2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandultraviolet control, a class that closes in one step, the flow of the coupling · Kiichi, Shiori and Rin

Ultraviolet control — what it means to close at infinitely many steps

• Ultraviolet control means showing "one step returns to the same form" at infinitely many steps without the constants degrading (§01).
• In four dimensions the sum of the one-step losses $g_k^2$ diverges like $\ln K$, so that part has to be absorbed exactly into the flow of the coupling, and the rest confined to sizes whose sum is finite (§02).
• For a group-valued field, of the four properties of the weight, "closes in one step under the expectation of the fluctuations" is missing, and the seams where the gauge is re-fixed also enter the estimates (§03, §04).
• Bałaban's four-dimensional theorem assumes "the effective coupling stays in $(0,\gamma]$", and the theorem that removes this is unpublished (§05).
• That assumption splits into "an estimate of the one-step remainder" and "continuity within the domain", and the real content is the former (§07).

The object is Wilson's lattice for $SU(2)$ on a four-dimensional torus of side $L_0$, with lattice spacing $a$. $L_0$ is fixed at a physical scale, and only $a$ is varied. As in 6-05 §01, we repeat blocking with factor $L$ and integrate from the finest scale upwards. The number of steps is

Proved here (from the definition). Example: if $L=2$ and $L_0/a=2^{20}$, then $K=20$, and each halving of $a$ adds one step. Writing $\rho_k$ for the weight (effective density) after step $k$, we have $Z=\int\rho_0=\int\rho_1=\cdots=\int\rho_K$.

The same form. In Bałaban's method, at each step the lattice is divided into a small-field region $\Omega$ and its outside, and one sums over all the ways of dividing it (6-05 §04). The form the effective density must keep is, in abbreviated notation,

Cited (the form follows the exposition for scalar fields, Dimock II §1, and the wording of the three-dimensional theorem; 6-05 §07). $A_\Omega$ is Wilson's action on the small-field region (written in terms of the background field), $E_k(X,\cdot)$ is a correction localised on a connected lump $X$, $R_k$ is the factor carried by the large-field region, and $g_k$ is the effective coupling at step $k$.

The three conditions. Ultraviolet control means keeping the form (2) at every step while satisfying the following three. The constants $r,C,\kappa,c,\gamma$ depend neither on the step $k$ nor on the lattice spacing $a$.

• (U1) Locality and analyticity for small fields. Each $E_k(X,\cdot)$ is analytic in a complex neighbourhood of radius $r$ in the background-field variables, and $|E_k(X)|\le C\,e^{-\kappa\,d(X)}$ ($d(X)$ is the length of a tree joining $X$).
• (U2) Rarity of large fields. $|R_k(\Omega^c)|\le e^{-c\,p(g_k)^2\,|\Omega^c|}$ ($|\Omega^c|$ is the number of large-field blocks, and $p(g)$ is the threshold factor of equation (7) in 6-05).
• (U3) The coupling stays small. The sequence of effective couplings $g_0,g_1,\dots,g_K$ lies entirely in $(0,\gamma]$.

The three put the content of ① into the words of this book, and none of them has been shown in four dimensions Assumption (the row ① of the ledger in 14-02).

The form of closing in one step. Let the class $\mathcal K(g;\,r,\kappa,c)$ be "the collection of effective densities that have the form (2) and satisfy (U1)(U2) at coupling $g$ with constants $r,\kappa,c$". What has to be shown is

The point is that the constants $r,\kappa,c$ are the same on both sides, and only the coupling may move. $b_0=22/(48\pi^2)$ is the one-loop coefficient for $SU(2)$ (6-07 §01), and $r_k$ is the remainder in the flow of the coupling.

Let us see the difference between a one-step estimate and an all-step proof in a single number. If a one-step estimate comes only in the form "constant $r_{k+1}\ge r_k(1-\delta_k)$", then what is left after $K$ steps is

Proved here (the inequality on the right follows from $\ln(1-x)\ge-2x$ for $0\le x\le\frac12$). If the sum of losses $\sum_k\delta_k$ is finite uniformly in $K$, the constant stays positive uniformly in $K$. If the sum grows with $K$, the constant degrades with the number of steps. The content of ultraviolet control is making this sum finite. Bałaban's three-dimensional theorem (equation (10) in 6-05 §07) and the four-dimensional theorem of 1989 (6-07 §03) carry out an induction of this form in finite volume Cited.

In four dimensions the coupling $g$ has no dimension of length, and the only thing that weakens the coupling at fine scales is asymptotic freedom (11-07 §01). Cutting the one-loop flow into steps, with $b:=2b_0\ln L\gt0$,

Cited (equation (9) of 6-05 §06, keeping only one loop). Below, we treat (4) as an exact equation and count the sum of losses line by line.

(a) The sequence of couplings. Fix the coupling at the coarsest step at $g_K=g_*$, and write in terms of the step $j=K-k$ counted from the coarse end ($j=1,\dots,K$). Using (4) $j$ times,

Proved here. The inequality on the left comes from $1/g_*^2+bj\le(1/g_*^2+b)j$ ($j\ge1$), and the one on the right just drops $1/g_*^2\gt0$. The coupling is smaller on the finer side, but only like $1/j$.

(b) The sum. Comparing with the integral of $1/x$, $\ln(K+1)\le\sum_{j=1}^K\frac1j\le1+\ln K$, so adding up (5),

Proved here. If there is a loss of size $g_k^2$ at each step, its sum keeps growing like the logarithm of the number of steps. Example: for $SU(3)$ and $L=2$, $1/b=1/(2b_0\ln2)\approx10.4$, and each doubling of $K$ raises the upper bound by $(\ln2)/b\approx7.2$. This matches the partial sums in 6-05 Table 4, which grow by about 7.2 with each doubling of $K$ (arithmetic).

(c) The product. If at each step a constant (the radius of analyticity, the rate of decay, the lower bound of the Hessian, and so on) shrinks by a factor $1-cg_k^2$, then from $1-x\le e^{-x}$ and the left of (6),

Proved here (in the range $cg_k^2\le1$). Since $K=\log_L(L_0/a)$, the constant goes to 0 like a power of $\ln(1/a)$. Slowly, but not uniformly.

(d) The size at which the sum is finite. If the loss is of size $g_k^{2+\delta}$ ($\delta\gt0$), then from the right of (5)

and this is finite uniformly in $K$ Proved here. For $\delta=2$ (a loss of $g_k^4$, the size next to one loop) it is $\pi^2/(6b^2)$.

Hence what is needed. Putting (7) and (8) together, the part that moves by size $g_k^2$ after one step cannot be carried as an "error". That part must be made the motion of the parameter of the class itself. Concretely, one defines the class $\mathcal K$ in a form that has the coupling $g$ as its parameter, reads off the coefficient of the leading term (the Wilson term, the direction of $F^2$) from the effective action after one step, and sets that to be $g_{k+1}$. In this way the part of size $g_k^2$ is absorbed exactly into the flow of the coupling (3). The losses left after absorption (such as the degradation of the constants of the localised corrections $E_k$) must be confined to sizes whose sum is finite, of the type (8) — typically $O(g_k^4)$, or irrelevant terms that shrink geometrically in the scale Proved here (a requirement that follows from the arithmetic of (7)(8), in the case where the one-step losses act multiplicatively).

On the side of the flow of the coupling things are different. The remainder $r_k$ in (3) may be allowed, in the variable $1/g^2$, to be of size $|r_k|\le Cg_k^2$ (the size of two loops); its sum grows like $\ln K$ as in (6), but since $1/g_k^2$ itself grows like $bK$, the sequence of couplings does not collapse. What is needed is that the sequence stays in $(0,\gamma]$, and that is handled by Lemma F in §07.

Contrast with three dimensions. In three dimensions $g^2$ has the dimension of length to the power $-1$, the dimensionless coupling at step $k$ has the form $g^2L^ka$ and is geometrically small on the fine side, and $\sum_kg_k^2$ is finite (it stays within 2 in 6-05 Table 4) Cited. Bałaban 1985 writes that the types that can diverge are finitely many, that the counterterm needed is a single constant, and that by semisimplicity there are no variables that grow in the effective action (relevant variables) (6-06 §05) Cited. In three dimensions the one-step loss $g_k^2$ can be carried as an error, and there is no need to follow the flow of the coupling. In four dimensions the $F^2$ term is exactly marginal and must be followed. This is the precise form, on the ultraviolet side, of root (R1) in 11-07 Table 3.

Example: in the one-link heat-kernel class the losses form a telescoping sum

We look at how "absorption" appears for the smallest field, taking as weight a one-link class function of $SU(2)$ (a function only of the angle $\theta$ from the identity) and a flow that convolves it with the heat kernel one step at a time (the candidate in 11-06 §07, and question F7 of 11-08). Every sentence in this section is a sentence about one link, the heat-kernel class, and the range written below, and says nothing about a four-dimensional step that includes many links and the couplings between plaquettes.

The heat kernel $p_\tau$ is a semigroup, $p_\tau*p_t=p_{\tau+t}$, so within the family of heat kernels the result of one step is again a heat kernel, and only the parameter $\tau$ (which plays the part of the inverse of the effective coupling) moves by $t$ Proved here (the semigroup property alone). So every quantity computed from a heat-kernel weight is a function only of the parameter at that moment, and when the changes over successive steps are added up only the difference of the end values remains (a telescoping sum). It does not become a product Proved here.

In coordinates divided by the centre $\mathbb Z_2$ ($SO(3)$, $\theta\in[0,\pi/2]$, with the branching set $\operatorname{tr}U=0$ at $\theta=\pi/2$), write $\kappa$ for the minimum of the Bakry–Émery curvature outside the layer $|\theta-\pi/2|\lt h$. For weights obtained by starting from Wilson's weight and convolving with the heat kernel, the curvature outside the layer was positive at every point of $\beta\in\{2,3,4,6,8\}$, $t\in\{0,0.05,0.1,0.2,0.5\}$, with layer half-width $h=\beta'^{-1/2}$ (for the undivided $SU(2)$ it was negative at every point of the same range) Reading (two independent implementations, by the character series and in position space, agree). The curvature outside the layer takes the form $\kappa/\beta=A(h)+B(h)/\beta$ (apart from exponentially small terms), and for Wilson's weight $A=\sin h$, $B=2$ Reading (there is a record of the derivation but it is not put in the book; for the heat-kernel weights it agrees with the numbers).

Applying this form to a four-dimensional step $\beta'=\beta-8b_0\ln L$ ((4) written with $\beta=4/g^2$), the leading term $A\beta$ moves with the coupling (the part that is absorbed), and the one-step change of $\kappa/\beta$ is $B\,(1/\beta'-1/\beta)=O(1/\beta^2)=O(g^4)$, a loss whose sum is finite, of the type (8) Reading (a reading obtained by applying the form above). The candidate $\kappa_{j+1}\ge\kappa_j(1-c/\beta_j)$ that 11-06 §07 put forward is a product of type (7), which goes to 0 like a power of the number of steps, but within this class it is too pessimistic.

There are, however, three reservations. (i) The rate of rarity of the layer is proportional to $\beta$, drops by $O(1)$ at each step, and runs out at $\beta=O(1)$ Reading. This is the one-link form of what Part 11 calls "the coupling window" (11-06 §07), not a new obstacle on the ultraviolet side. (ii) What the convexity of the divided coordinates can pay for is only the centre-invariant (adjoint) part; the $\mathbb Z_2$ sign of each link needs another currency. With many links, this sign part corresponds to centre vortices Reading (a physical estimate). (iii) A four-dimensional renormalisation step is not a one-link heat-kernel convolution; it involves many links and the couplings between plaquettes. Whether the form $A+B/\beta$ is preserved there has not been examined.

The edge variables $U_l$ of the lattice are elements of the group $G$. Group elements cannot be scaled, nor added and divided (6-05 §05 (i)(ii)) Cited. Since the group is compact the field is bounded, but a "large field" is the far side of the group (far from the identity), where the curvature of Wilson's one-link weight is negative (Proposition 3 of 11-06 §03) Proved here (proved in 11-06). Large fields can only be paid for with rarity, not with convexity.

The tools for flat fields. In the renormalisation of scalar fields, the rarity of large fields is put into the norm of a weight (regulator) $\tilde G(X,\phi)$, and closing under the expectation of one step is derived from the following four properties Cited (Appendix A.6 of the book by Bauerschmidt–Brydges–Slade; 11-06 §07).

1. It factorises into a product over disjoint sets.
2. $\tilde G(X,\phi+\phi')\le\tilde G^2(X,\phi)\,\tilde G^2(X,\phi')$ (for a sum of fields).
3. $\mathbb E_+\tilde G^t(X)\le2^{|B(X)|}$ (closes in one step under the expectation of the fluctuations).
4. $\tilde G^t\le\tilde G_+$ (bounded by the weight of the next step).

From these follows the one-step estimate $\|\mathbb E_+\theta K(X)\|_{\tilde G_+}\le2^{|B(X)|}\|K(X)\|_{\tilde G}$ (Lemma A.6.1). Property 2 uses the addition of fields $\phi+\phi'$, and property 3 uses the fact that the fluctuations are Gaussian variables independent of the background.

Property 2 on the group. Split the edge variable into background and fluctuation as $U_l=W_l\,e^{A_l}$ ($W_l\in G$, and $A_l$ an element of the Lie algebra $\mathfrak{su}(N)$). Let $l_1,\dots,l_4$ be the edges on the boundary of a plaquette $p$ and $s_i\in\{\pm1\}$ their orientation signs, set $U_{\partial p}=\prod_iU_{l_i}^{s_i}$, and define $W_{\partial p}$ in the same way. The norm is the operator norm.

Lemma R. For $G=SU(N)$,

Proved here Proof. (1) Split each factor into a $W$ part and a fluctuation part. If $s=+1$, $We^{A}=W\cdot Y$ ($Y=e^{A}$); if $s=-1$, $(We^{A})^{-1}=W^{-1}\cdot Y$ ($Y=We^{-A}W^{-1}$). In both cases $Y$ is a conjugate of $e^{\pm A}$. (2) In the product $W_1'Y_1W_2'Y_2W_3'Y_3W_4'Y_4$ ($W_i'=W_{l_i}^{s_i}$), move each $Y_i$ to the right of the $W'$ to its right by $YW'=W'(W'^{-1}YW')$; then $U_{\partial p}=W_{\partial p}\,Y_1''Y_2''Y_3''Y_4''$, and each $Y_i''$ is a conjugate of $e^{\pm A_{l_i}}$. (3) For unitary matrices, repeating $\|XY-1\|\le\|X-1\|+\|Y-1\|$ and $\|hYh^{-1}-1\|=\|Y-1\|$ (equation (8) of 6-05) gives $\|U_{\partial p}-1\|\le\|W_{\partial p}-1\|+\sum_i\|e^{\pm A_{l_i}}-1\|$. (4) $A$ is anti-Hermitian and so unitarily diagonalisable, and for an eigenvalue $e^{i\vartheta}$, $|e^{i\vartheta}-1|=2|\sin(\vartheta/2)|\le|\vartheta|\le\|A\|$. Hence $\|e^{\pm A}-1\|\le\|A\|$. (5) The squares follow from $(x+y)^2\le2x^2+2y^2$ and $(\sum_{i\le4}y_i)^2\le4\sum_iy_i^2$. ∎

Corollary (decomposition of the weight). Setting $G_c(X,U):=\exp\bigl(c\sum_{p\subset X}\|U_{\partial p}-1\|^2\bigr)$ ($c\gt0$), since on a four-dimensional lattice an edge lies on at most 6 plaquettes,

Proved here (sum the second of (9) over $p\subset X$; $8\times6=48$). This is the version of property 2 in which the sum of fields is replaced by "background × fluctuation", and on the fluctuation side the weight is of Gaussian type. It may be a known elementary fact (an estimate of the same form may appear in the large-field papers of the series, but we have not checked the text).

What is missing on the group is property 3. On a group the measure of the fluctuations depends on the background (propagators in a background field, the title of the foundational papers in 6-07 Table 1). So estimating $\mathbb E_+G^t$ requires regularity of the background, and in large-field regions where the background lies on the far side of the group, the fluctuations are not a small perturbation of the background. Not even the statement of the group version of property 3 has been written yet Assumption (set down as a component that is needed). (10) speaks only of property 2 and does not touch on closing in one step.

Models without gauge are also stuck. The two-dimensional $O(N)$ sigma model ($N\ge3$) has no gauge and is a model with a dimensionless coupling and asymptotic freedom, yet its ultraviolet limit has not been constructed even in finite volume (Rivasseau 2000 lists it as an open problem) Cited. The case $O(4)$ is the principal chiral model, whose field takes values in $S^3=SU(2)$. As of 2024, transplanting Bałaban's method has reached the variational problem and the background-field stage (Goswami, a family of two-dimensional nonlinear sigma models; Dybalski–Stottmeister–Tanimoto, two-dimensional $O(4)$: the equation for the critical point is derived by the calculus of variations on the Lie group and solved by a contraction mapping) Cited (both from abstracts). On the other hand, the fermionic two-dimensional Gross–Neveu model (the massive case) has been constructed (11-07 Table 3) Cited.

Reading from the contrast Reading (a physical estimate): the core of the wall can be read as lying in the combination "a field that takes values in a group (a nonlinear manifold)", "a dimensionless coupling (root (R1))" and "large fields", with gauge invariance a layer sitting on top of it. We do not say that gauge invariance is not an essential difficulty.

There is no convexity in the orbit directions. On a finite lattice the group of gauge transformations is a product of compact groups, and the action $V$ is gauge invariant, that is, constant along orbits.

Proposition O. Let $V$ be a smooth gauge-invariant function on the configuration space (a compact manifold), and $x_0$ a minimum point of $V$. For a tangent vector $\xi\ne0$ to the orbit through $x_0$, $\operatorname{Hess}V(x_0)(\xi,\xi)=0$. In particular, if there is even one such $\xi$, then $\operatorname{Hess}V\ge m\,g$ ($m\gt0$) does not hold.

Proved here Proof. Let $\gamma(t)=e^{t\eta}\cdot x_0$ ($\eta$ an element of the Lie algebra of gauge transformations with $\gamma'(0)=\xi$). Then $V(\gamma(t))=V(x_0)$ holds for all $t$. Since $dV(x_0)=0$, $\frac{d^2}{dt^2}V(\gamma(t))|_{t=0}=\operatorname{Hess}V(x_0)(\xi,\xi)+dV(x_0)(\nabla_{\gamma'}\gamma')=\operatorname{Hess}V(x_0)(\xi,\xi)$, and the left-hand side is 0. ∎

Tools that rely on convexity (Bakry–Émery, Brascamp–Lieb) cannot be used unless the orbit directions are removed Reading (a physical estimate; the same as the "theory" paragraph of 11-07 §05). To remove them one fixes the gauge, but there is no continuous global gauge fixing that works for all configurations at once (Gribov 1978, Singer 1978) Cited (bibliographic only; 6-05 §05).

A gauge works only at the size of a block. Taking the complete axial gauge on a two-dimensional block of side $\ell$, the plaquette condition $\|U_{\partial p}-1\|\lt\varepsilon$ gives for edges only $\|U_b-1\|\lt\ell\,\varepsilon$ (6-05 §05) Proved here (proved in 6-05). Once $\ell\varepsilon$ exceeds 2 this says nothing, so the gauge is chosen anew at each scale, within the size of a block. Bałaban's three-dimensional theorem introduces the axial gauge block by block by the Faddeev–Popov method (6-06 §05) Cited. We have not checked in the text how the gauge is chosen concretely in the four-dimensional papers (6-07 Table 3).

The seams of re-fixing. The gauge at step $k$ and the gauge at step $k+1$ are different, and the re-fixing is a field-dependent gauge transformation. Its Jacobian (the Faddeev–Popov determinant) and the terms that appear at the seams between neighbouring blocks also enter the effective action, and within the class of (3) they must be estimated as losses whose sum is finite in the sense of §02 Reading (a physical estimate; it states the form of the estimate that is needed). MRS, constructed with a continuum cutoff, also joins the axial gauge of the large-field region with the background-field-dependent gauge of the small-field region, and has seams of the same type (6-07 §06) Cited.

In terms of roots, §03 and §04 write (R3) of 11-07 Table 3 (there is no global gauge fixing, and large fields cannot be controlled by convexity) broken down within one step. In this chapter, which deals only with the ultraviolet, the coupling is small at every step, so root (R2) (there is no expansion connecting the ultraviolet and the infrared) does not appear Reading (a physical estimate).

Dividing Bałaban's series on gauge theories (Commun. Math. Phys., 1983–1989) into five stages by the work each takes on gives Table 1. The bibliography is the same as in 6-07 Table 1; the only text we have checked is the opening of the three-dimensional paper and the abstract and introduction of the last paper. The "work it takes on" in the other rows is read from the titles, not checked against the content.

Where it stopped. The theorem in the introduction of the last paper is as follows Cited (6-07 §03).

'Theorem 1. If the sequence of the effective coupling constants is contained in an interval ]0, γ] with a sufficiently small positive γ, then the effective densities ρk have the form, and satisfy all the conditions and bounds, described in Sect. 2.'

The assumption is (U3) itself, and the conclusion is the form (2) and the estimates (U1)(U2). The author says that the assumption can be removed because it follows from a basic inequality derived from Theorem 2 of an earlier paper, and continues: 'The proof of Theorem 2, which is based on second order perturbative calculations, is very awkward and long ... and has not been published yet' Cited. Within our search of the bibliography of the author's papers after 1989, we found none that contains the proof of Theorem 2 (the source table of 6-07). So the four-dimensional achievement is: on a finite-volume lattice, upper and lower bounds on the effective densities and the partition function uniform in the lattice spacing, assuming (U3) Cited (level 1 in 6-07 Table 4). No source includes estimates of correlation functions, uniqueness of the limit, the OS axioms, infinite volume or the gap in what was reached.

MRS, and rewriting in easier models. Magnen–Rivasseau–Sénéor (1993), with a continuum momentum cutoff, stated the ultraviolet limit of the Schwinger functions for four-dimensional $SU(2)$ in the trivial topological sector, with a fixed infrared cutoff and a regularised axial gauge, but wrote 'we do not provide a detailed proof' and said the details would run to 'several hundred pages' Cited (6-07 §06). Work rewriting the method in easier models has reached three-dimensional $\phi^4$ (Dimock I–III, up to convergence of the expansion and the stability estimates) and three-dimensional QED (uniform upper and lower bounds on the partition function) Cited (the source table of 6-05, 11-07 §05). Transplanting it to the two-dimensional sigma model has reached the point corresponding to stage 3 of Table 1 (§03), and work corresponding to stages 4 and 5 has not yet been written even for easier models (within our search) Cited (abstracts).

In one line, where it stopped is the estimate of the one-step remainder that removes (U3). In §07 we see that (U3) splits into that estimate and another, elementary part.

In examining ultraviolet control, the same kind of misreading happens again and again. The common pattern is writing a one-step estimate and reading it as closing the induction, which amounts to looking at just one $\delta_k$ without looking at the product $\prod(1-\delta_k)$ of §01. Table 2 divides it into ten logical patterns. Every row can be avoided by putting the one-step claim and the all-step claim on separate lines.

The top four rows of the table have the same form. Getting the all-step version from the one-step version needs the assumption that "the constants do not degrade over the steps", and unless this is made explicit on the side of the rules, a one-step proof looks like an all-step proof. The bundle of rules in §10 puts this caution into the way the rules are written.

Bałaban's assumption (U3), "the effective coupling stays in $(0,\gamma]$", 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 (11), $1/g_k^2\ge1/g_0^2-3bk/2\ge1/\gamma_0^2$ ($k\le n$), 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 (11) 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 what we saw in §02: "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.

On a periodic $T^4$ the classical minima at the last step (flat connections) form a continuous family, and the integrals of the last few steps degenerate. We put forward, as a proposal, a statement in which this incidental difficulty is removed by 't Hooft's twist. It moves the small twisted $T^2$ of the spatial road (12-01) to the whole four-dimensional torus.

Setting. $G=SU(2)$, a four-dimensional torus of side $L_0$, blocking factor 2, the number of points on a side of the last lattice $M$ (fixed), the number of steps $n$, and lattice spacing $a_n=L_0/(M\cdot2^n)$. Put the twist $n_{12}=1$ into Wilson's action (multiply one stack of plaquettes in the $(1,2)$ plane by the central element $z=-1$):

Define the blocking transformation $T$ by an averaging that is gauge covariant and commutes with central shifts, and set $\rho_0=e^{-S_{\rm tw}}$, $\rho_{k+1}=T\rho_k$.

Statement T⁴-UV (proposal, unproved). There are a $\gamma\gt0$ and constants $C,\kappa,c,\delta,r\gt0$ depending only on $M$ such that, for every $\gamma_0\in(0,\gamma)$ and every $n$, choosing the bare coupling $g_0=g_0(n,\gamma_0)$, the following hold Assumption.

• (T-U3) The sequence of effective couplings $g_0\lt g_1\lt\cdots\lt g_n=\gamma_0$ is defined throughout.
• (T-U1) $\rho_n$ has the form (2), and each $E_n(X,\cdot)$ is analytic in a complex neighbourhood of radius $r$ in the background-field variables, with $|E_n(X)|\le C\gamma_0\,e^{-\kappa d(X)}$.
• (T-U2) $|R_n(\Omega^c)|\le\exp(-c\,\gamma_0^{-\delta}|\Omega^c|)$.
• (T-Z) $e^{-CM^4}\le Z_n/Z_n^{\rm G}\le e^{CM^4}$ ($Z_n^{\rm G}$ is the sum of the Gaussian integrals around the isolated flat connections).
• (T-Obs) The three above hold with the same constants even when sources (of strength $\le s_0$) are inserted for smoothed observables whose support is larger than the scale of the last lattice $L_0/M$.

None of the constants depends on $n$ or on $a_n$. (T-U3) has the form of the conclusion of Lemma F.

Where the twist works. The flat connections on the last lattice are only those whose holonomies in directions 1 and 2 anticommute and whose holonomies in directions 3 and 4 are central elements, and up to gauge equivalence there are finitely many of them. The adjoint fluctuations around them are antiperiodic in direction 1 or 2, and there are no zero modes Assumption (a reading that extends to four dimensions the facts about the twisted part of $T^2$ in 12-01; the four-dimensional classification has not been checked against the original sources). At fine steps, on the other hand, the twist has no effect. The twist's plane can be moved by central gauge transformations and is not visible locally Reading (a physical estimate).

What the statement says and does not say. If shown, what it says is an estimate of the effective action uniform in the lattice spacing, and non-degeneracy at the last step; this amounts to Bałaban's Theorem 1 + the unpublished Theorem 2 ((T-U3)) + an extension to observables ((T-Obs)) + non-degeneracy at the last step from the twist. Subsequential limits of the smoothed observables follow from it by compactness. What it does not say is uniqueness of the limit, the OS axioms, recovery of rotations, non-triviality, or a gap uniform in the volume. The difficulty of (U1)–(U3) at the fine steps is exactly the same as for the periodic $T^4$; what the twist removes is only the incidental difficulty of the degeneracy of the last few steps Reading (a physical estimate). The role of T⁴-UV is not to reduce the difficulty but to state ① in a pure form without zero modes.

A test bench with the gauge taken out. The two-dimensional $O(4)$ sigma model (the principal chiral model of $SU(2)$) has a field taking values in $SU(2)$, a dimensionless and asymptotically free coupling, and large fields, and no gauge. Writing (U1)–(U3) on the twisted cylinder $\mathbb R\times S^1_L$ (no zero modes because of the twist) gives the four-dimensional ① with the gauge half of root (R3) taken out Reading (a physical estimate). As in §03, the ultraviolet limit has not been constructed for this model either, and transplantation has reached the variational problem and the background-field stage Cited. Writing stage 4 of Table 1 (one step of the small-field expansion) for this model is cheaper than rereading the second half of the four-dimensional series, and is a test that cuts out the "group-valued field × dimensionless coupling" part of ①. Two-dimensional results imply nothing about four dimensions.

Table 3 sets out in ten rows the directions being tried towards ①. "Smallest setting" is the smallest object on which the direction can first be tried, and "What can be said now" is the current state with labels. The last column is the risk that the direction ends up as a restatement of something already known. We do not give numbers for the prospects. No row closes ①.

What to read from Table 3 Reading (a physical estimate). The directions with even one component closed on paper are direction 3 (Lemma R) and direction 6 (Lemma F), and both lie outside the real issue (property 2, the bookkeeping part). The real issue — the group version of property 3, and the one-step remainder (S) including large fields — is the single point towards which directions 3 and 4 and direction 6 each head, and they meet where the dependence of the measure of the fluctuations on the background takes effect.

We rewrite (3) of §01 as a bundle of rules. Each row of Table 4 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) is rules 4 and 6 (the handling of the marginal direction), (R3) is rules 1, 2, 3 and 5 (group and gauge). Since (R2) does not appear in the purely ultraviolet ①, "the coupling window" of Part 11 does not appear in ① either. The rules for ① are the ultraviolet side of N1 + N4, which is exactly the half that 11-06 §07 judged "looks writable". The determinant in self-dual coordinates (direction 9 in Table 3) is another coordinate system for rules 4 and 6: it puts "extracting the local part of the determinant" into the rules in place of the projection onto the leading term.

Cautions when making rules (from §06).

• 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 a rule's inputs from the outputs of the same rule ("circularity" in Table 2).
• For each rule, write a single direction of inequality (upper or lower bound) ("direction of the inequality" in Table 2).

(a) ① ultraviolet control is not closed. This chapter shows none of (U1)(U2)(U3), in four dimensions or in two.

(b) (4)–(8) of §02 are arithmetic that treats the one-loop flow (4) as an exact equation, not theorems about the actual sequence of effective couplings. "A class that absorbs the losses into the flow is needed" is a requirement for the case where the losses act multiplicatively at each step.

(c) The result for the one-link heat-kernel class (§02) is a reading for one link, class functions, and the recorded ranges of $\beta$ and $t$, and implies nothing about a four-dimensional step that includes many links and the couplings between plaquettes.

(d) Lemma R is only the group version of property 2 of the four properties of the weight, and does not touch the real issue, property 3. It may be a known elementary fact.

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

(f) Statement T⁴-UV is a proposal and has not been shown. Even if shown, it says nothing about uniqueness, the OS axioms, non-triviality, infinite volume, or a gap uniform in the volume.

(g) "The core of the wall is group-valued field × dimensionless coupling × large fields" is an estimate from the known situation of two-dimensional models; we do not say that gauge invariance is not an essential difficulty.

(h) The content of the series in Table 1, apart from the opening of the three-dimensional paper and the abstract and introduction of the last paper, is a reading from the titles.

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

[01] PRV STEP   steps K; loss-product bound  (0 ≤ δ ≤ 1/2)
[02] CIT FORM   the form (2) of the effective density
[03] ASM UV     (U1)(U2)(U3)  (from a to L₀)
[04] PRV RUN    couplings (5), log sum (6)  (one loop)
[05] PRV LOSS   product → 0 (7), sum finite (8) ⟸ [01][04]
[06] CIT 3D     in three dimensions the sum is finite
[07] PRV SEMI   telescoping sum, heat-kernel class  (one link)
[08] RDG HK1    curvature A + B/β ⟸ [07]  (1 link, recorded)
[09] CIT BBS    the four properties of the weight
[10] PRV R      Lemma R, Corollary (10)  (SU(N))
[11] CIT SIGMA  two-dimensional O(N) not constructed
[12] RDG CORE   the core of the wall ⟸ [05][06][11]
[13] PRV ORB    Proposition O
[14] CIT GRIB   no global gauge fixing
[15] RDG SEAM   the seams of re-fixing ⟸ [05][13][14]
[16] CIT THM    Theorem 1, Theorem 2 unpublished
[17] CIT MRS    what MRS reached
[18] PRV F      Lemma F  (under (S)(C))
[19] ASM S      (S) is the main part ⟸ [16][18]
[20] ASM T4     Statement T⁴-UV ⟸ [02][18]  (proposal)
[21] RDG TW     the twist only at the last steps ⟸ [20]
[22] RDG O4     2D O(4) test bench ⟸ [11][12]
[23] RDG DIR    the ten directions ⟸ [08][10][18][20]
[24] RDG RULE   bundle of rules ⟸ [05][09][10][15][18]

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] and [20] are assumption rows, and both are in the form of questions that restate ①. Among the nodes of this chapter there is no node saying that ① has been shown, and no node for the continuum limit, confinement or the mass gap.

What this chapter can say

Sources and reproduction

Terms used in this chapter — ultraviolet control (①), the number of steps, the same form, a class that closes in one step, (U1)(U2)(U3), the sum of losses, absorption, the four properties of the weight, Lemma R, Proposition O, the seams of re-fixing, Lemma F, the one-step remainder (S), continuity within the domain (C), Statement T⁴-UV, the bundle of rules. Their meanings are in the A-02 glossary.

Next chapter: 14-00 Entrance — the promise of this part — the whole of the conditional statements, including ① of the ledger. The row for ① is in 14-02.

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