2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlFlux network, transverse contraction, Theorem F · Kiichi, Shiori and Rin

The flux network theorem — transverse contraction and the flow of the parameter from one projection

• At tree level restricted to commuting configurations, the minimal configuration with the angle of the single coarse plaquette fixed is determined by the projection $P$ and the unit flux $f$ alone, and $\beta'/\beta$, the map of the quartic term and the transverse contraction $\lambda$ all follow from it (Theorem F, §02).
• At $\lambda\le\max f^2$, in a box where the 4 plaquettes of the coarse plaquette carry $\frac14$ symmetrically, $1/(16d-8)\le\lambda\le1/16$. The Migdal–Kadanoff $1/16$ is the upper end of this (§04).
• The more the box is widened, made periodic or embedded in the lattice, the more $\sum f^2$ rises and $\lambda$ falls. For a 2×2 coarse plaquette embedded in the four-dimensional lattice, $\sum f^2\approx0.73$, $\lambda\approx0.026$ and the source is $\approx5\times10^{-4}$ (§05).
• With a 4×4 coarse plaquette, $\sum f^2$ falls to 0.29–0.31, smaller than the iterated $(0.72)^2$. The action is paid along the perimeter, and the source is diluted as $\approx8\times10^{-3}/L^4$ (§06).
• If all coarse plaquettes are fixed at once, the coarse action does not stay a plaquette action, the map of the quartic term becomes a matrix, and in coarse directions it exceeds $1/16$ (Theorem G, §07).

Lattice. Take the cubical complex of a $d$-dimensional box $[0,n_1]\times\cdots\times[0,n_d]$ (free boundary) or a $d$-dimensional torus (periodic). Let $C_k$ be the space of real-valued functions on the $k$-dimensional cells ($C_1$ are links, $C_2$ plaquettes, $C_3$ cubes), with inner product the sum of the products of the values on each cell, and let $d:C_1\to C_2$ be the map that makes plaquette angles from link angles (the coboundary).

Commuting configurations. Restrict all links of $SU(2)$ to rotations about one direction ($\sigma_3$) and write them by link angles $A\in C_1$; then the plaquette angles are $\phi=dA$. The action of a single image of the heat kernel is $\beta\sum_p\phi_p^2/2$, and Wilson's action is $\beta\sum_p(\phi_p^2/2-\phi_p^4/24+\cdots)$ (the same conventions as 15-04 §01).

The single coarse plaquette. Take a coarse loop $W$ (for example the 8 links of the outer boundary of a $2\times2$ square) and a surface $s\in C_2$ with boundary $W$ (the 4 plaquettes of that square, coefficients $\pm1$, support written $S$). The constraint that the angle of the holonomy of $W$ is $\theta$ is $\langle s,\phi\rangle=\theta$.

Allowed fields and the projection. Let $\mathcal A:=\operatorname{im}d\subset C_2$ be the space of plaquette angles that come from links, $P$ the orthogonal projection onto $\mathcal A$, and $u:=Ps\ne0$. Since a box with free boundary is contractible, $\mathcal A$ coincides with the space in which "the oriented sum over each cube is 0"; in the periodic case the condition that the flux through each coordinate 2-torus is 0 is added Proved here.

and call $f$ the flux. $f\in\mathcal A$, $\langle s,f\rangle=1$ (the unit flux), $f\perp\mathcal A_0$.

The "one step" of this chapter means: given the angle $\theta$ of the single coarse plaquette, integrate out all the remaining degrees of freedom, and read the resulting distribution of $\theta$ (the marginal distribution) as the weight of a single plaquette at the next step. It is a step that looks at the marginal distribution of one coarse plaquette, not a genuine blocking step (the joint distribution of all plaquettes of the coarse lattice is treated only in part, in §07).

Theorem F. In the setting above, under the conditions (a) restriction to commuting configurations, (b) the main term in $\beta\to\infty$ (tree level; the one-loop correction and $O(1/\beta)$ are not included), (c) free boundary or periodic, fixed, the following holds Proved here.

• (i) For each $\theta$ the minimiser of $\min\{\frac12\|\phi\|^2:\ \phi\in\mathcal A,\ \langle s,\phi\rangle=\theta\}$ is exactly one, $\phi^*=\theta f$, and the minimum is $\theta^2/(2\|u\|^2)$.
• (ii) The coarse action of the action $\beta\sum\phi^2/2$ is $\beta'\theta^2/2$ with $$\frac{\beta'}{\beta}=\sum_pf_p^2=\frac1{\|Ps\|^2}. \tag{2}$$ and the lower bound $\sum f^2\ge1/|S|$ holds. The upper bound "$\sum f^2\le1$" is not unconditional (at $|S|=1$, if that plaquette is a face of a cube, $\sum f^2\gt1$).
• (iii) Let the fine action be $V(\phi)=\sum_p(\beta\phi_p^2/2-\sigma\phi_p^4)$ ($\sigma\ge0$; for Wilson $\sigma=\beta/24$), and set the local coarse action $S'_r(\theta):=\inf\{V(\theta f+\psi):\ \psi\in\mathcal A_0,\ \|\psi\|\le r\}$. With $m:=\max_p|f_p|$ and $\|f^3\|:=(\sum_pf_p^6)^{1/2}$, if $\sigma(6\theta^2m^2+4|\theta|mr+r^2)\le\beta/4$ then $$\frac{\beta\sum f^2}{2}\theta^2-\sigma\sum f^4\,\theta^4-\frac{16\sigma^2\|f^3\|^2}{\beta}\theta^6\ \le\ S'_r(\theta)\ \le\ \frac{\beta\sum f^2}{2}\theta^2-\sigma\sum f^4\,\theta^4. \tag{3}$$ that is, the coefficient of $\theta^4$ in the coarse action is exactly $\sigma'=\sigma\sum f^4$ (excluding $\theta^6$ and higher).
• (iv) The normalised quartic deviation $\kappa:=\sigma/\beta$ maps in one step to $$\kappa'=\kappa\cdot\lambda,\qquad \lambda:=\frac{\sum_pf_p^4}{\sum_pf_p^2} \tag{4}$$ and $$\frac1{|S|^3\sum_pf_p^2}\ \le\ \lambda\ \le\ \max_pf_p^2. \tag{5}$$

Proof. (i) If $\phi\in\mathcal A$, then from $P\phi=\phi$ and the self-adjointness of $P$, $\langle s,\phi\rangle=\langle Ps,\phi\rangle=\langle u,\phi\rangle$. By Cauchy–Schwarz $\theta^2\le\|u\|^2\|\phi\|^2$, hence $\|\phi\|^2\ge\theta^2/\|u\|^2$. $\phi=\theta u/\|u\|^2=\theta f$ lies in $\mathcal A$, satisfies the constraint and gives equality. The constraint set is closed under midpoints, and $\|\phi\|^2$ on it is strictly convex by the parallelogram identity, so the minimiser is unique. If $\psi\in\mathcal A_0$ then $\langle f,\psi\rangle=\langle u,\psi\rangle/\|u\|^2=\langle s,\psi\rangle/\|u\|^2=0$.

(ii) (2) follows from the minimum $\beta\theta^2/(2\|u\|^2)$ and $\sum f^2=\|u\|^2/\|u\|^4$. The lower bound is Cauchy–Schwarz on $\langle s,f\rangle=\sum_{p\in S}s_pf_p=1$. If $|S|=1$, then $s$ itself is not closed (the sum over the boundary of a cube is not 0), so $u\ne s$, $\|u\|\lt\|s\|=1$, $\sum f^2\gt1$.

(iii) Every $\phi$ satisfying the constraint can be written uniquely as $\theta f+\psi$ ($\psi\in\mathcal A_0$). The upper bound is $\psi=0$. Lower bound: from $f\perp\psi$, $\sum(\theta f+\psi)^2=\theta^2\sum f^2+\|\psi\|^2$. Expanding the quartic term, $|\sum f^3\psi|\le\|f^3\|\|\psi\|$, $\sum f^2\psi^2\le m^2\|\psi\|^2$, $|\sum f\psi^3|\le mr\|\psi\|^2$, and bounding with $\sum\psi^4\le r^2\|\psi\|^2$, we get, with $x=\|\psi\|$ and $a=4\sigma|\theta|^3\|f^3\|$, $V(\theta f+\psi)\ge(\text{right-hand side})+\frac\beta4x^2-ax\ge(\text{right-hand side})-a^2/\beta$.

(iv) The coarse action has the same form as $\beta'\theta^2/2-\sigma'\theta^4+O(\theta^6)$, so $\sigma'/\beta'=(\sigma/\beta)\sum f^4/\sum f^2$. The inequality on the right is $f_p^4\le(\max f^2)f_p^2$ added up. The one on the left follows from the power mean $(\sum_{S}g)^4\le|S|^3\sum_Sg^4$ ($g_p=s_pf_p$, Cauchy–Schwarz twice) and $\sum_Sg=1$. ∎

The meaning of (iv). For two weights that arrived at the same $\beta'$ ($\kappa=1/24$ started from Wilson and $\kappa=0$ started from the heat kernel), the difference of the coefficients of $\theta^4$, normalised by $\beta'$, is multiplied by $\lambda$ in one step. This is the main term of the transverse contraction in the ledger of 15-03 §04. It agrees with the ratio of total-variation distances measured at finite $\beta$ (0.0443, 0.0451, 0.0463, 0.0469 for $\beta_T=1,2,3,4$ at $2\times2\times1$) approaching $7/144=0.0486$ from below, but we have not shown that the ratio of total-variation distances converges to $\lambda$ Reading.

In a box with free boundary the orthogonal complement of $\mathcal A$ consists of 2-cycles, and since the box is contractible it coincides with sums of boundaries of cubes. Hence $\|Ps\|^2=\min_{c}\|s-\partial c\|^2$ ($c$ is a function on cubes; Dirichlet's principle) Proved here. In three dimensions, consider the dual resistor network whose nodes are the cubes, whose edges of resistance 1 are the interior faces, with the faces of $S$ as edges to a source S and the other outer faces as edges to a sink T; then $\phi\in\mathcal A$ says that the oriented sum at each cube is 0 (Kirchhoff's current law), the constraint says that the amount of current from S to T is $\theta$, $\sum f^2$ is the effective resistance between S and T, and $f$ is the unit current (Thomson's principle) Cited (a restatement of standard facts about electric circuits).

Example: the $2\times2\times1$ slab. Each cube receives the current $\frac14$ from the bottom face and passes it to the 2 side faces on the outer rim (in parallel, resistance $\frac12$) and to the top face (resistance 1). The resistance of a column is $1+(\frac12\parallel1)=\frac43$; with four in parallel, $$\sum f^2=\frac13,\qquad \sum f^4=4\cdot\frac1{4^4}+12\cdot\frac1{12^4}=\frac7{432},\qquad \lambda=\frac7{144} \tag{6}$$ Proved here. The flux is $\frac14$ on the bottom faces, $\frac1{12}$ on the top faces and the outer side faces, and 0 on the inner walls. In four dimensions the network is a potential not on cubes but on 3-cells, and the picture is harder to draw, but the projection formula (1) can be used as it is, independently of dimension.

(a) Monotonicity under inclusion. For complexes $K\subset K'$ containing $W$, $\sum f^2(K)\le\sum f^2(K')$ Proved here. Proof: restricting a configuration on $K'$ to $K$ does not change the holonomy of $W$, and the action (a sum of non-negative terms) decreases, so the minimum is larger for the larger complex. ∎ In three dimensions this is the same content as Rayleigh's monotonicity (a box with free boundary is the result of shorting the surrounding cubes to the sink, and shorting lowers the effective resistance). So a small box with free boundary makes $\beta'/\beta$ look small.

(b) Both sides of the contraction. When the 4 plaquettes of the coarse plaquette carry $f=\frac14$ symmetrically (when the box is invariant under the 90° rotation about the coarse plaquette; this holds for all boxes in Table 1), $\sum f^4\ge4\cdot4^{-4}=\frac1{64}$. Moreover, the trial configuration "$\theta/8$ on each of the 8 links of $W$, 0 elsewhere" gives $\theta/4$ to each plaquette of the coarse plaquette and $\theta/8$ to each of the other $2d-3$ plaquettes containing a link of $W$, so from Thomson's inequality (i), in any box (even in the infinite lattice) $\sum f^2\le\frac14+\frac{2d-3}8$. Together, $$\frac1{16d-8}\ \le\ \frac1{64\sum f^2}\ \le\ \lambda\ \le\ \max f^2=\frac1{16} \tag{7}$$ Proved here (the equality on the right, $\max f^2=\frac1{16}$, holds when the largest flux is on the coarse plaquette; we have confirmed this in all boxes computed, but there is no general proof Reading). In four dimensions $1/56\le\lambda\le1/16$, in three $1/40\le\lambda$. The equality $\lambda=1/16$ holds when the flux sits only on the faces of $\frac14$, and the 16 plaquettes of Migdal–Kadanoff (15-04 §02) are exactly this. For a step of the type in which the coarse plaquette is made of 4 fine plaquettes, the Migdal–Kadanoff $1/16$ is the upper end of the transverse contraction, and the more parallel paths there are, the more thinly the flux is divided and the smaller $\lambda$ becomes.

In the boxes computed, $\lambda\cdot\sum f^2=\sum f^4$ stays within 1.04–1.22 times $1/64$, and the movement of $\lambda$ is determined almost entirely by the reciprocal of the denominator $\sum f^2$ (the slope of the invariant curve) Reading (an observation).

Table 1 lists, box by box, the three numbers of one step for a 2×2 coarse plaquette. $\sum f^2$ and $\lambda$ are the main terms of the flux network (for the three-dimensional free boundary and the boxes free only on the outer rim they are rational, and the two formulations agree Proved here; in four dimensions and the periodic case they are floating-point, and the two formulations agree to about $10^{-12}$ Reading), and the source is the numerical value of the coefficient of $\theta^4$ at one loop, started from the heat kernel, with a single image, at $\beta\to\infty$ Reading. "Free only on the outer rim" means that only the outer faces on the rim of the slab are free boundary, with cubes left around the coarse plaquette; "interior" means the coarse plaquette is embedded in the interior of the box; "periodic" is a torus with only one coarse plaquette fixed and the other plaquettes free.

What to read from Table 1 Reading.

• Direction. The more lattice is added around the coarse plaquette (free boundary → free only on the outer rim → interior → periodic), the more $\sum f^2$ rises (as in §04 (a)) and the more $\lambda$ falls. In four dimensions $\sum f^2$ goes from 0.44 ($2^4$) to 0.73 (the limit embedded in the lattice), and $\lambda$ from 0.039 to 0.026. The free boundary created "a sink for the flux" and made $\beta'$ look small and $\lambda$ look large.
• The flow of the parameter. The main term of $\beta'$ is $\beta\sum f^2$, and the shift at one loop is a $O(1)$ independent of $\beta$ (started from the heat kernel): $\beta/3-0.146$ for $2\times2\times1$, $180\beta/407-0.258$ for $2^4$, $1023\beta/2815-0.220$ at the centre of $4\times4\times2$ (all $+O(1/\beta)$). It is the same type as the Migdal–Kadanoff $-\frac14$.
• The source. It is positive and converges to a value independent of $\beta$: $4.7$–$5.5\times10^{-4}$ in four-dimensional boxes (1.8–2.1 times Migdal–Kadanoff), and $4.1$–$6.5\times10^{-4}$ in three-dimensional slabs. For the Villain action of $U(1)$ it is 0: this reads as saying that the source needs both "parallel", where the weights of two discs are multiplied on a closed surface, and the Haar weight $\theta/\sin\theta$ of a non-commutative group. An open two-dimensional region is series only, so it has no source.

We widen the coarse plaquette to $L\times L$ (the outer boundary $W$ has $4L$ links). $L=2$ is one step, and $L=4$ is the "genuine two steps" that take the factor 4 in the plane directions at once. The "iterated two steps" compared with it are the approximation of applying the one-step map once more to the coarse lattice (feeding the marginal distribution of one coarse plaquette back in as the action of the next step); we count $\beta''/\beta=(\sum f_1^2)^2$, $\lambda$ as $\lambda_1^2$ and the source as $\sigma_1(1+\sum f_1^4)$.

(a) Upper bound. The trial configuration "$\theta/(4L)$ on each of the $4L$ links of $W$, 0 elsewhere" gives $2\theta/(4L)$ to the 4 corner plaquettes of the coarse plaquette, $\theta/(4L)$ to the $4(L-2)$ edge plaquettes and 0 to the inner plaquettes (the sum over the coarse plaquette is 1, matching the holonomy of $W$), and the outer plaquettes containing each link of $W$ number at most $2d-3$. Hence $L\ge2$, and in any box (even in the infinite lattice) $$\sum f^2(L)\ \le\ \frac{d-1}{2L}+\frac1{2L^2},\qquad \lambda(L)\ \ge\ \frac{L^{-6}}{\sum f^2(L)}\ \ge\ \frac2{L^4\bigl((d-1)L+1\bigr)} \tag{8}$$ Proved here (the right side from Stokes, i.e. the sum of $f$ over the $L^2$ plaquettes of the coarse plaquette is 1, and the power mean). In four dimensions $L=2$ is $7/8$ and $1/56$ (the same as §04), and $L=4$ is $13/32$ and $1/1664$.

(b) The genuine two steps lower $\beta$ much more than the iteration does. By monotonicity under inclusion, one step embedded in the four-dimensional lattice has $\sum f_1^2\ge0.717961$ (the value in the interior of $4^4$; the two formulations agree Reading), hence $(\sum f_1^2)^2\ge0.5155\gt13/32\ge\sum f_2^2$ Proved here (the input 0.717961 is a computed number). The iteration falls only by about $L^{-0.48}$ in $L=2^k$, while the genuine $L\times L$ is $O(1/L)$ by (8), so the ratio tends to 0 with the number of steps. The $\beta'/\beta$ of the marginal distribution of one coarse plaquette falls in four dimensions with the perimeter type ($O(1/L)$), and cannot be expressed by repeating the square of one step. In the two-dimensional plane alone the two agree (for a 4×4 coarse plaquette, $\sum f^2=1/16=(1/4)^2$, $\lambda=1/256=(1/16)^2$).

(c) The perimeter type. In the four-dimensional boxes computed, 44–51% of $\sum f^2$ sits on "the plaquettes in transverse directions touching $W$" ($2(d-2)=4$ per link of $W$), and this proportion hardly changes when the coarse plaquette goes from 2×2 to 4×4 Reading. This reads as the action being paid not over the area of the coarse plaquette but along the perimeter of $W$.

What to read from Tables 2 and 3 Reading (an observation of four points, transverse thickness 4, one implementation). The source is $\approx8\times10^{-3}/L^4$, and $c_2$ is $\approx0.46$–$0.50/L$. If read as $\theta\sim L^2F$, $\theta^4$ is an operator of dimension 8, and the coefficient falling as $L^4$ fits the same counting as the Migdal–Kadanoff $b^{d-8}=1/16$. So the source of a genuine blocking looks not like "a number reborn in a fixed band at each step" but like the type in which "the shift of $\theta^4$ born at the finest scale is diluted by about $1/16$ at each step", and the fixed point $\sigma/(1-\sum f^4)\approx5\times10^{-4}$ of the iterated ledger lies, in this way of measuring, on the side of an upper bound. For $\lambda$ it is the other way round: the genuine two steps are 3.4–3.7 times the iterated $\lambda_1^2$, and the iteration makes the contraction look fast. But at $2.4$–$2.6\times10^{-3}\ll1$ the denominator $1-\lambda$ of the ledger stays almost 1.

The periodic rows of Table 1 were networks in which "only one coarse plaquette is fixed and the surrounding coarse plaquettes are free". The main term of a genuine blocking step is the network in which the lattice is divided into $2^d$ blocks and the angles of all plaquettes of the coarse lattice are fixed at once, and this is a different network. Conditions of this section: restriction to commuting configurations, tree level (the main term in $\beta\to\infty$), a periodic lattice $(\mathbb Z/L)^d$ ($L$ even), $2^d$ blocks, and the coarse angle $\theta_P:=\langle s_P,\phi\rangle$ is the linearisation of the angle around the 2×2 outer boundary spanning the coarse plaquette $P$.

Notation. Let $S$ be the matrix with the $s_P$ in rows, $D_c$ the coboundary of the coarse lattice, and $C$ the map to coarse links ($(CA)_\ell$ is the sum of the angles of the 2 fine links that make up the coarse link $\ell$); then $SD=D_cC$. Let $u_P:=Ps_P$, $G:=SPS^{\mathsf T}$ ($G_{PQ}=\langle u_P,u_Q\rangle$), $B:=G^+$ (the Moore–Penrose pseudoinverse), and $U$ the matrix with the $u_P$ as columns.

Theorem G. Under the conditions above Proved here:

• (i) The constraint $S\phi=\theta$ ($\phi\in\mathcal A$) is non-empty if and only if $\theta\in\operatorname{im}D_c$ (Bianchi sum 0 for each coarse cube, and coarse flux 0 through each coordinate 2-torus). In particular, $\theta$ that twists only one coarse plaquette and keeps the others at 0 is not allowed.
• (ii) For $\theta\in\operatorname{im}D_c$ the minimiser is unique, $\phi^*=F\theta$, $F:=UB$, and the minimum is $\frac12\theta^{\mathsf T}B\theta$. The coarse action is $$S'(\theta)=\frac\beta2\,\theta^{\mathsf T}B\,\theta,\qquad \beta'_{PQ}=\beta B_{PQ}\quad(\operatorname{im}D_c\ \text{: as a quadratic form on it}). \tag{9}$$
• (iii) Written in the coarse link variables $a$, $\min\{\frac12\|dA\|^2:CA=a\}=\frac12a^{\mathsf T}Qa$, $Q=D_c^{\mathsf T}BD_c$. The Lagrange multipliers are exactly the gradient of the coarse action (the divergence of current at each coarse link).
• (iv) Since $\theta$ are not independent coordinates, the off-diagonal entries of $B$ and the "diagonal / cross" components of the quartic form depend on the representation. Quantities that do not depend on the representation are: the spectrum of $B$ on $\operatorname{im}D_c$, the ratio by direction $\beta'(\theta):=\theta^{\mathsf T}B\theta/\sum_P\theta_P^2$, $\mu(\theta):=\sum_q(F\theta)_q^4/\sum_P\theta_P^4$, $\lambda(\theta):=\mu(\theta)/\beta'(\theta)$ (an oriented version of the $\lambda$ of Theorem F).
• (v) For all $\theta\in\operatorname{im}D_c$, $\beta'(\theta)\ge\frac14$.

Proof. (i) $\theta=SDA=D_c(CA)\in\operatorname{im}D_c$. Conversely, putting $a_\ell$ on the first fine link of each coarse link and 0 elsewhere gives $CA=a$. (ii) The minimum-norm solution lies in $\operatorname{im}U$, and with $\phi=Uc$, $Gc=\theta$. Since $\operatorname{im}G=S(\operatorname{im}d)=\operatorname{im}(D_cC)=\operatorname{im}D_c$, a solution exists, and $c=B\theta+\ker G$, $U\ker G=0$, $\|\phi^*\|^2=\theta^{\mathsf T}BGB\theta=\theta^{\mathsf T}B\theta$. Uniqueness is the same strict convexity as in Theorem F. (iii) The inner minimum is invariant under coarse gauge directions, so substitute $\theta=D_ca$ and read off the KKT conditions. (v) The sets of 4 plaquettes of the coarse plaquettes do not overlap one another, so $\sum_q\phi_q^2\ge\sum_P\sum_{q\in s_P}\phi_q^2\ge\sum_P\theta_P^2/4$ (Cauchy–Schwarz on each coarse plaquette). ∎

A smooth trial. For a configuration in which the fine field has the same value $\theta/4$ on the $2^d$ plaquettes of each coarse plaquette, the energy ratios are $2^{d-4}$, $\lambda=1/16$ (independent of $d$) Proved here (arithmetic). This configuration cannot be uniform on the torus, and has meaning only as a trial in the limit of long wavelength. In Migdal–Kadanoff (all coarse plaquettes independent, with $P$ taken as the identity) $\beta'/\beta=2^{d-4}$, $\lambda=1/16$.

What to read from Tables 4 and 5 Reading. (1) The coarse action does not stay a plaquette action. In four dimensions the eigenvalues spread from $\frac14$ up to nearly 1 at long wavelength, and already at quadratic order terms of loops of length 2 or more are born (this seems to correspond to the quadratic part of the classical fixed-point action, but we have not checked against the original sources). (2) The long-wavelength $\beta'/\beta$ is larger than the 0.731 for one plaquette fixed, and moves towards the Migdal–Kadanoff side of 1. (3) The map of the quartic term is a matrix that mixes with neighbours, and in coarse directions it exceeds $1/16$ (0.091 for one coarse link, 0.27 for the sup found). In the smooth direction, in three dimensions it heads towards $1/16$ (0.0615 at $k=\pi/12$), while in four dimensions it is 0.047 up to $k=\pi/4$ and convergence is not yet visible. The Migdal–Kadanoff $\lambda=1/16$ survives in a genuine step only as the value in the long-wavelength direction. (4) The values found are all below 0.3, but the pseudoinverse representative $F$ is non-local, and the Riesz–Thorin upper bound grows with the box (4.2 in four-dimensional $16^4$); we have not obtained an upper bound uniform in the volume. So we do not say "the transverse contraction is smaller than 1".

Candidate coordinates for writing it as a matrix. If the class of quartic deviations is taken to be "$|Q(\theta)|\le c\sum_P\theta_P^4$" (quartic forms bounded by $\ell^4$), the tree-level step preserves this class with $c\mapsto c\cdot\sup\mu$ Proved here ($|Q(F\theta)|\le c\sum(F\theta)^4\le c\sup\mu\sum\theta^4$). The diagonal class of plaquettes (the Migdal–Kadanoff class) is not preserved, but the cone of $\ell^4$ is.

The ledger of 15-03 §04 said that if the one-step map contracts transversely around the class ($\|\rho_{k+1}\|\le\lambda\|\rho_k\|+\varepsilon(\beta_k)$, $\lambda\lt1$), the deviation is bounded not by a sum but by the supremum $\sup_j\varepsilon(\beta_j)/(1-\lambda)$ Proved here (under the linear inequality with quadratic terms dropped). This chapter showed that this $\lambda$ and the main term $\beta'/\beta$ of the flow of the parameter can be counted beforehand on paper from the same projection $P$ for the marginal distribution of one coarse plaquette (Theorem F). The numbers to ask about for one four-dimensional step are two, "the size of the source" and "$\lambda$"; the former is a one-loop quantity and the latter a tree-level one. However, when all coarse plaquettes are fixed, $\lambda$ becomes a matrix and exceeds $1/16$ in coarse directions (§07), so to show the $\lambda\lt1$ of the ledger for a genuine step, an upper bound, uniform in the volume, on an operator norm that does not depend on the representation is needed Reading.

(a) Theorem F and Theorem G are statements at tree level restricted to commuting configurations (the main term in $\beta\to\infty$). They do not include that the global minimum, including non-commuting configurations, is commuting; the one-loop correction (the source, the shift $O(1)$ of $\beta'$); or large fields (outside $\|\psi\|\le r$).

(b) The numbers in Tables 1–3 are numbers for a step that looks at the marginal distribution of one coarse plaquette, not for a genuine blocking step. The iterated two steps are the approximation of feeding back the marginal distribution of one plaquette.

(c) The source, $c_2$ and the ratios at finite $\beta$ are computer values, and the source is at one loop, started from the heat kernel, with a single image, at $\beta\to\infty$. The one-loop values for the 4×4 coarse plaquette and for large boxes are from one implementation. The source $\approx8\times10^{-3}/L^4$ is an observation of four points; there is no limit or closed form.

(d) The equality on the right of (7), $\max f^2=1/16$, and that the largest flux is on the coarse plaquette in the boxes computed, are box-by-box confirmations; there is no general proof.