2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerationsthe length at which conservation breaks, coordinates, two-dimensional test beds · Kiichi, Shiori and Rin

Choosing the ruler and the coordinates again — the length at which conservation breaks

Assumption H is the power statement $\lim\ln C_P/\ln\xi=2$ ((13) of 10-01), comparing the relaxation time $C_P$ with the square of one length $\xi$. Part 10 took this $\xi$ to be the correlation length $1/m_{\rm lat}$ and used the bookkeeping length $\xi_{\rm RG}$ in its place (10-01 §07). But in the pattern of 11-01, what fixes the slowness is not the correlation length but "the scale of the events that break the global quantity $Q$". So we split the candidates for what to compare with into three.

(a) The static correlation length $\xi$. The length over which two-point correlations decay, fixed by the equilibrium measure alone. In four dimensions $\xi=1/m_{\rm lat}$. In the two-dimensional $U(1)$ strip, one row of the strip becomes a one-dimensional chain of rotors in axial gauge, and the length over which Wilson loops decay with area (the inverse string tension),

plays this role ($I_\nu$ is a modified Bessel function. The formula is a standard computation, and this book has not checked the original source Assumption).

(b) The length at which conservation breaks $\ell_*$. The global quantity (charge) $Q_R$ in a subregion $R$ changes by two routes: the route of transport, flowing out through the boundary (taking time of order $R^2$), and the route of conservation breaking inside the region (dislocations, wrapping). The length at which the two balance is $\ell_*$. The crossing length $R_*$ of (7) in 11-01 was its static version. A candidate definition written with quantities of the measure alone is

Estimate. Putting in the estimate $\operatorname{Var}/\mathcal E\approx\chi_tR^2/(hR^2+d\lambda_W)$ of (7) in 11-01 gives $S=\chi_t/h$, and the ratio reaches half of it when $hR^2=d\lambda_W$, that is, $\ell_*=R_*$ Assumption (on top of the estimate (7)). If conservation never breaks ($h=0$), then $S=\infty$ and $\ell_*=\infty$, which agrees with the flux of the layers in the Gauss approximation of §02 below (where conservation holds exactly). Strictly, a definition of the form "the supremum of $R$ with $\operatorname{Var}/\mathcal E\le KR^2$", comparing the ratio with $R^2$, does not fix a length: under the estimate (7) the ratio divided by $R^2$ decreases in $R$, so the set of $R$ satisfying the condition is a half-line open upwards or empty, and the supremum is infinite or undefined. (2) avoids this by cutting at half the saturation value.

(c) The time for the global quantity to move by one $\tau_{\rm glob}$. A time, not a length. Its representative is the integrated autocorrelation time $\tau_{\rm int}(Q)$ of the topological charge $Q$ in the periodic box, and for reversible dynamics $\tau_{\rm int}(Q)\le C_P$ ((1) of 11-01).

The three can take different values within the same model. In the 0+1-dimensional double well, the slow quantity (the number of kinks) and the correlation length are born from the same kinks, and the three differ only by logarithms (11-01 Table 3). In the two-dimensional strip of §03, (a) grows linearly in $\beta$ while (b)(c) grow exponentially, and the three move apart. Fixing the model does not automatically fix which ruler H is a sentence about.

What this section can say: (2) is a candidate definition, and the four-dimensional value of $\ell_*$ is not in this book. The $\xi_{XY}$ of (1) is used only as the static length of the strip; it does not stand in for the four-dimensional correlation length.

The denominator of the Poincaré constant $C_P=\sup\operatorname{Var}(f)/\mathcal E(f)$ is a Dirichlet form, and it is not fixed until one decides the variables whose gradients it measures. We saw in Part 2 that changing variables by a constant factor changes $C_P$ by a constant factor ("Differences other than constants" in 2-10: a factor $\beta$ depending on whether one measures by link angles or by the field). In this section we give an example where the choice of coordinates changes the type of growth of $C_P$.

Stage. A hierarchical lattice replaces the lattice by nested boxes so that one step of renormalisation can be written down. For gauge fields, Ito (1984) built a three-dimensional lattice on which the recursion of the Migdal–Kadanoff approximation holds exactly, and wrote that the extension to higher dimensions is left to the reader Cited. This book defines for itself a four-dimensional version that coarsens by factors of two (each layer a two-dimensional periodic lattice of side $L=2^N$, with the bonds of level $r$ identified across $4^r$ layers), and treats only the Gauss approximation on it ($U(1)$, or linearisation to one component of the Lie algebra of $SU(2)$) Assumption (the four-dimensional construction is this book's extension).

Tree coordinates. Taking the values of the plaquettes (the holonomies at the nodes) as variables, in the Gauss approximation the configuration space is a linear subspace $S$, the equilibrium measure is the density $e^{-\beta|x|^2/2}$ on $S$, and the Hessian of the Langevin potential is $\beta I$ on $S$.

Proved here Proof. It is an Ornstein–Uhlenbeck process with unit metric and drift matrix $\beta I$, so the gap of the generator is $\beta$. ∎ Lean HierLink.tree_no_decay (bundle LatticeGaugeHierarchicalLink; assumption: $0\lt\beta$; only the arithmetic that the ratio is 1). Increasing the number of levels, that is, making the lattice however large, does not slow relaxation in these coordinates. Even if the correlation length grows under the non-abelian flow, the coarse levels are expected to stay close to Haar with $C_P=O(1)$ Estimate, and in tree coordinates H is false or empty.

Link coordinates. What H in Part 10 names is Langevin in the link variables (10-01 §04). The link generator in the Gauss approximation is $\beta\,D\Pi D^*$ ($D$ is the exterior derivative from links to plaquettes, $\Pi$ the projection of the identification), and its smallest positive eigenvalue is

Proved here (paper: splitting functions of the layers into sectors by the hierarchical projection, the block operator for sector $k$ becomes the Neumann Laplacian of the two-dimensional dual lattice of side $2^{k+1}$, and the minimum is the value for sector $k=N-1$ with side $2^N$) Reading (agreeing with the direct full spectrum for $N\le2$, and with a separate implementation by sectors for $N=3,4$). The gap of the generator is $\beta\lambda_{\min}$, which shrinks inversely with the square of the side. Lean HierLink.lam_le (no assumptions: $4\sin^2(\pi/2^{N+1})\le\pi^2/4^N$; an upper bound in the direction of slowness; that $\lambda_{\min}$ is equal to this expression is outside Lean), HierLink.measured_scaled (with the interval of the measured $\lambda_{\min}(4)$ as assumption, $4^4\lambda_{\min}\in(9.83,9.84)$).

Labels of Table 1: the tree-coordinate column is Proved here by (3). Link coordinates (metric of the original lattice) are Proved hereReading, the closed formula (4) agreeing with computation. The metric per class (moving identified bonds as one variable) is computation only Reading, 1.0–1.74 times the metric of the original lattice. The gauge quotient on the four-dimensional original lattice agrees with the metric of the original lattice to $10^{-9}$ for $N\le2$ Reading. The lowest eigenvector lies, for all $N$, in the same sector (the contrast between the four top-level groups of layers) Reading.

The slowness comes from conservation of flux. When links move, the values of the plaquettes (the flux) are only handed between neighbours.

Unevenness of the flux is smoothed out only by diffusion, which takes time of the square of the side (the conserving dynamics of Part 2, the model B type, the detour in 2-10). The decisive point is the last column of Table 1. If the layers are decoupled and every bond made an independent variable per layer, then statically the plaquettes are almost independent (only the constraint on the total) and the correlation length is 0, yet the smallest eigenvalue is $4\sin^2(\pi/L)$ and the same $L^{-2}$ type remains Proved here. Hence this slowness does not come from the correlation length. The static correlation of plaquettes in the Gauss approximation decays masslessly, of order $-\beta^{-1}R^{-4}$; $\beta$ does not flow (the Gauss-approximation face of the coupling being dimensionless in four dimensions, 11-07), and the correlation length is only cut off by the volume Proved here.

Consequence. Assumption H is not a sentence about the measure alone but a sentence about the pair (measure, Dirichlet form) Estimate. Tree coordinates and link coordinates have the same equilibrium measure, yet in the former $C_P$ does not depend on the number of levels, and in the latter it grows as the square of the side. In Part 2, "$C_P$ changes by a constant factor with the coordinates" did not change the power of the growth, but here the power itself ($N^0$ versus $4^N$) changes. H in Part 10 names Langevin in the link variables, so it already meets this point. Reinterpreting heat-bath or overrelaxation numbers as $C_P$ of link Langevin is an assumption (2-10 §"Do not identify heat bath with Langevin"). And even in link coordinates, conservation holds exactly in the Gauss approximation, so $\ell_*$ of (2) is infinite, and comparing "$C_P\propto L^2$" with the correlation length has no meaning. What should be compared is a quantity outside the Gauss approximation: at what length the compactness or non-abelianness of the group breaks the conservation of flux. The next section is the first measurement of it.

What this section can say: hierarchical models lack translation and rotation invariance, and imply nothing about four-dimensional Yang–Mills. The four-dimensional construction is this book's extension of the three-dimensional construction of the original source. The numbers are numbers for the Gauss approximation, with $N\le4$.

Model. We cut one layer of the previous section into a narrow strip. Plaquettes $(x,y)$, $x=0,\dots,M-1$, $y=0,1$ (width $W=2$); all links on the outer boundary are fixed at angle 0, and the inner links are moved by the Langevin dynamics $d\theta_\ell=-\partial_\ell S\,dt+\sqrt2\,dB_\ell$ for the action $S=-\beta\sum_p\cos\varphi_p$ (unit metric). Gauge transformations at inner vertices do not change the plaquette angles $\varphi_p$, so on gauge-invariant functions the link generator reduces to a diffusion on $\varphi$, whose metric is the Neumann Laplacian of the dual graph of the plaquettes Proved here. The equilibrium measure is the product of independent $e^{\beta\cos\varphi}$ per plaquette conditioned on $\sum_p\varphi_p\equiv0\ (\mathrm{mod}\ 2\pi)$, and there is almost no static correlation between plaquettes.

In the Gauss approximation conservation holds exactly. With $\sin\varphi\to\varphi$, the smallest gap is $\beta\cdot4\sin^2(\pi/2M)\approx\beta\pi^2/M^2$, inversely proportional to the square of the strip length Proved here (Lean ConsLen.neumann_gap_lower, ConsLen.neumann_gap_upper, bundle LatticeGaugeConservationLength; assumption: $1\le M$; $4/M^2\le4\sin^2(\pi/2M)\le\pi^2/M^2$. That the gap equals this expression is outside Lean).

If the group is compact, conservation breaks. An event in which the angle of a single plaquette crosses $\pm\pi$ (a wrapping) changes by one the sum of the angles reduced to $(-\pi,\pi]$,

(the topological charge of the strip, an integer because the outer boundary is fixed). A wrapping is a rare event crossing a barrier of height $2\beta$. We measure two quantities: $\lambda_X:=1/\tau_{\rm int}(X)$ for the flux dipole $X=\sum_pw_p[\varphi_p]$ ($w_p=\cos(\pi(x+\tfrac12)/M)$, the slowest mode of the Gauss approximation), and $\lambda_Q:=1/\tau_{\rm int}(Q)$ for the topological charge. For reversible dynamics $\tau_{\rm int}\le C_P$ ((1) of 11-01), so both are read as upper bounds on the gap. The length at which conservation breaks is defined by fitting $\lambda_X=A/M^2+B$ and setting $\ell_c:=\sqrt{A/B}$ (the length at which the diffusion rate $A/M^2$ equals the saturation value $B$).

Every label in Table 2 is Reading. "Paper estimate" is the value, by numerical integration, of a capacity estimate with a test function (the length at which the two terms of an upper bound become equal, using a test function whose return from wrapping is optimised by capacity; for large $\beta$, $e^\beta\sqrt{\pi/(6\beta)}$) Assumption (an estimate on top of the independence approximation), and is of the same order as the measured $\ell_c$.

Slopes. With a weighted straight line over $\beta=2,\dots,6$:

• The slope of $\ln\ell_c$ is $a=0.916(21)$. Split into intervals: $0.988(34)$ for $\beta=2$ to $4$, $0.834(46)$ for $4$ to $6$.
• The slope of $-\ln\lambda_Q$ is $b=1.355(18)$. Split into intervals: $1.286(28)$ and $1.520(57)$ (difference $3.7\sigma$).
• The slope of $\ln$ of the combination $\pi/\sqrt{E\lambda_Q}$ is $0.884(9)$, and $0.879$ to $0.884$ on every interval.

Reading Lean ConsExp.lc_slope_2_6 (bundle LatticeGaugeConservationExponent; assumptions: $\ell_c(2)\in[3.09,3.49]$, $\ell_c(6)\in[125,153]$, the measured $1\sigma$ intervals; conclusion: $a\in[3/4,1]$), ConsExp.lamQ_slope_2_6 (with the $1\sigma$ intervals of $\lambda_Q(2)$ and $\lambda_Q(6)$ as assumptions, $b\in[5/4,3/2]$). The bundles guarantee only interval arithmetic.

Not proportional to the static length. The static length $\xi_{XY}\approx2\beta$ is linear in $\beta$ and grows only about fourfold over the five points. The length at which conservation breaks, $\ell_c\propto e^{0.92\beta}$, grows 42-fold, and the ratio $\ell_c/\xi_{XY}$ opens from $1.2$ to $12.8$ Reading. Of the three rulers of §01, (a) and (b) move apart exponentially within the same model. When the strip is longer than $\ell_c$, relaxation stops depending on the strip length and is fixed by the time for $Q$ to move by one.

Conditions: two-dimensional $U(1)$, width 2, outer boundary frozen, gauge quotient of link Langevin with unit metric, $\beta=2,\dots,6$, strip length $M\le256$, $\tau_{\rm int}$ as a reading of an upper bound on the gap, fit of the form $A/M^2+B$ (for $\beta=2$, $\chi^2/\text{dof}=3.3$, and the form does not fit). Nothing is said about the exponent as $\beta\to\infty$.

The $\rho$ column of Table 2 shows that the two quantities are one process. The relaxation of the flux dipole $X$ falls as $M^{-2}$ at the rate $\mu_1(M)/E$ ($\mu_1\approx\pi^2/M^2$), with the Gauss stiffness replaced by $1/E$, while the relaxation rate $\lambda_Q$ of the topological charge $Q$ does not depend on the strip length. The length at which the time for diffusion to cross a length $\ell$ equals the time for $Q$ to move by one is

Proved here (algebra) Lean ConsExp.lc_eq_iff (assumptions: $0\lt E$, $0\lt\lambda$, $0\lt\ell$), and its ratio $\rho$ to the measured $\ell_c$ is $0.80$ to $1.03$ over the five points Reading. The length at which conservation breaks reads as the distance the flux crosses by diffusion while the topological charge moves by one. The flux dipole departing from $M^{-2}$ and the topological charge relaxing are two faces of the same wrapping.

A single point at $\beta=8$ supplements this reading. Up to strip length 64, $X$ keeps $M^{-2}$ (ratio to the Gauss stiffness $0.86$ to $1.03$), but at strip length 32, $\tau_{\rm int}(Q)=5.3(1.7)\times10^3$ (run length $10^6$, $\operatorname{Var}Q=0.145$), and a slow mode at about $1/380$ of the value for $X$ appears Reading. Because $\operatorname{Var}Q$ is small (a sector where $Q\ne0$ is rare), this point is not included in the slopes.

What this section can say: $\ell_c$ is the length of the bend in the relaxation of $X$, not the length of a bend in the gap itself (including $Q$, the upper bound on the gap at $\beta=4$ already saturates at $M\approx8$ to $16$). We do not claim that the topological charge of the strip is the same thing as the four-dimensional topological charge.

The paper prediction before measuring was "$\lambda_Q\propto\beta e^{-2\beta}$" (the Kramers rate for the barrier $2\beta$), so the slope $b$ should have been near 2. The measured $b=1.355(18)$ is gentler, and moreover split before and after $\beta$ (bottom of Table 2). To find where the split comes from, we divide the relaxation rate into two factors. $Q$ can be regarded as a chain of $\pm1$ jumps with a restoring force, so from the total jump rate $WM\nu$ ($\nu$ is the rate of wrapping per plaquette) and the variance $\operatorname{Var}Q\approx WM\,E/(4\pi^2)$ comes

this form Assumption (the chain picture; $\kappa$ is a correction for the raw count. It is a different quantity from the comparison constant $\kappa$ of Proposition B (11-A1)). The $\nu$ in the numerator is the raw rate (a quantity of the dynamics), and the $E$ in the denominator is the static fluctuation (a quantity of the measure). Taking logarithms gives

Proved here (algebra) Lean ConsExp.lamQ_log_split (assumptions: $0\lt\kappa$, each quantity positive).

Counting the raw number of wrappings $\nu_{\rm raw}$ (the number of times $\pm\pi$ is crossed), we found the following Reading.

• $\kappa=\lambda_QE/(2\pi^2\nu_{\rm raw})$ is constant at $0.167,\ 0.165,\ 0.158,\ 0.167$ for $\beta=3,4,5,6$ (the raw count also counts back-and-forth trips near $\pi$, so the effective crossings are about $1/6$ of it).
• The slope of the raw rate $-d\ln\nu_{\rm raw}/d\beta$ is $1.73,\ 1.75,\ 1.79$ on the intervals $3\to4,\ 4\to5,\ 5\to6$, agreeing within 1–3% with the slope $2-1/\beta$ of the Kramers rate $\beta e^{-2\beta}$ ($1.71,\ 1.78,\ 1.82$ at the midpoints of the intervals).
• The slope of the static factor $\ln(1/E)$ is $0.38,\ 0.27,\ 0.21$, decreasing with $\beta$ (approaching the Gauss value $1/E\approx\beta$).

From (9), $b=(\text{slope of the raw rate})-(\ln(1/E)\ \text{slope})$, and since the latter decreases with $\beta$, $b$ increases with $\beta$. This is what the split really is. The paper prediction had dropped the static factor $1/E$. The corrected form is $\lambda_Q\propto\beta e^{-2\beta}/E$ (for large $\beta$, $\propto\beta^2e^{-2\beta}$, $b\to2-2/\beta$), in the same direction as the measured $b$ Estimate. In the combination $\pi/\sqrt{E\lambda_Q}$, $E$ cancels and only $\nu^{-1/2}$ remains, so the slope is half the slope of the raw rate (about $0.88$) and constant over all intervals. That is the third slope at the bottom of Table 2.

Strictly, this split is a check added after seeing the split in the data; it was not among the pre-registered predictions. Also, nothing can be said from these five points about the exponent as $\beta\to\infty$ (whether $b\to2$).

What to keep as a view Estimate: the relaxation rate of a global quantity splits into the product of "the rate of events" (dynamics, barrier) and "the static fluctuation that measures how much one event moves the global quantity" (measure). The one-loop accounting that the original sources write for the four-dimensional topological charge (11-07 §02) also counts only the exponent of the density of events. Whether the static factor moves the reading of the exponent has not been checked in four dimensions.

In the strip, measuring $\tau(Q)$ with the static length $\xi_{XY}$ as the ruler, $\tau(Q)/\xi_{XY}^2$ is about $0.4$ at $\beta=4$ and about $24$ at $\beta=8$ (strip length 32) Reading. Since $C_P\ge\tau_{\rm int}(Q)$, $C_P/\xi_{XY}^2$ also grows with $\beta$. "$C_P\lesssim\xi^2$", with only the static correlation length as the ruler, tends to fail with $\beta$ inside this toy (row (c) of the pattern in 11-01). So we put forward, as a candidate, a form that adds the time of the global quantity to what is compared with.

Estimate. This is one direction for correcting H. The other direction is to restrict the side of test functions to families local in time and space (11-03 §02), where, as long as the box is smaller than $\ell_*$, the time of the global quantity is expected not to enter (§07).

(10) has three weaknesses. (i) If $\tau_{\rm glob}$ is defined by $\tau_{\rm int}(Q)$ of the same dynamics, (10) is close to a restatement of $\tau_{\rm int}(Q)\le C_P$ and has no content. To give it content, $\tau_{\rm glob}$ must be fixed independently of the dynamics. Candidates are the split of §05 (barrier rate and static fluctuation, $\tau_{\rm glob}\approx E/(2\pi^2\kappa\nu)$), or making the diffusion time $\ell_*^2/D$ from the static $\ell_*$ of (2) Estimate. (ii) The strip numbers see only the lower side of (10), $C_P\ge\tau_{\rm glob}$, and not a single number bounds $C_P$ from above. At $\beta=4$, $\tau(Q)\approx17$ against $\xi_{XY}^2\approx46$, and whether $C_P$ is on the side of $\xi_{XY}^2$ has not been measured. (iii) In four dimensions the candidate for $\tau_{\rm glob}$ is the topological charge in the periodic box, whose measurements are the heat-bath and HMC numbers of the original sources, and reinterpreting them for Langevin is an assumption (11-01 §06).

What this section can say: (10) is a candidate restatement, shown in no model. Nor do we propose replacing H of Part 10 with (10). The order of the account is: because H was set up sharply, we could name the places where a single thing to compare with is not enough.

Why this model. The two-dimensional $CP^{N-1}$ model shares two points with four-dimensional Yang–Mills — a dimensionless coupling, and instantons with no scale (11-07) — and lets one vary $N$, which plays the role of the size of the group. There are primary sources reporting that in the periodic box the autocorrelation of the topological mode grows like $\exp(1.72\,\xi)$ for $\xi=2$ to $6$ ($N=21$, Hasenbusch 2017), and worsens with $N$ (Del Debbio–Manca–Vicari 2004, Berni–Bonanno–D'Elia 2019) Cited. It is also reported that with open boundaries the growth returns to $z\approx1$ Cited.

Setting. The action is Hasenbusch's form $S=-\beta N\sum_{x,\mu}(\bar z_{x+\mu}z_x\lambda_{x,\mu}+{\rm c.c.}-2)$ ($z_x\in\mathbb C^N$, $|z_x|=1$, $\lambda\in U(1)$), and the topological charge is the sum of plaquette angles $Q=\frac1{2\pi}\sum\theta_{\rm plaq}$. The dynamics is heat bath (sites and links) mixed with $n_{\rm ov}$ overrelaxation steps; it is not Langevin. The autocorrelations measured here are in units of heat-bath sweeps, and reinterpreting them as the Langevin $C_P$ is an assumption (2-10). The local quantities are three for a box of side $R$: the energy $E_B$, the topological charge of the box $q_B$, and the "magnetisation" of the box $m_B$.

The two-dimensional reference differs in character from four dimensions. The reference of the ratio form $K_{\rm rel}$ of 11-03 (the box value for the massless free field) diverges in two dimensions not only with the box size $R$ but logarithmically in the lattice side $L$ (the constant part of the continuum kernel $-\tfrac1{2\pi}\ln|z|$ sits on the box average) Proved here. Removing the box average makes it finite, about $0.0548$ by extrapolation in $1/R^2$ Reading. Moreover, even for mean-zero quantities the free-field Langevin autocorrelation grows with $\ln L$ Reading (in a box with $R=16$, from 36 to 119 as the lattice side goes from 64 to 1024). So the ratios of this test bed are ratios of autocorrelations with the lattice side fixed at $L=64$, $K_{\rm rel}(c;\text{qty}):=\tau(\text{qty}_B(c\xi))/\tau^{\rm free}(D_B(c\xi))$ ($D_B$ is the first Neumann mode of the free-field box, with the same algorithm and the same $L$) and $K_{\rm glob}:=\tau(Q^2)/\tau^{\rm free}(D_B(4\xi))$, quantities different from the static ratios of 11-03.

Positive control. With two runs (run length 70000 sweeps) at $N=21$, $\beta=0.70$, $L=44$, $n_{\rm ov}=6$, it passed with $\tau_{\rm int}(Q^2)=160(26)$ (0.80 times $199(18)$ of Hasenbusch's Table II, inside the band $[139,259]$ fixed in advance), an energy difference of $-9\times10^{-6}$, and a $\xi$ difference of $-0.1\%$ Reading.

Every label in Table 3 is Reading. $\tau(Q^2)$ for $N=21$, $n_{\rm ov}=0$ has a run length of only about $11\tau$ and is treated as a lower bound.

What can be read.

1. The freezing of the topological charge splits with $N$. At $\xi\approx4$, $\tau(Q^2)$ for $N=21$ against $N=2$ is $154$ against $0.50$ with overrelaxation (about 300 times), and $3874$ against $0.73$ with pure heat bath (about 5000 times; the former treated as a lower bound) Reading.
2. The ratios of local quantities are small regardless of $N$. For the pure heat-bath pairs, $K_{\rm rel}$ for the energy and the magnetisation is almost the same at $N=2,10,21$ (0.03–0.35 for $E_B$, 0.25–0.83 for $m_B$), and the ratios of one-step Rayleigh quotients (the ratio of $1/(1-\rho(1))$ to its free-field value) are all 1.1 or below Reading. The verdict at the freezing point was on the side of "freezing of topology does not break the local ratios up to box $4\xi$". Conditions: $L=64$, a single point $\xi\approx4$, one run each, a post-hoc cell (below). There is only one freezing point, $(N=21,\ \xi\approx4)$, and $K_{\rm glob}=55$ has a $1\sigma$ width of 15–95 from the error of one run, so whether it exceeds the decision threshold 30 is within that error. It is a weak reading at one point.
3. $\ell_*$ grows with $N$. Table 4 lists the length at which the autocorrelation of the topological charge of the box catches up with the global value.

Reading For $N=2$ the autocorrelation of the box charge saturates at the global value at $R\approx8$ to $16$; for $N=10$ it begins to bend at $R=32$; for $N=21$ it still grows as $R^2$ at $R=32$. The box charge takes time $R^2$ through transport across the boundary (the route (b) of §01), and the length at which it catches up with the global value is the reading of $\ell_*$. In this model and this range, freezing does not break the ratios for boxes smaller than $\ell_*$, and rides only on quantities on scales beyond $\ell_*$ Estimate. That in Table 3 only $K_{\rm rel}(q_B)$ of the box charge splits with $N$ ($0.022\to0.83\to2.27$ at $c=4$) we read as a reflection of this difference in catching-up length. In the language of 11-03, the local H asks about the transport coefficient $d$ and the global H about the coefficient of breaking of conservation $h$ (11-03 §05), so this is one two-dimensional face of the prospect that local ratios are unaffected by freezing. It also agrees in direction with the conclusion of 11-04 that what the one-sided bridge needs is the local side (11-04), but the ratios here are ratios of autocorrelations, not the ratio of the bridge's threshold (a static ratio).

What should be disclosed.

• The registered verdict could not be defined. The verdict registered in advance was the ratio for dynamics mixed with overrelaxation ($n_{\rm ov}\gt0$), but overrelaxation makes the autocorrelation of the free-field reference oscillate, and $\tau^{\rm free}(D_B)$ came out at $0.33$ to $0.54$, below 1. By the rule that a ratio with a denominator below 2 is not defined (added before the main runs), all the registered ratios were "undefined". Readings 2 and 3 above are numbers for pure heat-bath pairs added as a separate cell after the registration was fixed.
• The dynamical exponent $z_Q(N)$ cannot be read. For each $N$ there is only one point in $\xi$. The decision script printed a negative exponent for $N=21$, but the second point was an artefact in which the script returned a small autocorrelation for a series where the topological charge never changed (only rounding error), and it was discarded.
• The topological charge for $N=2$ carries an ultraviolet label. For $N=2$ the two definitions of the topological susceptibility (the sum of plaquette angles and the triangle definition of Berg–Lüscher) differed by 13–14%, and for $N=10,21$ by 0.1–3% Reading. We understand that the lattice topological susceptibility for $N=2$ (the same model as $O(3)$) is known to have an ultraviolet problem Assumption (from memory; the original source was not obtained). $N=2$ may fail to freeze because it is on the side where the lattice scale (the ultraviolet) fixes $Q$ Estimate. If so, the speed for $N=2$ is not "weak freezing" but a shadow of "$Q$ not being a topological charge in the continuum sense", and the analogy with $SU(2)$ (11-07) carries this reservation.
• Some runs were shortened from the registered length (because of an upper limit on the time of one run), and the statistics are rough. The negative control with open boundaries was not run.

What this section can say: numbers for the two-dimensional $CP^{N-1}$, this action and these algorithms (heat bath and overrelaxation), $L=64$, a single point $\xi\approx4$; nothing is said about four dimensions, $SU(N)$, Assumption H or the mass gap.

(a) Two dimensions and four. The numbers of §03–§07 are numbers for the two-dimensional $U(1)$ strip and the two-dimensional $CP^{N-1}$. Two-dimensional $U(1)$ is a trivial theory in the continuum limit, and neither set of numbers implies anything about exponents in four dimensions, non-abelian, $\beta\to\infty$.

(b) Readings of upper bounds. $\lambda_X$ and $\lambda_Q$ are readings as upper bounds on the gap (lower bounds on $C_P$), not the gap itself. $\ell_c$ is the length of the bend in the relaxation of the flux dipole and depends on the form of the fit $A/M^2+B$.

(c) Candidates. The $\ell_*$ of (2) and the thing to compare with of (10) are candidates, shown in no model. The strip numbers see only the lower side of (10).

(d) The hierarchical lattice. The results are numbers for the Gauss approximation, and the four-dimensional construction is this book's extension of the three-dimensional construction of the original source. Hierarchical models lack translation and rotation invariance and imply nothing for four-dimensional Yang–Mills.

(e) Conditions of the test bed. The $CP^{N-1}$ ratios are ratios of autocorrelations of heat-bath dynamics, conditional quantities with $L=64$ fixed, and different from the static ratios of 11-03. For each $N$ there is one point in $\xi$, one freezing point, and the verdict on local ratios is a weak reading at one point on a cell added after registration; the registered verdict could not be defined. $z_Q(N)$ has not been read.

(f) Afterthought. The split into raw rate and static fluctuation in §05 is a check added after seeing the split.

(g) The analogy with $SU(2)$. Using the split of freezing with $N$ as a two-dimensional analogy of the borderline reading for $SU(2)$ is an estimate, and $N=2$ carries an ultraviolet label (11-07).

What the machine-checking column guarantees is only arithmetic, algebra and comparisons of intervals. That eigenvalues equal closed formulas, the spectral representation of $\tau_{\rm int}\le C_P$, the fits and the Monte Carlo lie outside Lean (10-13).

Dependency graph (premise numbers are smaller than conclusion numbers):

[01] ASM GA    Gauss approx., 4D extension of hier. lattice     N ≤ 4
[02] PRV TREE  λ₁ = β in tree coordinates ⟸ [01]                Lean tree_no_decay
[03] PRV LINK  λ_min(N) in link coordinates ⟸ [01]              Lean lam_le (upper bound)
[04] RDG CHK   cross-check for N ≤ 4 ⟸ [03]                     Lean measured_scaled
[05] PRV FLUX  L⁻² even with layers decoupled ⟸ [01]
[06] EST PAIR  H is a sentence about a pair (measure, form) ⟸ [02][03][05]
[07] PRV NG    Gauss gap of the strip                           Lean neumann_gap_lower/upper
[08] PRV TI    τ_int ≤ C_P ((1) of 11-01)                       Lean tau_int_le_CP
[09] RDG BAND  five points of the strip ℓ_c, λ_Q, a, b ⟸ [07][08]
               Lean lc_slope_2_6, lamQ_slope_2_6
[10] PRV LC    algebra of (7)                                   Lean lc_eq_iff
[11] RDG RHO   ρ = 0.80–1.03 ⟸ [09][10]
[12] PRV SPL   the split (9)                                    Lean lamQ_log_split
[13] RDG KR    raw rate and κ (added afterwards) ⟸ [09]
[14] EST EF    the paper prediction dropped 1/E ⟸ [12][13]
[15] RDG TQ    τ(Q)/ξ_XY² goes 0.4 → 24 ⟸ [08][09]
[16] EST MAX   compare with max(ξ², τ_glob) ⟸ [06][11][15]
[17] RDG PC    positive control for CP^{N−1}
[18] RDG SPLIT τ(Q²) splits with N ⟸ [17]
[19] RDG LOC   local ratios small regardless of N ⟸ [17]        post-hoc cell, one point
[20] RDG QB    catching-up length grows with N ⟸ [17]
[21] EST UV    is N = 2 on the side where UV fixes Q? ⟸ [18]
[22] EST ELL   freezing rides only on quantities beyond ℓ_* ⟸ [19][20]

How to check: see by eye that on every "⟸" the numbers on the right are smaller than the number on the left. There are no circles. The conclusion nodes of this chapter, [06][16][22], are all "Estimate", and there is neither a four-dimensional node nor a node for the mass gap. The two-dimensional readings [09]–[22] are premises of none of the four-dimensional rows.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the three rulers, the length at which conservation breaks $\ell_*$ ($\ell_c$), the Dirichlet form (coordinates), tree coordinates and link coordinates, conservation of flux, wrapping, the topological charge of the strip, the flux dipole, raw rate and static fluctuation, the time for the global quantity to move by one, two-dimensional test beds. Their meanings are in the A-02 Glossary and 11-A1.

Next chapter to read: 11-03 The sector H and the local H — two forms that restate H to avoid the pattern. What is specific to four dimensions (why the two lengths split) is treated in 11-07.

← 11-01 One form that obstructs H — a nearly conserved global quantityContents11-03 Two ways to repair H — the sector H and the local H →