2026-10-22 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlSystematic attempts · Kiichi, Shiori and Rin

Systematic attempts — giving names to the recurring patterns of the record

This chapter is this book's study. It is not a component of the proof on the main line of argument. §01–§03 and §05 are restatements in known frameworks, and §04 is at the level of an estimate.

How to read this chapter. The record of computations in Part 15 contains the same kind of handwork again and again. There are five kinds: measuring the norm of a three-step composite and writing "it exceeds 1 in one step but contracts in three"; taking the absolute value of a signed kernel to build an upper bound; applying a local kernel to a map that is determined only on fields satisfying the Bianchi identity; setting the leading terms of the strong-coupling and weak-coupling series side by side to look for the point where they meet; and pulling a degree-$p$ cone back step by step and bounding it by the weighted Stein bound. This chapter is an attempt to give each of them a definition and axioms (what can be said under what assumptions). Many of the results are restatements of known facts, and we say so row by row for the candidate theorems of each section. What remains after the restatement — quantities that do not correspond directly to a known framework, and regularities shown by the numbers of the record that have not been made into theorems — are the questions of this chapter. §01–§03 can be read independently of one another. §04 shares no tools with §01–§03; it is an attempt to restate the band of ⑦. §05 extends Candidate 1.4 of §01 to the language of cones and can be read alongside §01.

The non-negative matrix that lists the norms of the blocks of an operator split into blocks transfers estimates of powers of the operator to estimates of powers of a non-negative matrix. On the side of non-negative matrices the Perron–Frobenius theory applies as it stands, so "in how many steps does it contract" and "with which weight must we measure for it to contract in one step" can be written in the language of matrices. Everything up to this point is known; the role of this section is to bring the estimates of the record into line with this framework.

Definitions

Definition 1.1 (dominant matrix). Let $I$ be a finite set, $E_i$ ($i\in I$) normed spaces, and $T$ an operator on $E=\bigoplus_iE_i$ split into blocks $T_{ij}:E_j\to E_i$. A non-negative matrix $B\in\mathbb R_{\ge0}^{I\times I}$ is a dominant matrix of $T$ if for all $i,j$ we have $\|T_{ij}\|\le B_{ij}$. The smallest one is the matrix of block norms $N(T)_{ij}=\|T_{ij}\|$.

• Products and powers: $N(ST)\le N(S)N(T)$ (entrywise). Hence if $B,C$ are dominant matrices of $S,T$, then $BC$ is a dominant matrix of $ST$ and $B^n$ is a dominant matrix of $T^n$ Lean (norm_mul_apply_le, dominant_mul, dominant_pow; the power version assumes that the identity has norm 1).
• The class is also closed under sums and non-negative scalar multiples Reading.
• The max-plus side: for the maximal cycle mean $\lambda_{\rm mp}(A)$ of $A=\log B$ (entrywise), $e^{\lambda_{\rm mp}}\le\rho(B)\le|I|\,e^{\lambda_{\rm mp}}$ Cited (Friedland 1986). In the record, the max-plus eigenvalue bounded the optimal weighted norm from below to within 1–2.5% Computed.

Definition 1.2 (weighted block norm). For a positive weight $w\in\mathbb R_{\gt0}^I$, $\|x\|_{w,\infty}=\max_i\|x_i\|/w_i$, and the $\ell^2$ version is $\|x\|_{w,2}^2=\sum_i\|x_i\|^2/w_i^2$. The operator norm is $\|T\|_{w,\infty}=\|W^{-1}TW\|_\infty$ ($W={\rm diag}(w)$).

Definition 1.3 ($k$-step contraction, transient constant). For a non-negative matrix $B$ put $\kappa_k(B):=\|B^k\|$; when $\kappa_k\lt1$ for some $k$ we say that it "contracts in $k$ steps". The transient constant is $C_k(B):=\max_{r\lt k}\|B^r\|$. By Gelfand's formula $\rho(B)=\lim_n\|B^n\|^{1/n}$, $\rho(B)\lt1$ is equivalent to $\kappa_k\lt1$ for some $k$ Cited.

Definition 1.4 (one-step weight). For $B\ge0$, $w\ge0$, $s\gt0$, $k\ge1$ we call

the one-step weight built from $w$.

Candidate theorems and labels

Candidate 1.1 (from $k$-step contraction to a weight that contracts in one step). If $B\ge0$, $w\ge0$, $s\gt0$, $k\ge1$ and $B^kw\le s^kw$ (entrywise), then $w\le w'$ and $Bw'\le s\,w'$ Lean (liftWeight_step; lemmas mulVec_pow_succ, pow_mulVec_nonneg).

Combined with Candidate 1.2 (the one-step estimate), this says that if "the $k$-step composite contracts by a factor $s^k$ in the weight $w$", then "measured in the weight $w'$ it contracts by a factor $s$ in every step". The weight ratio $w'/w$ lies in $[1,\ \sum_{r\lt k}s^{-r}\max_i(B^rw)_i/w_i]$, so the constant of re-measurement does not depend on the number of steps (Candidate 1.3). The content is the constructive half of "$\rho(B)\lt s$ ⇔ there is $w'\gt0$ with $Bw'\le sw'$", known from Varga's analysis of iterative methods and from the line of Berman–Plemmons on the characterisation of M-matrices Cited.

Correspondence with the numbers of the record

• The matrix of the finite types of the W coordinates (quadratic including differences, dimension 6) had spectral radius 0.36, a one-step norm above 1, and 0.05–0.13 in three steps Computed. It is a typical case of $\rho\lt1$ with a one-step norm above 1, and the $C$ of Candidate 1.4 (a) corresponds to the transient constant. Candidates 1.1 and 1.2 read this as: taking the Perron weight, it contracts in one step (in the record, 0.546 in one step with the Perron weight) Computed.
• The weighted Stein bound of the $\ell^4$ cone (15-05) was 4.45 in one step and 0.10 in three steps Computed. Put in the form of Candidate 1.1 it would read "if it is 0.10 in three steps, a weight that contracts to $0.10^{1/3}\approx0.46$ in one step can be built", but this requires the three-step estimate to come from powers of a single linear dominant operator. The cone is a quartic quantity obtained through Riesz–Thorin, and the kernel $K_k$ differs from step to step, so it is not a power. What can be used in this case is only the form of a product over steps (weighted_iter) Reading.
• The pair of finite types and remaining returns has the form of the two-by-two aggregation of Candidate 1.5. Putting the remaining block in the lower right and checking $bc\lt(1-a)(1-d)$ has not yet been done (F-SYS4).
• The Dobrushin column sum used in the coarse chain of the double well in Part 10 is Candidate 1.2 with the weight $w=\mathbf 1$. Taking the Perron weight could improve the contraction constant Reading.

A question proper to this pattern — the loss from discarding signs

A dominant matrix discards signs. For the actual operator $T$ and a dominant matrix $B$, define the loss at step $k$ as

Then the loss is not necessarily bounded in the number of steps. $T=\begin{pmatrix}1&-1\\1&-1\end{pmatrix}$ has $|T|=\begin{pmatrix}1&1\\1&1\end{pmatrix}$ as a dominant matrix, $T^2=0$, and every entry of $|T|^{n+1}$ is $2^n$. Hence $L_n=+\infty$ for $n\ge2$ Lean (absEx_eq, signedEx_sq, absEx_pow). Conversely, if $T$ is non-negative ($T=|T|$) or sign-similar, $T=S|T|S$ ($S$ a diagonal matrix of $\pm1$), then $L_k=1$ in the $\infty$ norm with the Perron weight Reading.

How far the choice of signs can reach towards the estimate of $\rho(|A|)$ is treated by Rump's sign-real spectral radius (Rump 1997) Reading. We have not found a source that treats how the loss behaves over steps (not found, which does not mean there is none). In the record the loss grows only slowly, 7.6 in one step and 9.3 in two, and the conditions for making this a theorem are not determined by the information of the dominant matrix alone (the example above shows this). We take up this question in §02.

A closer look at the loss in the record showed that it was of a different type from the counterexample of §01. The Stein bound of the record takes the upper bound after composing the step kernels. The counterexample of §01 is a quantity that composes after taking absolute values. In the former, the upper bound of the composite for the nilpotent example $T^2=0$ is also 0. In this section we see from the numbers that the loss with the upper bound taken after composition hardly moves over steps, and we write an inequality that bounds it by a geometric quantity within a single row (the row alignment).

Definitions

Definition 2.1 (composite kernel and loss). Write the composite of the step kernels $K_0,K_1,\dots$ as $T_k=K_0K_1\cdots K_{k-1}$. Let the true ratio be $\sup_\theta\|T_k\theta\|_4^4/\|\theta\|_4^4$, and the Stein bound the weighted $\ell^2$ quantity of the form of (11) of 15-05 (weighted by the $\ell^1$ norms $A_r$ of the rows of the composite kernel); we put

The true ratio is replaced by a lower bound from maximisation (the larger of a quasi-Newton method and a maximisation over period-2 fields), so the loss in the tables is an estimate of the loss from above.

Definition 2.2 (row alignment). The Stein bound of a translation-invariant $T$ has the form $\sup_q\lambda_{\max}\bigl(\sum_rA_r^2\hat T_r(q)^{\rm H}\hat T_r(q)\bigr)$; let $(q^*,u^*)$ be its maximiser. Put $y_r=\hat T_r(q^*)u^*$, $M=\|u^*\|_\infty$, and call

the alignment of row $r$ in the worst direction ($\kappa_r\le1$ is coherence_le_one Lean). $\langle\kappa^{-2}\rangle$ is the average weighted by $|y_r|^4$.

The numbers of the record — the loss stops, and falls a little

The true ratio and the Stein bound decrease at the same rate (both step ratios tend to about 0.07), so the loss stops. The shape is the same on larger tori (Table 3). The earlier recorded two-step value 9.3 was due to a weak lower bound of the true ratio, 0.108; with period-2 fields the lower bound becomes 0.1405 Computed. For $k\ge2$, 99.8–100% of the energy of the maximiser of the true ratio was at $|q|_\infty=\pi$, sitting on period-2 points of type $(\pi,\pi,\pi,0)$ Computed.

The rightmost column is the product of the one-step upper bounds of each step (4.45, 5.75, 6.50, 6.81) divided by the lower bound of the true ratio; it grows by a factor of 60–90 per step, 7.5 → 182 → $1.1\times10^4$ → $1.0\times10^6$ Computed. Only the side that composes after taking absolute values grows exponentially. This column shows how much is gained by taking the upper bound on the composite kernel.

The three hypotheses failed, and they failed in the same way

As reasons for the loss stopping we tried three: (a) composition reduces the proportion of the negative part; (b) the cancellation takes place within a cell; (c) the signs align at the maximiser of the true ratio. None of them holds in the form written. All three, however, failed in the same way: "it does not change over steps".

That all three quantities hardly move over steps can be read as showing not that "composition brings the kernel closer to positive" but that "composition brings the profile of the kernel (its normalised shape) closer to a single one" Reading. A profile positive at the centre and negative around it, a negative proportion of 0.43 and an alignment of 0.6 are kept independently of the step. This is close to the view of the limit function of subdivision Reading. Note that the signs "positive at the centre, negative around it" depend on the sign convention for the components of the direction; what does not depend on the convention is the negative proportion and the alignment.

The inequality that held — loss ≤ inverse square of the alignment

Candidate 2.1. Suppose $A_r\ge0$, $\kappa\ge0$, $M\ge0$, $|v_a|\le M$ for every component, and $\kappa A_rM\le|y_r|$ for every row. Then

Lean (stein_le_inv_sq_mul_test; lemmas stein_form_le, sum_four_le). The bracket on the left is the value of the Stein quadratic form and the right-hand side is the value of the test, so it reads "upper bound ÷ test value $\le\kappa^{-2}$".

On the lattice, using as the test the real plane wave in the worst direction $\theta={\rm Re}(e^{i\varphi}u^*e^{-iq^*\cdot X})$, we get

$\Phi$ is the phase factor; it is 1 if $2q^*\not\equiv0$ and $4q^*\not\equiv0$, and also 1 at period-2 points ($2q^*\equiv0$) if $u^*$ is taken real Reading (the average $3/8$ of $\cos^4$ cancels between numerator and denominator; this step and the Stein formula itself are not in Lean).

• For $k=2,3,4$ on the $4^4$ torus, the maximiser of Stein was a period-2 point, and its plane wave was itself the maximiser of the lower bound of the true ratio (test values 0.1396, 0.01487, 0.00109). Then Stein ÷ test value $=\langle\kappa^{-2}\rangle=$ 5.55, 5.13, 5.31, and the shape factor and the phase factor were exactly 1 Computed.
• On larger tori the maximiser of Stein moves away from period 2, and $\langle\kappa^{-2}\rangle$ becomes 22.9, 10.6, 10.2 ($k=1,2,3$). As an upper bound of the loss it is bounded and falls over steps, but it does not agree with the loss itself (6.4 and so on) Computed.

How to read it: on the $\ell^2$ side the Stein bound remembers all the signs; what it forgets is only the "row $\ell^1$" on the $\ell^\infty$ side. So the loss reduces to the amount of cancellation within a row, $1/\kappa^2$ Reading. This gives a conditional statement — if the alignment in the worst direction is bounded below over steps, the loss is bounded over steps — but that the alignment is bounded below is not shown (F-SYS1).

A pattern that did not apply — positive part dominant

For a sign field $s$, if the negative part of each row is at most $\delta$ times the positive part, then $(Ts)_i\ge\frac{1-\delta}{1+\delta}r_i$ ($r_i$ the row $\ell^1$), and combined with the $\ell^4$ Schur test $\sum(Tx)^4\le A^3B\sum x^4$, loss $\le U\bigl(\frac{1+\delta}{1-\delta}\bigr)^4$ ($U$ the non-uniformity factor of the row $\ell^1$). Under composition $\delta$ degrades additively (alignment $1-\delta_1-\delta_2$) Lean (coherent_of_neg_le, schur_four, loss_le_of_coherent, comp_coherent).

The theorem is correct, but it does not apply to the kernels of the record. The $\delta$ read from the $\ell^4$-weighted average alignment is about 0.46, 0.39, 0.39, 0.35 and is not small; $U$ grows as 3.0, 4.5, 8.8, 15.8; and the minimal alignment per row is almost 0 (the assumption of $\delta$ on every row is not satisfied) Computed. The Schur test itself was also two orders of magnitude coarser than the Stein bound (Schur ÷ lower bound 437 → 745 → 1453 → 1777) Computed.

The W coordinates — the absolute value after composition stops at a rank-one value

The loss is the ratio of $\rho$'s ($\rho(|\cdot|)/\rho$). The value 1.10 at which the side taking the absolute value after composition stops almost agrees with the rank-one value $\sum|u_i||v_i|/|\sum u_iv_i|=1.085$ built from the left and right dominant eigenvectors $u,v$ Computed. If the dominant eigenvalue has a gap, $T^k/\rho^k\to uv^{\rm T}/(v^{\rm T}u)$, and since $|\cdot|$ and $\rho$ are continuous it converges to this value Reading. The side composing after taking absolute values grows as $(\rho(|M|)/\rho(M))^k$ for powers of a single matrix (here by a factor of 1.070 each time), and is bounded only when $\rho(|M|)=\rho(M)$, which for irreducible matrices means $M=e^{i\alpha}D|M|D^{-1}$ (Wielandt) Cited. That the absolute value after composition is at most the composition after absolute values (row $\ell^1$) is abs_comp_le, row_l1_comp_le Lean.

Representing by a local kernel a map determined only on fields satisfying the coarse Bianchi identity — this handwork came up as the choice of "representative" in Theorem G′ of 15-05. In this section we separate the overall shape of the representatives (an affine space), the quantities that do not depend on the representative (the value of the zero mode, and the order of folding restricted to the constraint), and the quantities that do (the overall order of folding and the tail norm). The content is a restatement of the order of multigrid transfer operators and of the sum rules of subdivision (conditions of Strang–Fix type).

Definitions

Definition 3.1 (constrained map and extension). We call a quadruple $(E,V,W,f)$ of finite-dimensional spaces $E,W$ over a field $K$, a subspace $V\le E$ and a linear map $f:V\to W$ a constrained map, and a $g:E\to W$ with $g|_V=f$ an extension. Extensions exist (the restriction map ${\rm res}_V$ is surjective), any two extensions agree on $V$, and the set of extensions is the affine space $g_0+\ker{\rm res}_V$, of dimension $(\dim E-\dim V)\cdot\dim W$ Lean (exists_extension, restrict_surjective, IsExtension.eqOn, isExtension_iff_mem_coset, finrank_freedom). The freedom is 0 only when $V=E$ ($W\ne0$; freedom_eq_zero_iff).

Definition 3.2 (lattice representative). The coarse plaquette field is $\theta:\mathbb Z^4\to\mathbb C^6$, the constraint subspace is $V=\operatorname{im}D_c$ (fields satisfying the coarse Bianchi identity), and the output is a fine plaquette field with $96=16\times6$ components per coarse cell. A translation-invariant kernel $K:\mathbb Z^4\to\mathbb C^{96\times6}$ whose symbol satisfies $\hat K(q)\hat D(q)=\hat R(q)$ for all $q$ (with $R=FD_c$) is called a representative of $F$, and one that has finite range, exponential decay or $\ell^1$ is called a local representative. For $q\ne0$, $\hat D(q)$ has rank 3, the symbol at each $q$ is an instance of the template "$E=\mathbb C^6$, $\dim V=3$, $W=\mathbb C^{96}$", and the freedom is $(6-3)\cdot96=288$-dimensional.

Definition 3.3 (zero mode, moments). We call $\hat K(0)=\sum_ZK(Z)$ the zero mode and $M_\alpha(K)=\sum_ZZ^\alpha K(Z)$ the moments.

Definition 3.4 (order of folding). For $\eta\in\{0,1\}^4$, $p_\eta=(q+2\pi\eta)/2$ and the fine position $s\in\{0,1\}^4$ within a coarse cell, the folded component is $\hat C_\eta(q)[o,a]=\sum_se^{-ip_\eta\cdot s}\hat A_{(s,o),a}(q)$. We write $\mathrm{ord}(K)$ for the minimal order to which $\hat C_\eta(q)$ with $\eta\ne0$ vanishes as $q\to0$, and $\mathrm{ord}^V(K)$ for the minimal order of $\hat C_\eta(q)P(q)$, multiplied by the projection $P(q)$ onto the constraint subspace.

Quantities that do not depend on the representative

The zero mode is determined by continuity alone (Candidate 3.1). If $K$ is an $\ell^1$ representative (continuous symbol), $\hat D(q)=D_1(q)+O(|q|^2)$, $\hat R(q)=R_1(q)+O(|q|^2)$ and $\sum_\nu\operatorname{im}D_1(e_\nu)=\mathbb C^6$, then $\hat K(0)$ is the unique solution of $\hat K(0)D_1(e_\nu)=R_1(e_\nu)$ ($\nu=1,\dots,4$) Reading. The last step (maps that agree on a family of subspaces whose sum spans the whole are equal) is eq_of_eqOn_iSup, correction_eq_zero_of_iSup Lean. The value in the record is $\hat K(0)=U/4$, $U=\mathbf 1_{16}\otimes I_6$ ($\theta/4$ uniformly at every position), with deviation $\le3.1\times10^{-10}$ over four steps Computed. On the torus $\hat D(0)=0$, so the zero mode is unconstrained, and for the symmetric representative on finite tori $\max|\hat A(0)-U/4|=0.250$, $1.47\times10^{-2}$, $2.55\times10^{-5}$ at $L_b=8,16,32$ Computed. This is the numerical trace of the constraint of locality being lost in a finite box.

The minimal-norm representative is not local (Candidate 3.2). $\hat K_{\rm MP}(q)=\hat R(q)\hat D(q)^+$ (the Moore–Penrose representative) does not lie in $\ell^1$ unless the limits along different directions agree, because the projection $P(q)$ is a function of $q$ homogeneous of degree 0 and leaves a memory of direction at $q=0$ Reading. The freedom of local representatives can be seen as the room to repaint this memory of direction into a continuous function. The decay of the kernel has not been measured (F-SYS9).

The order restricted to the constraint does not depend on the representative (Candidate 3.3). With the singular value decomposition $\hat D=U\Sigma V^{\rm H}$, $\hat C_\eta(q)U(q)$ can be written using only the folding of $\hat R$, so $\mathrm{ord}^V(K)=\mathrm{ord}^V(F)$ is an invariant of $F$. Restriction does not lower the order of vanishing, so for every representative $\mathrm{ord}(K)\le\mathrm{ord}^V(F)$ Reading.

A4 attains $\mathrm{ord}(K)=\mathrm{ord}^V(F)=2$ on the whole $6\times6$. That is, within the types of the record, A4 is a representative that reaches the ceiling.

Tail norm and order — D ≤ 4 is the upper limit

Candidate 3.4. In the $n$-step map of the W coordinates $(Tf)(q)=16^{-n}\sum_mG_m(q)^{\rm H}f(p_m)G_m(q)$ ($m$ runs over $16^n$ paths, $G_m$ is a product of foldings), suppose that the representatives of all steps satisfy $\mathrm{ord}(K_k)\ge k_0$ and that $\hat C_0(0)=4I$. (a) If $D\le2k_0$, $T$ is bounded in the norm on the symbol side $\|\cdot\|_D=\sup_{q\ne0}\|\cdot\|_2/|\hat q|^D$. (b) If $D\gt2k_0$ and the leading coefficient $S$ of the paths with $m\ne0$ is non-zero, then for a peaked input near $q_0$ it diverges at the rate $|q_0|^{2k_0-D}$ Reading (a line of argument assuming constants uniform in the number of steps; uniformity over infinitely many steps is not treated). In the version restricted to the constraint, $k_0$ is replaced by $\mathrm{ord}^V(F)$.

From $R$ ($\mathrm{ord}^V=2$) it was bounded at $D=4$, $q^{-2}$ at $D=6$, and $q^{-3}$ at $D=7$ Computed. All cases in the record fit the types (a)(b). Since $\mathrm{ord}^V(F)=2$, the $D=6$ tail on $\operatorname{im}D$ diverges however the representative is chosen, and this reads as: within the types of the record, $D\le4$ is the upper limit for the tail of sup$/|\hat q|^D$ Reading.

The 2 in "$2\cdot\mathrm{ord}$" comes from the map of the W coordinates being a congruence $G^{\rm H}fG$ with the same representative on both sides. In multigrid the orders of prolongation and restriction can be chosen separately, and there is the rule "order of prolongation + order of restriction ≥ order of the equation" (Hemker 1990) Reading. Here a single representative plays both roles.

Representatives with matched first moments, and the sum rules

Candidate 3.5 (order and shifted jets). Putting $J_s(q):=e^{-iq\cdot s/2}\hat K_s(q)$ ($\hat K_s$ the symbol of the 6 rows at position $s$), $\mathrm{ord}(K)\ge k$ ⇔ the Taylor polynomial of $J_s(q)$ up to degree $k-1$ does not depend on $s$ Reading. The line of argument is $\hat C_\eta(q)=\sum_s(-1)^{\eta\cdot s}J_s(q)$ together with the orthogonality of the 15 non-trivial characters of $(\mathbb Z/2)^4$ (inner product 0 with all of them ⇔ constant).

• $k=1$: $\hat K_s(0)$ does not depend on $s$ ($U/4$ has this form).
• $k=2$: the first moments are $M_s=(s/2)\hat K_s(0)+B$ ($B$ independent of $s$). From $\hat K_s(0)=I/4$, $M_{s,\mu}=s_\mu I/8+B_\mu$. This agrees with the target moments used to build A4, $M^*_{(s,o),\mu,a}=s_\mu\delta_{oa}/8+B_\mu[o,a]$. The $8$ is (position within the cell $s/2$) × ($1/4$).

The target $B$ is determined from the $q^2$ term of $R$. Against 24 unknowns per row (4 directions × 6), the part determined by $R$ had rank 20, leaving 4 dimensions of freedom Computed. This count is of the type of a finite-dimensional "extension with conditions": if $({\rm res}_V,C)$ is surjective, there is a $g$ satisfying both $g|_V=f$ and $C(g)=y$, and the remaining freedom is $\dim E\cdot\dim W-\dim V\cdot\dim W-\dim Y$ Lean (exists_extension_with_conditions, finrank_freedom_with_conditions, conditions_count_le). The remaining 4 dimensions were fixed by linear programming to minimise the increase of $\ell^1$; the moments matched the target to within $1.1\times10^{-16}$, and the row $\ell^1$ went from 2.26 to 2.34 (first step) Computed.

Composition of representatives and compatibility

If $(E_1,V_1,E,f_1)$ and $(E,V_2,W,f_2)$ are compatible ($f_1(V_1)\subseteq V_2$), the composite of representatives is a representative of the composite, and its value on $V_1$ does not depend on the choice of representatives. If they are not compatible ($W\ne0$), the value of the composite changes with the choice Lean (comp_extension_eqOn, comp_isExtension, comp_depends_of_not_le). The chain of W coordinates in the record composes representatives, but whether the output of the representative at step $k$ lies in the constraint subspace of step $k+1$ has not been checked (F-SYS8). The difference between the chain on $\operatorname{im}D$, 0.3426, and the $6\times6$ chain, 0.3446, is a clue to this Computed. As examples of norms changing with the representative: the zero-mode correction moved the three-step Stein bound from 0.09743 to 0.09868, and the signed Stein bound went from 4.25 for A3 to 4.48 for A4 (first step) Computed.

Dictionary

• $\mathrm{ord}_\eta(K)$ corresponds to the high-frequency order of the representative viewed as a prolongation (Hemker 1990). That the definition by symbols and the definition by preservation of polynomials are equivalent is stated in a later note, and the condition on the zeros of the symbol in Toeplitz multigrid is in Donatelli (arXiv:0807.2565) Reading. $\hat C_\eta$ is the same as the 16 components of the harmonic subspaces of local Fourier analysis Cited.
• Candidate 3.5 is exactly the standard argument that turns the sum rules of subdivision and wavelets into polynomial generation (Strang–Fix 1973, Jia 1998, Charina–Conti arXiv:1202.3931, Daubechies's vanishing moments) Cited. The part $s_\mu I/8$ of the target of A4 is a restatement of the sum rule of order 2.
• Two points fall outside the sum rules: (i) the part where $B$ is determined by the $q^2$ term of $R$ (a condition of the type of order of consistency of differences); (ii) the ceiling $\mathrm{ord}^V(F)$ set by the constraint $V=\operatorname{im}D$. That the masks are not scalars but $96\times6$ matrices is within the range of sum rules for vector-valued subdivision Reading.

There is no payoff theorem yet. This section is an attempt to restate ⑦ the crossover band of the ledger (13-06) in the language of two valuations, those of the strong-coupling expansion and the weak-coupling expansion. The definitions and the computation of the leading terms are done, but there is as yet no theorem that this view newly says about the band. The labels are mainly Assumption and Computed.

Definitions

The space of couplings $\mathcal K$: the space of character coefficients $(a_r)$ of the plaquette weight $f(U)=1+\sum_rd_ra_r\chi_r(U)$. Its coordinates are $\beta$ and the shape coordinates.

Two valuations. $v_s$ (strong coupling) is the order on the side $a_r\to0$ (the order in $\beta$ for the Wilson action), and $v_w$ (weak coupling) is the order in $1/\beta$ on the side $\beta\to\infty$. Terms of type $e^{-c\beta}$ are treated as $v_w=\infty$ (another branch of the transseries).

Two coordinates. For an observable $O$ with values in $[0,1]$ ($O(0)=0$, $O(\infty)=1$; the plaquette $P$, for example), the coordinate on the strong-coupling side is $O$ and the coordinate on the weak-coupling side is $1-O$. We write the leading coefficients as $O\approx s_1\beta$ (strong coupling) and $1-O\approx w_1/\beta$ (weak coupling).

Tie sets. (i) the complementary tie $\{s_1\beta=w_1/\beta\}$; (ii) the same-coordinate tie $\{s_1\beta=1-w_1/\beta\}$; (iii) the error tie $\{K$-term optimal truncations have equal errors$\}$. We call the region between the two roots of (ii), $\{s_1\beta\le1-w_1/\beta\}$, the lens.

The invariant κ and the lens

Candidate 4.1 (a scale-independent quantity). Changing the scale of the coupling as $\beta\mapsto c\beta$ gives $s_1\mapsto s_1/c$, $w_1\mapsto cw_1$, so

does not depend on the scale. The complementary tie is at $\beta^*=\sqrt{w_1/s_1}$, where the value of the leading terms is $\sqrt\kappa$. The same-coordinate tie is at the roots of $s_1\beta^2-\beta+w_1=0$, whose discriminant is $1-4\kappa$, so the lens opens only when $\kappa\le\frac14$. The geometric mean of the two roots is always $\beta^*$. A tie for the string tension $\sigma=-\log u$ exists only when $\kappa\le1/e$ Reading (the elementary fact of the discriminant of a quadratic).

Looking at the leading term on one side only, the difference between groups is absorbed into the scale. Looking at the leading terms on both sides at once, a change of scale is only a translation in the $(\log O,\log(1-O))$ plane, and the height $\sqrt\kappa$ at which they cross does not move. The view is that the information about the group remains in "the height of the crossing" Assumption.

Examples

The specific heat $\beta^2u'$ has a peak (0.680) at $\beta=2.48$ for $U(1)$, while for $SU(2)$ it rises monotonically to $3/2$ Computed. This difference goes in the same direction as the difference in the sign of $w_2/w_1^2$ ($-1/6$ and $+1/2$), and can be read as: the difference between groups lives not in the leading terms of the valuations but in the sign of the next term Assumption. That $u(\sqrt6)=0.5004\approx\frac12$ for $SU(2)$ does not hold for $U(1)$ (0.446), so it is not a general law.

The lens of the leading terms alone in four dimensions. Using equipartition $w_1=\dim G/d$, $\kappa=s_1\dim G/d$.