2026-10-15 · chapter Yang–Mills and the mass gapPart 13 The crossover bandthe four faces, the Z₂ glass, candidates for the band · Kiichi, Shiori and Rin

Candidate new viewpoints

The candidates on this page are set out after passing the selection criteria of 13-08 §07 as checkpoints. In particular, we name the following four for each candidate.

• (d) Does it use non-commutativity essentially? A candidate that gives the same conclusion for $U(1)$ is wrong. This is why we write "where it breaks for $U(1)$" for each candidate.
• (g) Is it not a restatement of one loop? A quantity that at one loop becomes asymptotic freedom itself is not enough on its own.
• (k) Does it not fail at small scales? Claims are put in a form restricted to outside the lattice artefacts, such as $\ell\ge\ell_{\min}$.
• (m) Does it distinguish from the transition of $U(1)$ within the band ($\beta\approx1.01$)?

For each candidate we write (i) the local quantity that carries it, (ii) why we expect it to hold for $SU(2)$ (the place where non-commutativity enters), (iii) where it breaks for $U(1)$ and $\mathbb Z_2$, (iv) the check against the ledger (13-08), (v) whether there are numbers. The labels are "Proved here" for elementary facts, "Reading" for numbers, and "Reading (physics)" for estimates.

There are four candidates for the local quantity in which to write the non-abelian inequality that crosses the band. We call them "faces" in the sense that it suffices to show any one of them, uniformly in the scale, in a form that fails for $U(1)$. For every face, a comparison across scales (for example $f(2\ell,2N,\beta)\le Kf(\ell,N,\beta')$) is a local version of the same difficulty as the global renormalisation comparison, and does not become easier. What changing the face gains is that the object of the proof becomes "a statement about a finite region with its boundary held fixed".

Face 1 The softness of the twist direction $\kappa_w$ — joining sectors by transport

The foothold is an identity for the free energy given by a map that carries one sector to the other. Take a measurable bijection $T$ (Jacobian $J_T$) from the configuration space of sector 0 to that of sector 1; let $\mu_0,\mu_1$ be the Gibbs measures of the two sectors and $\Delta F=-\ln(Z_1/Z_0)$. Then

holds [Proved here].

Proof. By the change of variables, $Z_1=\int e^{-S(Tx)}J_T(x)\,dx$, so setting $G:=S\circ T-S-\ln J_T$ we have $Z_1/Z_0=\mathbb E_{\mu_0}[e^{-G}]$. The measure $\tilde\mu\propto e^{-G}\mu_0$ is $\mu_1$ pulled back by $T^{-1}$, and $\mathrm{KL}(\mu_0\|\tilde\mu)=\mathbb E_{\mu_0}[G]+\ln\mathbb E_{\mu_0}[e^{-G}]$. Since $T$ is a bijection, $\mathrm{KL}(\mu_0\|\tilde\mu)=\mathrm{KL}(T_\#\mu_0\|\mu_1)$. Rearranging gives (1). ∎

The first term of (1) is "the action cost of the map", and the second is the remainder. If the map $T$ is deterministic, the remainder (the relative entropy) grows with the area of the vortex sheet, so an area law of type $e^{-\sigma A}$ does not come out [Reading] (physics) (C3 of 13-08 §05). The goal of the transport road is therefore not the area law but the qualitative $\Delta F(L,L)\to0$ (the intermediate goal of 13-06 §08).

(ii) Why we expect it to hold for $SU(2)$ [Reading] (physics). The "adapted" map that carries sectors along the generator of the twist is a triangular change of variables with $J_T=1$: a map that preserves volume while changing the shape of the fluctuations. It is the non-perturbative version of the principle of 12-07 — that what hides the antiscreening $+4$ is not preserving volume but not changing the shape of the fluctuations. $\kappa_w$ is the size of the fluctuations of the action cost of that map, and we expect it to decrease with $w$ if the soft direction of Nielsen–Olesen works within a band of width $w$.

(iii) Where it breaks. $U(1)$ has no spin term, and $\kappa_w\ge c\beta$ remains. $\mathbb Z_2$ has no continuous path joining the sectors and can use only thin sheets ($2\beta$ per unit area). (iv) The ledger: there is a risk of hitting item 25 of Table 1 (spin and the index theorem are a restatement of asymptotic freedom at one loop); the dividing line is whether the non-perturbative part can be cut out as an estimate of the variance $\kappa_w$. (v) Numbers: none.

Face 2 The flux mixing of the box $\delta(L_0)$ — a finite-box criterion

A candidate for a gauge version of a Dobrushin–Shlosman-type finite-size criterion. The form of the goal is "if $\delta(L_0,\beta)\le\varepsilon_d$, then the area law"; the criterion itself has not been written down [Reading] (physics). It does not use correlation inequalities (GKS), so it does not hit the obstacle that $SU(2)$ has no GKS (items 2 and 4 of Table 1).

For $SU(2)$ we expect $\delta(L_0)$ to go to 0 exponentially with $L_0$. In the Coulomb phase of $U(1)$, in the Villain model in the Gaussian approximation, the relative entropy stays at $O(\beta)$ independent of $L_0$ (a count at the level of estimates) [Reading] (physics). Caution: if the ordinary Dobrushin criterion is used with a frame, the continuous field follows the boundary and the influence coefficient becomes nearly 1 (C7 of 13-08). $\delta(L_0)$ is a quantity that changes only the centre flux precisely to avoid this following. No numbers.

Face 3 The twist cost of the column $f(\ell)$ — the same quantity as the Wilson loop of the box

Given the skeleton $\Sigma$ (the outer frame of the columns), the interiors of the columns are conditionally independent, so

holds [Proved here] (the same framework as Lemma 5 of 12-03 §03, with no new content). What is new is how $f(\ell)$ is read. The electric flux in a box with its outside held fixed is the same quantity as the area law of the Wilson loop of the box [Reading] (physics). This closes a loop: the Tomboulis–Yaffe inequality (from $\theta$ to the Wilson area law, 13-01 §04) and (2) (from the Wilson area law of the box to $\theta$).

Two things are missing to go round the loop (open): the comparison between periodic and fixed boundary conditions, and the loss of $\ln N$. The estimate is that even if the loss remains, $\ln(1/\theta(L,L))\ge cL^2/\ln L$ comes out, and through Tomboulis–Yaffe this gives $\langle W\rangle\le2\exp(-cA/\ln A)$ (decay faster than the perimeter law) [Reading] (physics).

There are numbers (13-09 §07) [Reading]. On $T^2_4\times\ell\times\ell$ with the outer frame held fixed, $\mathbb E|\theta_B|$ for $SU(2)$ ($\beta=2.3$) at $\ell\ge5$ is 1/5 to 1/7 of that for $U(1)$ ($\beta=1.3$), and the $\ell$ needed to keep $\theta$ small hardly grows for $SU(2)$, staying at 6–7 for $N\le8$, while it grows for $U(1)$. At these sizes, the difference between fixed and periodic boundary conditions does not seem to matter. The values for $SU(2)$, however, are close to the statistical floor, and $e^{-\sigma\ell^2}$ cannot be told apart from a power.

Face 4 The pressure margin — a computer-assisted pincer

We write the chessboard density as $\kappa_\Lambda=\exp(|B|(p_{G,\Lambda}-p_\Lambda))$ ($p_G$ is the pressure of the system with the twist pattern, $|B|$ the volume of a tile). From above: if the pattern is reflection-symmetric, the periodic transfer matrix is positive, and the monotonicity of Schatten norms gives $p_{G,\Lambda}\le p_{G,L}$ ($\Lambda$ a multiple of $L$). From below: $p_\Lambda\ge p_\infty$, together with a lower bound $p_{\rm lb}$ on the infinite-volume pressure taken from a dual certificate of the loop-equation bootstrap (the method of Kazakov–Zheng; checking not yet done). Combined, $\kappa_\Lambda\le\exp(|B|(p_{G,L}-p_{\rm lb}))$ becomes uniform in $\Lambda$, and an explicit area law for the tension comes out at fixed $\beta$ — that is the estimate [Reading] (physics).

This face does not cross the band (uniformly in the lattice spacing). What it could give is confinement at fixed $\beta$ for four-dimensional $SU(2)$ at intermediate coupling (for example $\beta=1.0$). In the range we searched, no proof of that was found. Combined with the reduction of §05 (candidate A), a result of this kind at fixed $\beta$ becomes a component of the band as it stands. $\beta=1.2$ for $U(1)$ should not produce a small $\kappa$, and can be used as a control. No numbers.

Writing an element of $SU(2)$ as $U=\sigma V$ ($\sigma\in\mathbb Z_2$, $V$ a section of $SO(3)$), the partition function of a twisted sector can be written $Z_n=\int d\omega\,Z_n(\omega)$. $Z_n(\omega)$ is the partition function of a $\mathbb Z_2$ gauge theory with couplings $\beta J_p(\omega)$ fixed by the $SO(3)$ background $\omega$ ($J_p=\tfrac12\mathrm{tr}\,V_p$, with random signs) and twist $n$. Setting $X(\omega)=\ln Z_0(\omega)/Z_1(\omega)$ and $P_n(d\omega)\propto Z_n(\omega)d\omega$, we have

[Proved here].

Proof. The left side follows from $Z_0/Z_1=\int Z_1(\omega)e^{X(\omega)}d\omega/Z_1=\mathbb E_{P_1}[e^X]\ge e^{\mathbb E_{P_1}X}$ (Jensen). Likewise the right side from $Z_1/Z_0=\mathbb E_{P_0}[e^{-X}]\ge e^{-\mathbb E_{P_0}X}$. ∎

The right-hand side of (3) is the twist free energy of a $\mathbb Z_2$ random-plaquette gauge model quenched in the $SO(3)$ background. This representation alone exists in the same form for $U(1)$, so it does not satisfy criterion (m). The distinction between groups enters dynamical quantities: the frustration of the background ($\mathbb Z_2$ monopoles) and the density of soft plaquettes [Reading] (physics).

Result of the diagnosis (13-09 §08) [Reading]. On $8^4$, for $\beta\gtrsim2.0$ the mean of the effective $\mathbb Z_2$ coupling is 1.1–1.9, well above the transition point 0.44 of pure $\mathbb Z_2$, and $p_{\rm eff}$, the frustration converted to independent signs, is 0.017–0.09, below the Nishimori multicritical point 0.11. In the upper half of the band, the glass is on the side of the ordered phase. The $SO(3)$ field already distinguishes the sectors ($|X|$ is large), and the sandwich (3) becomes very loose. Confinement there can be read as carried by the $SO(3)$ part (thick centre vortices). This fits the physical picture that centre vortices are thick at weak coupling [Reading] (physics).

For $\beta\lesssim1.5$, $p_{\rm eff}\ge0.16$, soft plaquettes are 28–49%, and the glass is on the disordered side. The glass road is therefore restricted to the lower half of the band. We have discarded it for the upper half (C4 of 13-08 §05). A candidate use in the lower half is a variant of the flux mixing of the box (face 2): condition the outside on the $SO(3)$ part only, flip only the centre flux at the boundary (the $\mathbb Z_2$ sign), and measure the total variation distance $\delta_c(L_0,\beta)$ of the inner law. The $SO(3)$ frame does not move, so the obstacle of the continuous field following the boundary (C7) is avoided. The only coupling left is the $\mathbb Z_2$ glass. In the upper half of the band $\delta_c$ should also be close to 1, and this variant belongs to the work of pushing up the lower end of the band (13-06 §04) [Reading] (physics).

The space $\mathbb R^4$ with one straight line removed is conformal to $H^2\times S^2$ [Proved here]: with $r$ the distance from the line and $t$ the coordinate along the line,

and the bracket is the product of the hyperbolic plane $H^2$ (the upper half-plane model) and the unit sphere. If a 't Hooft line is placed on the line, every sphere surrounding the line carries $\mathbb Z_2$ flux 1 (an exact analogue of $n_{12}=1$). And the sphere $\{t\}\times S^2_r$ corresponding to a point $(t,r)\in H^2$ winds an odd number of times with a loop $C$ in the half-plane if and only if $(t,r)$ lies inside the region enclosed by $C$ [Proved here] (the intersection of $C$ with the three-dimensional ball spanned by the sphere is the intersection of the segment $\{t\}\times[0,r)$ with $C$). This avoids the topological obstacle that stopped the road using local closed surfaces in $\mathbb R^4$ (item 8 of Table 1: a closed surface meets a plane an even number of times) by using the line as "the outside". Near the line ($r\to0$) is the region of small spheres (weak coupling), far away ($r\to\infty$) is the infrared: scale turns into position.

The promise went only this far; on examination the result was negative [Reading] (physics).

• A container at height $r$ with hyperbolic size 1 has Euclidean size of order $r$ and carries a sphere $S^2_r$. This is the band quantity $\theta(r,r)$ itself (with $S^2$ in place of $T^2$).
• Hyperbolic nesting: the hyperbolic area of a unit container at height $2r$ (about $[-r,r]\times[r,3r]$) is $2r\,(1/r-1/(3r))=4/3$, and only 1–2 unit containers at height $r$ fit inside it [Proved here] (computation of an integral). Hyperbolic area does not change under shape-preserving dilation, so nesting does not increase the number of sub-containers; what comes out is about $\theta(2r)\le\theta(r)^{1.3}$, not the exponent 4 of planar nesting.
• The hyperbolic area of a loop in the half-plane (from height $d$ to $d+\ell$, length $T$) is $T(1/d-1/(d+\ell))\approx T/d$, and a uniform $\theta_0$ independent of height gives only the perimeter law. The area law needs $\theta\le e^{-c(r\Lambda)^2}$ for unit containers at height $r$: the same condition as the band.

Conclusion: the conformal frame puts renormalisation into the language of space as "change of scale = translation in $H^2$", but kinematics alone does not give the area law, and the core (the growth of disorder through asymptotic freedom) remains in the same place. Its value lies in the clarity of a framework that states the $\mathbb R^4$ stage (stage 6 of 9-06) without going through slabs. The relation between loops in the measure with a 't Hooft line and loops in $\mathbb R^4$ without the line is open, and there is a precedent of the comparison principle failing (13-08 §04).

We examined eight candidates for the band, and put three of them here (the rest are in §06). The reasons for the choice: candidate A is a reduction that can be written reliably on paper, and it moves "where the band is really hard"; candidate B is a two-dimensional checkpoint that can be tried more cheaply than four dimensions, and is one of the few forms that does not fall into the type in which the source of monotonicity and the mechanism of disorder point in opposite directions (the lesson of 13-08 §01); candidate C can be tried directly with the existing numerical tools for the column, and lets us look at criterion (m) directly.

Candidate A Reduction by the compactness of the band

(i) In one line. The band is a bounded closed interval $I=[\beta_a,\beta_b]$ in $\beta$, so the renormalisation comparison within the band holds by lowering $\beta'$ only a little from $\beta$, and the band can be crossed in finitely many steps. All that is needed is "confinement at fixed $\beta$ (two-sided asymptotics with errors of perimeter order)" at each $\beta$ in the band.

This is set out with a proof as Lemma 1 of 13-06 §06 [Proved here] (under assumptions (A1)(A2), $N\ge N_1$, $M\ge M_0$). Here we write only its meaning as "a viewpoint".

(ii) The local quantity that carries it. Two-sided estimates of the vortex free energy in large volume at fixed $\beta$. As local quantities, the area rate $\sigma_N(\beta)$ of the column bias at $\ell\gg\xi(\beta)$, and the error $p$ of perimeter order.

(iii) Where non-commutativity enters [Reading] (physics). The lemma itself does not depend on the group. Non-commutativity enters entirely through the assumption (confinement for $SU(2)$ within the band). This candidate is a reduction, not a tool, and serves to move the place of the difficulty. After the reduction, what remains within the band is a problem at fixed $\beta$ (estimates uniform in the volume, non-convex large fields, absence of a phase transition), and the difficulty across scales gathers at the upper end of the band (where the correlation length $\xi(\beta)$ grows sharply in lattice units) and at the seam with ultraviolet control.

(iv) Where it breaks. For $U(1)$ ($\beta\in(1.01,2.7)$) and $\mathbb Z_2$ (on the weak-coupling side of the transition) the assumption is false, so no conclusion follows either. Criterion (m) is carried by the assumption.

(v) Check against the ledger. Unlike item 6 of Table 1 (a proof by contradiction extending continuously from a small $T^2$), the only continuity used is the continuity in $\beta$ of finite-volume quantities. It is in the same spirit as the idea of considering the step function of the continuum-limit coupling on a bounded interval, but here we stop at the lattice $\beta$ and do not pass through the continuum limit.

(vi) Relation to known results. On the strong-coupling side, a statement of exactly this form exists at the level of a strand (Proposition S of 13-06 §03). A proof of confinement for the four-dimensional $SU(2)$ lattice at $\beta\approx1$ within the band was not found in the range we searched (the front line on the strong-coupling side is at small $\beta$, 13-08 §07). Chatterjee (arXiv:2006.16229; abstract checked, text not yet checked) gives a result of the form: if the centre is non-trivial and correlations decay exponentially under arbitrary boundary conditions, then centre symmetry is not broken, and if it is not broken there is confinement. This could be an entrance to qualitative confinement at fixed $\beta$, but it does not give the perimeter-order precision of assumption (A1).

(vii) What is needed next. A road that gives the upper side of (A1) for each $\beta$ by finite-box criteria (faces 2 and 4), and a road that gives the lower side (that vortices are not too cheap = an upper bound on the string tension) by a variational trial vortex. Both are problems at fixed $\beta$, and boxes of size about $\xi(\beta)$ suffice, but at the upper end of the band $\xi(\beta)$ exceeds 10 in lattice units, and finite-box computations in four dimensions stop on scale (item 3 of Table 1). No numbers.

Candidate B The log-concavity of the Weyl measure, and the height–azimuth decomposition

(i) An elementary fact [Proved here]. The density of the uniform measure on the unit sphere $S^{n-1}\subset\mathbb R^n$ projected onto one coordinate $t$ is

Proof. The slice at height $t$ is a sphere $S^{n-2}$ of radius $\sqrt{1-t^2}$, whose area is proportional to $(1-t^2)^{(n-2)/2}$, and the width (length along the sphere) of the band of height $dt$ is $dt/\sqrt{1-t^2}$. Multiplying gives (5). ∎ Hence for $n\ge3$, $\rho_n$ is log-concave (flat at $n=3$), and at $n=2$ it is log-convex (the arcsine distribution). For $SU(2)\cong S^3$ ($n=4$), the density of $t=\tfrac12\mathrm{tr}\,U$ is $\propto\sqrt{1-t^2}$ (the Weyl measure); for $U(1)=S^1$ it is $\propto1/\sqrt{1-t^2}$.

In the two-dimensional $O(3)$ model, writing the spin as $s=(\sqrt{1-t^2}\cos\varphi,\ \sqrt{1-t^2}\sin\varphi,\ t)$, the uniform measure on $S^2$ is $dt\,d\varphi/(4\pi)$ ($n=3$ of (5)), and $s_i\cdot s_j=\sqrt{(1-t_i^2)(1-t_j^2)}\cos(\varphi_i-\varphi_j)+t_it_j$, so the model of the azimuths $\varphi$ conditioned on the heights $t$ is an XY model with couplings $J_{ij}=\beta\sqrt{(1-t_i^2)(1-t_j^2)}\ge0$ [Proved here]. There is no frustration.

(ii) The local quantity that carries it. The effective azimuthal coupling $J_{\rm eff}(b)$, measured in the coarse variables of a $b\times b$ block, and the fluctuations of the height.

(iii) Why we expect it to hold [Reading] (physics). The conditioned azimuth model is ferromagnetic, so the Ginibre inequality for XY (checking not yet done) can be used in the right direction: lowering the couplings lowers the correlations. What exponential decay needs is also an upper bound on correlations, so the source of monotonicity and the mechanism of disorder point the same way. The disorder comes from the fluctuations of the height, that is, from the measure (5) with $n\ge3$. $O(2)$ has no height.

(iv) Where it breaks. $O(2)$: there is no height, and the azimuths remain XY with coupling $\beta$ (the KT phase). Four-dimensional $U(1)$: there is no degree of freedom for the direction of the class ($S^2$). $\mathbb Z_2$: there is no continuous azimuth.

(v) Check against the ledger, and risks. Unlike items 22 (one-point Bakry–Émery) and 26 (renormalisation of curvature) of Table 1, it uses not curvature but the log-concavity of a projection and a conditional Ginibre inequality. Unlike item 10 of Table 1 (GKS for $\mathbb Z_2$), the conditioned model has no frustration. There are two risks. (a) There is an argument of Patrascioiu–Seiler (checking not yet done) for the opposite conclusion (that $O(3)$ may have a KT-type phase) through embeddings of the equatorial direction, and this candidate stands at the site of that dispute. (b) At the level of a single site, the typical values of $J_{ij}$ exceed the KT threshold, and the comparison says nothing. If it works, it works only at the block level, where there is no direct Ginibre comparison, and there is the risk of detour type 7 (transfer).

(vi) To four dimensions. One needs a representation in which the remaining $U(1)$, conditioned on $n$ in $U=t+i\sqrt{1-t^2}\,n\cdot\sigma$ ($n\in S^2$ the direction of the class), is ferromagnetic, but this has not been constructed. Showing exponential decay at all temperatures for the two-dimensional lattice $O(n)$ ($n\ge3$) (Polyakov's conjecture) is unsolved ((j) of 13-08 §07).

(vii) Numbers. None. The form of the first test is fixed: measure $J_{\rm eff}(b)$ for the two-dimensional $O(3)$ ($\beta=1.5,1.7$), the control $O(2)$ (the KT phase and the disordered phase), and the auxiliary $O(4)$ (whose height density $\sqrt{1-t^2}$ is the same as for $SU(2)$); if $J_{\rm eff}$ stays above the KT guide even at sizes at which correlations of $O(3)$ have already decayed, discard this candidate. If $J_{\rm eff}$ falls with $b$ for $O(2)$, the measurement is lowering the coupling mechanically, and the result is not used. A block-level Ginibre comparison does not exist as a theorem, so even if it passes it remains a watch on a necessary condition.

Candidate C The local step function of the column

(i) In one line. Replace the renormalisation comparison of the global $\Delta F$ by a comparison of column quantities with the outer frame held fixed:

($K$ and $c$ independent of $\ell$ and $N$). (ii) The local quantity that carries it. The conditional twist free energy $\Delta F_B$ and the bias $\theta_B=\tanh(\Delta F_B/2)$ of the column $T^2_N\times\ell\times\ell$ (the outer frame held fixed at an equilibrium configuration). Two quantities are compared: the median over frames of $\Delta F_B$ (with sign) and $\mathbb E|\theta_B|$.

(iii) Why we expect it to hold [Reading] (physics). In the existing numbers (13-09 §07), the $\ell$ needed to keep $\theta$ small hardly grows with $N$ for $SU(2)$, and grows for $U(1)$. Non-commutativity enters where the vortex can localise in a lump of the size of $T^2$ (a fractional instanton), because its action, by the Bogomolny bound, does not depend on the size of $T^2$ (12-02).

(iv) Where it breaks. In the Coulomb phase of $U(1)$ the flux spreads uniformly and the column bias decreases only as a power (about $\ell^{-1.4}$ over $\ell=5\to8$), so the local $\beta'$ should come out at or above $\beta$. For $\mathbb Z_2$ the sheet tension remains, likewise.

(v) Check against the ledger. Unlike item 27 of Table 1 (comparison of boundary conditions and geometries), it compares the same object at two scales. Nor does it use the correlation inequalities of item 2 of Table 1. Because of criterion (k), small $\ell$ (3 or less) may fail through lattice artefacts, so the claim takes the form $\ell\ge\ell_{\min}$. It is a local version of the step function of the coupling defined with twisted boundary conditions (the method of de Divitiis et al.; checking not yet done).

(vi) Numbers. There are as yet no numbers measuring (6) itself. The form of the test is fixed: fine side $(N,\ell)=(4,4)$, coarse side $(2,2)$ (and, if time allows, $(4,6)$ and $(2,3)$), with $SU(2)$ at $\beta=2.3$, and $U(1)$ in the confining phase at $\beta=0.9$ and in the Coulomb phase at $\beta=1.3$. The prediction: $\beta'\lt\beta$ for $SU(2)$; for $U(1)$, $\beta'\lt\beta$ at $\beta=0.9$ and $\beta'\ge\beta$ at $1.3$ (the sign flips with the phase). If the sign does not separate between the two $U(1)$ points, this observable does not distinguish phases, and the candidate is lowered a grade. $(2,2)$ has large artefacts, so it is treated as a sign until confirmed with $(4,6)/(2,3)$.

(vii) Difficulty. Showing (6) is a local version of the same difficulty as the global renormalisation comparison (the opening of §02). What is gained is that the object of the proof becomes "a statement about a finite region with its boundary held fixed", and that assumption (A1) of candidate A can be written in the language of columns.

Of the eight candidates examined, we give the remaining five in one line each, with the reason none was given priority over the three of §05. All are labelled "Reading (physics)"; those that include an elementary identity have "Proved here" added in their row.

• (a) None of the candidates closes the band. For none of them have we shown a theorem at any $\beta$ in the band.
• (b) Candidate A is a reduction, and we do not say it has made the band easier. Assumption (A) contains confinement at intermediate coupling on the four-dimensional $SU(2)$ lattice (unsolved).
• (c) Even if the two-dimensional test of candidate B passes, it implies nothing for Polyakov's conjecture or for four dimensions. A block-level Ginibre comparison does not exist as a theorem.
• (d) We do not say that two-dimensional results lift to four dimensions. Only the form of the statement lifts.
• (e) The $SU(2)$ expectations of the four faces are all estimates. Only the column face $f(\ell)$ has numbers, and those are tendencies on small lattices.
• (f) The negative verdict on the conformal frame is a judgement at the level of estimates, that "kinematics alone does not give the area law"; it does not rule out roads that add other inputs in that frame.
• (g) We do not say that the candidates on this page are not combinations of known facts. The reduction of candidate A and the log-concavity of the projection in candidate B may be written somewhere.

The candidates on this page are all "Reading (physics)". Paper proofs are set out only for the elementary facts that serve as footholds for the candidates.

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

[01] PRV TRN     the transport identity (1)
[02] RDG KLA     remainder of a deterministic map grows with area ⟸ [01]  physics
[03] PRV COL     the column decomposition                             framework of 12-03
[04] RDG FNUM    numbers for the column bias                          13-09 §07
[05] RDG FACE    expectations of the four faces ⟸ [01][02][03][04]    physics
[06] PRV JEN     the Jensen sandwich (3) of the glass
[07] RDG GLS     the glass road is restricted to the lower half ⟸ [06]  8⁴, physics
[08] PRV CONF    conformal equivalence and the parity criterion (4)
[09] PRV HYP     the hyperbolic nesting area
[10] RDG NEG     the conformal frame is negative ⟸ [08][09]           physics
[11] PRV RED     the reduction of candidate A                         13-06 Lemma 1, assumption (A)
[12] PRV WEYL    projected density (5), height–azimuth decomposition
[13] RDG MON     the mechanism of candidate B ⟸ [12]                  physics
[14] RDG LST     the prediction (6) of candidate C ⟸ [03][04][11]     physics
[15] PRV SGN     θ = E_P[tanh(X/2)]

The check 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. The conclusion-side nodes ([05], [07], [10], [13], [14]) are all "Reading (physics)", and the "Proved here" nodes are either elementary facts serving as footholds for the candidates, or a reduction whose proof is in another chapter. This page has no node for a statement that crosses the band.

What this chapter can say

Sources and reproduction

Terms used in this chapter — the four faces, the adapted stiffness $\kappa_w$, the flux mixing of the box $\delta(L_0)$, the twist cost of the column $f(\ell)$, the pressure margin, the $\mathbb Z_2$ glass, frustration, the conformal frame, the compactness of the band, the Weyl measure, height and azimuth, the local step function of the column, the crossover band (the band). Their meanings are in A-02 Glossary.

Next chapter: 13-11 Ultraviolet control — what it means to close at infinitely many steps — the foundation under the two roads. What closing in one step means, and where the series stopped.

← 13-09 The measured recordContents13-11 Ultraviolet control — what it means to close at infinitely many steps →