lower limit as $M\to\infty$, the constant term vanishes and $\sigma_{\rm eff}(L')\ge\sigma_{\rm eff}(L)$. ∎

So the strong form includes that the slab tension does not decrease in $L$. Confinement (the area law) needs only $\inf_L\sigma_{\rm eff}(L)\gt0$, so the strong form is a stronger assumption than confinement. At small $L$ the tension is expected to grow as $\sigma_{\rm eff}\propto\Lambda^{11/3}L^{5/3}$, and at large $L$ to approach the tension $\sigma_\infty$ of $\mathbb R^4$ Reading (physics); but if at $L\sim1/\Lambda$, $\sigma_{\rm eff}$ goes once above $\sigma_\infty$ and then comes back (overshoots), the strong form with constant $C$ is false, while confinement can still hold. The weak form allows this kind of overshoot, as long as the sum of the fractions lost is finite.

On the numerical side, on the slab with $SU(2)$, $\beta=2.5$, $M=16$, the Creutz ratio of the $4\times4$ loop is $0.059,\ 0.061,\ 0.076,\ 0.071$ for $N_T=4,6,8,10$ (errors $0.009$–$0.002$, 2000 measurements): it increases and saturates, and no significant overshoot is seen Reading. The Creutz ratio contains the short-distance Coulomb term, so we read only the trend in $N_T$, not the values. The lattice is at a single $\beta$ and the range of $N_T$ is narrow, so this is not a reading about the band as a whole. The table of measurements is in 13-09.

The form that widens the transverse directions in one direction only (making $T^2$ a rectangle $L_1\times L_2$ and enlarging only $L_1$) is expected not to be treatable in the same way as the square form, because the small direction of the two behaves like the time of finite temperature Reading (physics). The chain of this chapter uses only the square form $L_1=L_2$.

We state (5) and (6) restricted to $L\ge L_{\min}(a)$, where lattice artefacts can be neglected. $L_{\min}(a)$ is kept constant in physical units, and in lattice units $L_{\min}/a\to\infty$. The goal of the band (1) is a statement about the continuum limit uniform in $a$, so this restriction is natural.

That the restriction is needed also shows in the numbers. For $SU(2)$, $\beta=2.5$, $M=8$, going $N_T=2\to4$ changes the vortex free energy as $-0.01\pm0.12\to0.63\pm0.18$ and the bias as $\theta\lesssim0.06\to0.31$, a ratio of about 5 Reading. This sits at the boundary, within errors, of the $Cs^2=4C$ that the strong form allows at $s=2$. It can be read as follows: on the twisted $2\times2$ $T^2$ with $N_T=2$, the vortex enters cheaply as a thin vortex on the lattice rather than as a smooth fractional instanton, and the slab tension comes out large. The strong form cannot be tested with values from small lattices.

The algebra of step 3 of Theorem 1 is two theorems in the Lean 4 file Doubling.lean (namespace Doubling). We read the statements in the words of the text.

Doubling.iterate: for a real number $c$ and a real sequence $t:\mathbb N\to\mathbb R$, under the assumptions $0\le c$, $0\le t_k$ for all $k$, and $t_{k+1}\le c^3t_k^4$ for all $k$, we have $c\,t_k\le(c\,t_0)^{4^k}$ for all $k$.

Doubling.decay: under the assumptions $0\lt c$, $0\le t_k$, $t_{k+1}\le c^3t_k^4$, we have $t_k\le(c\,t_0)^{4^k}/c$ for all $k$.

The $K$ of the text is $c^3$ in Lean. What machine checking guarantees is only the algebra of these real inequalities. That $t_k$ is the sector bias, and that the recursion (9) holds — that is, (A)(B)(C) — lie outside Lean. The axioms of the two theorems are, in the record, the three standard ones (propext, Classical.choice, Quot.sound), and they have passed this book's re-check (A-04).

Theorem 1 reduced the correct form of the band to three assumptions. The three are different in character.

• (A) The small-T² base point is a statement on the weak-coupling side, and a line of showing it under ultraviolet control and uniform acceptance of insertions has been considered (Part 12, 14-02).
• (B) Frame decorrelation is replaced by the chessboard density $\kappa_{\rm cb}$ if $L$ is fixed (13-05), and the form uniform in $L$ becomes the problem of the uniform slab gap.
• (C) The lateral-widening inequality is the only place in this reduction where non-abelian dynamics enters. It fails for U(1) and $\mathbb Z_2$, and is expected to hold for SU(2) in the dilute regime with the margin $5/3=11/3-2$ (13-03, [Reading] physics). It does not follow from structural inputs alone, such as reflection positivity or duality (13-04).

What is open is to show (C) (or (C′)) non-perturbatively through the region between the dilute regime and the confinement regime — $L\sim1/\Lambda$. The difficulty of the band has gathered into this single inequality.

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

[01] CIT NEST   nesting in the directions of the plane (2)            T² fixed
[02] ASM BASE   small-T² base point θ(L₀;T²_L₀) ≤ 1 − ε₀
[03] ASM DEC    frame decorrelation θ(sR;T²_L) ≤ C_D θ(R;T²_L)^(s²)
[04] ASM WID    lateral widening, strong form θ(R;T²_L′) ≤ C(L′/L)² θ(R;T²_L)
[05] ASM WID′   lateral widening, weak form (finite total loss)
[06] PRV S⇒W    the strong form is a special case of the weak form ⟸ [04]
[07] PRV ITER   t_(k+1) ≤ c³t_k⁴ ⇒ ct_k ≤ (ct₀)^(4^k)                 Lean Doubling.iterate, decay
[08] PRV THM1   Theorem 1: ln(1/θ(R_k,R_k)) ≥ γR_k² ⟸ [01][02][03][04][07]
[09] PRV THM2   Theorem 2: reduction with the weak form ⟸ [03][05]
[10] PRV MON    Proposition 3: strong form ⇒ σ_eff does not decrease ⟸ [04]
[11] RDG SHAPE  shape of σ_eff (grows at small L, saturates at large L)  physics
[12] RDG CREU   the Creutz ratio increases and saturates              β = 2.5, M = 16
[13] RDG ART    N_T = 2 → 4 gives a ratio of about 5                  β = 2.5, M = 8
[14] RDG LMIN   restriction to L ≥ L_min(a) is needed ⟸ [13]
[15] RDG SEP    (C) fails for U(1), Z₂; margin 5/3 for dilute SU(2)   13-03

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 premises of the theorem nodes [08] and [09] always include the assumption rows [02]–[05]. The assumption rows do not contain the quantity of the conclusion (the area exponent of $\theta(R_k,R_k)$): (A) is a statement at one small scale, (B) a statement of nesting with $T^2$ fixed, and (C) a statement of comparison with the container fixed. The numerical rows [12] and [13] and the estimate rows [11] and [15] are premises of no theorem.

What this chapter can say

Sources and reproduction

Terms used in this chapter — nesting in the directions of the plane, the lateral-widening inequality (strong form, weak form), the small-T² base point, frame decorrelation, the doubling iteration, jump, the slab tension $\sigma_{\rm eff}$, overshoot, the scale of lattice artefacts $L_{\min}(a)$. Their meanings are in A-02 Glossary and A-06 Index of assumptions.

Next chapter: 13-03 The only place where non-abelian enters — the $5/3=11/3-2$ that decides the sign of the lateral-widening inequality.

← 13-01 The correct form of the bandContents13-03 The only place where non-abelian enters →