2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controldecomposition, Lemma F, the bookkeeping of the transverse contraction · Kiichi, Shiori and Rin

The decomposition statement — a class that closes with finitely many parameters, and three remainders

• The ultraviolet wall can be stated in a form that splits it into "a class that closes with finitely many parameters (Cls)" and "the remainder of the flow, the remainder of the shape, and continuity" (§02, a statement only).
• (U3), "the effective coupling stays in $(0,\gamma]$", follows from the estimate (S) of the one-step remainder and continuity within the domain (C) by monotonicity and an intermediate-value argument alone (Lemma F, §03). The real content is (S).
• If the one-step map contracts transversely around the class by $\lambda\lt1$, the departure from the class is bounded by $\sup_j\varepsilon(\beta_j)/(1-\lambda)$, uniformly in the number of steps (§04).
• The class is taken as "the one-parameter attracting class", and the shape deviation of $O(1/B)$ is carried by (Rem-shape). We do not state it as a class that keeps the ratio of the quadratic couplings at 2 (§02; grounds in 15-06).
• The way of splitting is the same as in the decomposition of the band, with the difference that in the ultraviolet there is nothing to pay the one-step loss with (§06).

The goal of ultraviolet control was to show at every step the formula (3) of 15-00 §01 — if the effective density at step $k$ lies in the class $\mathcal K(g_k;r,\kappa,c)$, then the effective density at step $k+1$ lies in the class $\mathcal K(g_{k+1};r,\kappa,c)$ with the same constants. As we saw in 15-01 §01, in four dimensions the part that moves by an amount of size $g_k^2$ in one step cannot be carried as an error, and must be made a motion of the parameters of the class. This section sets up the frame in which that "motion of the parameters" and "the rest" are written separately.

Take a family of weights $\mathcal C=\{w_\theta:\theta\in\Theta\}$ indexed by finitely many parameters $\theta=(\theta^1,\dots,\theta^m)\in\Theta\subset\mathbb R^m$, and suppose that one step of blocking $T$ takes the form

Here $\Phi:\Theta\to\Theta$ is the flow of the parameters and $r_\theta[\rho]$ the remainder (gathering the localised irrelevant terms and the large-field factors); the size of the remainder is measured by the norm of the weight $G_k$ of 15-07. (1) is a form of definition: for any $T$ it can be written once the choice of $\Phi$ and $r$ is fixed. The content lies in "whether $\Phi$ and $r$ can be chosen so as to have estimates uniform in the step".

A rule of thumb for counting the parameters. For pure gauge theory in four dimensions, the local terms that remain under power counting, given gauge invariance and the lattice symmetries, are two, the vacuum energy (a constant) and the coefficient of $F^2$, and we read $m=2$, $\theta=(E,\beta)$, $\beta=4/g^2$ Reading (power counting; the four-dimensional version of the sentence in 15-01 §01 that in three dimensions "the counterterm is a single constant"). In the smallest systems, however, the action after one step does not fit into a constant multiple of a single quadratic form, and "shape coordinates" such as the ratio of the couplings by loop shape move (15-06). Under power counting these coordinates correspond to irrelevant directions that act at the square of the lattice spacing; if the one-step map contracts them, then there is effectively one parameter, $\beta$, and the shape deviation is paid for by the transverse contraction $\lambda$ — that is the reading Reading. The (Cls) of §02 puts this reading into the form of a statement.

From the following four, we set down a statement of the form that (U1)(U2)(U3) follow uniformly in the number of steps Assumption (the form of the statement only; none of the four is proved in four dimensions).

• (Cls) The class closes. The number of parameters $m$ is finite independently of the step, and the leading part of $T$ returns into the family (every change of the leading part goes into $\Phi$). The class is taken not as the family of constant multiples of the heat kernel or the Wilson action, but as the one-parameter attracting class of the one-step map — a single coupling $B$ (in the direction of $\beta$) and a fixed shape of $O(1/B)$ determined by the curvature of the group. The shape coordinates (the coefficients of the action of order $\psi^4$ and higher, the ratio of the couplings by loop shape, and so on) are directions that contract at each step, and the shape deviation from the class is carried by (Rem-shape).
• (Rem-flow) The remainder of the flow. The $\beta$ component of $\Phi$ is $\beta_{k+1}=\beta_k-8b_0\ln L+O(1/\beta_k)$. This is (S) of §03 written in the variable $\beta$.
• (Rem-shape) The remainder of the shape. $\sum_k\eta_k\lt\infty$, of type $\|r_{\theta_k}[\rho_k]\|_{G_{k+1}}\le(1+\eta_k)\,2^{|B|}\,\|\rho_k\|_{G_k}$, and $\rho_k$ itself is either bounded by a supremum through the transverse contraction (§04) or stays within the finite size of a sum.
• (C) Continuity within the domain (§03).

Of the implications, that (U3) follows from (Rem-flow) and (C) is Lemma F Proved here; that the product of constants $\prod_k(1+\eta_k)$ is bounded by (Rem-shape) is the arithmetic of products Proved here ($1+x\le e^x$); that the departure is uniform in the number of steps under the transverse contraction is a geometric series Proved here (§04, assuming the one-step inequality); and the part in which localisation and rarity follow from the norm of the weight is a reading from the framework of Appendix A for flat fields Reading (15-02 §04). As for the correspondence between components and conditions: the parameter $\beta$ of (Cls) is the "coupling" of (U3) itself (Rule 4 of §05), (Rem-flow) is (U3) (Rule 6), the localised part of (Rem-shape) is (U1) (Rule 3), and its large-field part is (U2) (Rule 5).

Why we do not state it as "a class that keeps the ratio at 2". In the smallest system (four plaquettes in a row, 15-06) one finds a curve of actions that keeps the ratio of the two quadratic couplings at its tree value 2 (actions mixing the fundamental and adjoint representations, with adjoint coupling $\rho\approx-1/4$). But the one-step map acts in the direction that contracts the shape coordinates, and sends that curve in one step to a place $O(1)$ away from it, so the curve does not close under the flow (15-06; SU(2), four in a row, the one step defined on the diagonal, at the level of the leading term) Reading. On the attracting class of the flow, the ratio is $2+\frac2{3B}+O(B^{-2})$, and the leading term of the deviation is a one-loop quantity coming from the curvature of the group Proved here (at the one-loop level, the same system). So the ratio 2 is not a coordinate that can be tuned by a parameter, but a quantity that remains in the attracting class. Stating (Cls) not as "a class that keeps the ratio at 2" but as "the one-parameter attracting class + a contracting shape" is consistent with the smallest systems we measured Reading (limited to this system).

An example with remainder 0. In the one-link heat-kernel class $\{p_\tau\}$ of 15-04 §01 (one parameter), $T(p_\tau)=p_{\tau+t}$ exactly and the remainder is 0 Proved here (the semigroup property). That the losses form a telescoping sum is a restatement of this "remainder 0", and can be read as another face of the same fact as the fluctuations independent of the background (Lemma H of 15-07) Reading.

Bałaban's assumption (U3), "the effective coupling stays in $(0,\gamma]$" (15-02 §02), is a sentence of the form "it can be satisfied by choosing the bare coupling $g_0$ well". But the effective coupling $g_{k+1}$ is defined only while the earlier steps are small (since it is a coefficient read off from the effective action). Making this explicit through the domain, (U3) splits into two conditions.

Lemma F. Suppose $b\gt0$, $C\ge0$, $\gamma_1\gt0$ satisfy $C\gamma_1^2\le b/2$ and $\gamma_1^2\lt2/(3b)$, and fix $n\ge1$. Suppose that for each $g_0\in(0,\gamma_1]$ the sequence $g_0,g_1,\dots$ is defined inductively in the form "if $g_k\in(0,\gamma_1]$ then $g_{k+1}$ is determined", and satisfies the following two.

• (S) The remainder of one step. $\dfrac1{g_{k+1}^2}=\dfrac1{g_k^2}-b+r_k$, $|r_k|\le C\,g_k^2$.
• (C) Continuity within the domain. On $D_k:=\{g_0:\ g_0,\dots,g_k\in(0,\gamma_1]\}$, $g_0\mapsto g_{k+1}(g_0)$ is continuous.

Then for every $\gamma_0\in(0,\gamma_1)$ there is a $g_0$ with $g_0\lt g_1\lt\cdots\lt g_n=\gamma_0$ (all defined).

Proved here Proof. (i) If $g_k\le\gamma_1$, then from (S) and $Cg_k^2\le b/2$

The left-hand side is at least $1/\gamma_1^2-3b/2\gt0$, so $g_{k+1}$ is a positive finite number, and from the right, $g_{k+1}\gt g_k$. (ii) Set $J:=\{g_0\in(0,\gamma_0]:\ g_0,\dots,g_n\ \text{all defined and}\le\gamma_0\}$. By the monotonicity in (i), $J=\{g_0\le\gamma_0:\ g_n\ \text{is defined and}\ g_n\le\gamma_0\}$. If $g_0^2\le(1/\gamma_0^2+3bn/2)^{-1}$, then repeating the left of (2), $1/g_k^2\ge1/g_0^2-3bk/2\ge1/\gamma_0^2$ ($k\le n$), so $J$ is non-empty. (iii) Let $s:=\sup J$ and take a sequence $g_0^{(j)}\to s$ of points of $J$. Since $s\le\gamma_0\lt\gamma_1$, $g_1(s)$ is determined, and by (C) $g_1(s)=\lim_jg_1(g_0^{(j)})\le\gamma_0$. Repeating the same for $k=2,\dots,n$ gives $s\in J$. (iv) If $g_n(s)\lt\gamma_0$, then from $n\ge1$ and (i), all $g_k(s)\lt\gamma_0\lt\gamma_1$. By (C), $g_0,\dots,g_n$ are also defined in a right neighbourhood of $s$ and are $\lt\gamma_0$, contradicting $s=\sup J$. Hence $g_n(s)=\gamma_0$, and by monotonicity $g_0\lt\cdots\lt g_n=\gamma_0$. ∎

A variant: if (S) is replaced by $|r_k|\le Cg_k^2+\eta_k$ ($0\le\eta_k\le b/4$), the same proof goes through with the constants of (2) replaced by $7b/4$ and $b/4$ and the positivity condition by $\gamma_1^2\lt4/(7b)$ Proved here. Non-universal scale-dependent terms are absorbed as a constant shift in $1/g^2$.

What Lemma F says. (U3) splits into the estimate of the one-step remainder (S) and continuity within the domain (C), and the rest (monotonicity and the intermediate-value argument) is elementary. (S) is the statement, on the side of the flow of the coupling, of "absorb the part of size $g_k^2$ into the flow, and confine the remainder to $O(g_k^2)$ in the variable $1/g^2$".

What Lemma F does not say. (S) has not been shown in four dimensions with the non-abelian large-field contributions included. This corresponds to the content of the unpublished Theorem 2 Assumption (since the text of Theorem 2 does not exist, the correspondence is a reading of one sentence in the introduction). (C) is the statement that "the effective action at each step depends continuously on the bare coupling, and the effective coupling is extracted as a continuous functional of it". Since these are integrals over finitely many compact groups, the line of argument goes through by the dominated convergence theorem, but since the way the effective coupling is extracted is fixed inside the small-field expansion, it depends on the analyticity (U1) of that expansion Assumption (outline only). We do not say that Lemma F has made (U3) easier. It only separates the bookkeeping part from the main part. A proposal for estimating (S) with large fields included is in 15-08.

We write the requirement of (Rem-shape) that "$\rho_k$ stays uniform in the number of steps" in the language of measured quantities. Set up the space of weights $X$, one step $T:X\to X$, and a measured quantity $\Phi_{\rm obs}:X\to\mathbb R$ (for example the curvature outside the layer $\kappa/\beta$, or the total variation distance); take a class $\{w_\beta\}$ and a read-off $b:X\to\mathbb R$ ($b(w_\beta)=\beta$), and define the function of the class $F(\beta):=\Phi_{\rm obs}(w_\beta)$. With $u_k:=T^k(u_0)$, $\beta_k:=b(u_k)$, and the departure from the class $\delta_k:=\Phi_{\rm obs}(u_k)-F(\beta_k)$, the identity

holds Proved here (from the definitions). The first term is the part absorbed into the movement of the parameters, and only the difference of the end values remains (the telescoping sum). The second term is the loss, and if $r_k\ge|\delta_{k+1}-\delta_k|$ then $|\delta_K-\delta_0|\le\sum_{k\lt K}r_k$ Proved here.

The case of transverse contraction. Let $T(w_\beta)=w_{\tau(\beta)}(1+r_\beta)$, $\|r_\beta\|\le\varepsilon(\beta)$, and assume transverse stability $T(w_\beta(1+\rho))=T(w_\beta)(1+\rho')$, $\|\rho'\|\le\lambda\|\rho\|$ (a linear inequality with the quadratic terms dropped). Writing $u_k=w_{\beta_k}(1+\rho_k)$, we have $\|\rho_{k+1}\|\le\lambda\|\rho_k\|+\varepsilon(\beta_k)$, and

Proved here (a geometric series, under the linear inequality above). Even if the remainder is $O(1/\beta)$, the departure is bounded by a supremum rather than a sum, uniformly in the number of steps. If $\lambda=1$ then $\|\rho_K\|\le\|\rho_0\|+\sum_{k\lt K}\varepsilon(\beta_k)$, and if $\varepsilon(\beta)=C/\beta^2$ and $\beta_k\ge\beta_*+ck$ (growing linearly at one loop towards the fine side, 15-01 §01), then

is finite Proved here (the telescoping sum of $1/(u+c)^2\le1/(cu)-1/(c(u+c))$). For $\varepsilon(\beta)=C/\beta$ the sum grows like $\ln K$ and is not uniform (the same type as the logarithmic divergence of 15-01). So there are two routes: "uniform if the one-step loss is $O(1/\beta^2)$" and "uniform even with a remainder of $O(1/\beta)$ if things contract geometrically to a fixed shape", and the latter goes through on a weaker assumption.

The shape deviation when the flow makes the coupling smaller. In one step of the smallest system (15-06), the tree coupling changes in one step as $B_{n+1}=B_n/5$, and the departure $r_n$ of the shape coordinates (in units of $q/B$) satisfies

Proved here (by induction; the one-step inequality is assumed, and the existence of its constant $C$ has only been seen in the numbers of the smallest system). If $\lambda\lt1$, the initial deviation dies out geometrically, and what remains is always the $O(1/B_n)$ of that step. In the smallest system, $\lambda$ was at most $17/125$ in the shape coordinates (15-06, four in a row, leading term) Reading. That (6) is written with $B_n$ decreasing towards the fine side is a convention of that system; in the ultraviolet direction (where $B$ is larger on the finer side), $C/B_n$ becomes smaller at finer steps.

Implication for the decomposition: if the class of (Cls) is taken to be the attracting curve, then what must be asked of one step in four dimensions becomes two numbers, "the size of the source $\varepsilon$" and "the transverse contraction $\lambda$" Reading. How these two can be measured in the smallest systems is in 15-04 (the hierarchical model) and 15-05 (the flux network).

We rewrite the form that closes in one step of 15-00 §01 as a bundle of rules. Each row of Table 1 is a candidate for a rule needed to build "a class that closes in one step", set against which of the rule candidates in 11-08 Table 2 it fills: N1 (axiomatising scale), N4 (the class of measures that pay with rarity), N5 (an axiom of asymptotic freedom), N6 (rules on curvature). Every row is a candidate; we have checked neither consistency nor that they can restate existing results Reading (a physical estimate).

Correspondence with the roots Reading (a physical estimate): (R1) (the dimensionless coupling) goes with Rules 4 and 6, and (R3) (no global gauge fixing, and large fields cannot be controlled by convexity) with Rules 1, 2, 3 and 5 (15-01 §03). In ① taken in the ultraviolet alone, (R2) (no expansion connecting the ultraviolet and the infrared) does not appear.

Cautions in turning these into rules (from the patterns of misreading in 15-00).

• Write a "one-step" version and an "all-step" version of each rule separately, and make explicit on the side of the rules the assumption that gets the all-step version from the one-step version (the constants do not degrade over the steps).
• Do not take the input of a rule from the output of the same rule (the "Circularity" pattern).
• Write one direction of inequality (upper bound or lower bound) for each rule.

In 13-06 §01 the crossover band was split into β-monotonicity (the vortex free energy is monotone in $\beta$) and the renormalisation comparison (the comparison with the doubled lattice) Assumption. Table 2 sets the two decompositions side by side.

What to read from Table 2 Reading (a physical estimate). The two decompositions are of the same type (a one-step comparison that distinguishes the group, and a tuning that does not), and they split in the same way. The difference lies in whether there is something to pay the one-step loss with. The band has a doubly exponential smallness at the strong-coupling end, so the one-step constant $K$ may be crude; the ultraviolet has nothing to pay with, so the one-step loss itself can only be confined to the finite size of a sum, to the movement of the parameters, or to the supremum given by the transverse contraction.

Table 3 lists the questions that come out of this decomposition, in the form of 11-08 Table 1. The numbering carries over from Table 4″ of 13-11. The questions of the other chapters of Part 15 (F-UV4 onwards) and the overall index are in 15-09.

(a) ① ultraviolet control is not closed. None of the decomposition statements (Cls), (Rem-flow), (Rem-shape), (C) is shown in four dimensions.

(b) Lemma F only splits (U3) into (S) and (C), and does not show (S) (the content of the unpublished Theorem 2).

(c) The bookkeeping of §04 is arithmetic under the assumption of a linear inequality with the quadratic terms dropped and of the one-step inequality. We do not show that $\lambda\lt1$ and the estimate of the remainder hold for one step in four dimensions.

(d) $m=2$ is a reading from power counting, not shown for a non-perturbative class. "The one-parameter attracting class" is a reading of the smallest system — four in a row, the diagonal definition, the leading term (the limit of large coupling) — and says nothing about blocking in four dimensions.

(e) The bundle of rules is a set of candidates; we have checked neither consistency nor that they can restate existing results. The contrast with the band is an estimate, and shows none of the band's β-monotonicity, the renormalisation comparison, or the ultraviolet remainders.

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

[01] RDG CLS    the class and the one-step map, m = 2                  power counting
[02] PRV F      Lemma F                                                under (S)(C)
[03] ASM S      (S) is the main part ⟸ [02]                           four-dimensional non-abelian
[04] PRV TEL    the identity and the telescoping sum
[05] PRV LEDG   bookkeeping of the transverse contraction ⟸ [04]     one-step inequality assumed
[06] RDG ATTR   ratio-2 curve does not close, attracting ratio ⟸ [01]  four in a row, leading term
[07] ASM DEC    decomposition of the UV wall ⟸ [01][02][05][06]      form of the statement
[08] RDG RULE   bundle of rules ⟸ [02][07]
[09] RDG BAND   contrast with the band ⟸ [07]
[10] ASM Q      questions F-UV1–3 ⟸ [03][05][07]

The way to check it 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. [03], [07] and [10] are assumption rows, and all are in the form of questions that split ①. Among the nodes of this chapter there is no node saying that ① has been shown.

What this chapter can say

Sources and reproduction

Words used in this chapter — class, the flow of the parameters $\Phi$, remainder, (Cls), (Rem-flow), (Rem-shape), continuity within the domain (C), the one-step remainder (S), Lemma F, the bookkeeping of the transverse contraction, the transverse contraction $\lambda$, the source, the attracting class, the bundle of rules, something to pay with. Their meanings are in A-02 Glossary.

Next chapter to read: 15-04 The one-link heat-kernel class and the hierarchical model — how the decomposition of this chapter looks in the smallest setting.

← 15-02 A map of the sources — Bałaban's programme, the BBS chain, two rows absent from the sourcesContents15-04 The one-link heat-kernel class and the hierarchical model →