2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerations — beyond Assumption H, and other viewslocal Poincaré constant, ratio form, sector H · Kiichi, Shiori and Rin

Two ways to repair H — the sector H and the local H — two forms that restate H to avoid the pattern

A section readable at [Y3]. Assumption H is a sentence about "every test function", so if the relaxation of a quantity of the pattern of 11-01 (a winding number, the topological charge) grows faster than $\xi^2$, that alone breaks the upper half H↑ ((3) of 11-01). Such a quantity is global, local neither in time nor in space. So we restrict the test functions in two ways and restate H.

The local H. Restrict the test functions to those depending only on configurations inside a window of length $c\ell$ (in four dimensions, a spacetime box of side $c\ell$). $\ell$ is a ruler given from outside, independently of the gap; in four dimensions it is the bookkeeping length $\xi_{\rm RG}$. The supremum $C_P^{\rm loc}(c)$ is fixed by the marginal distribution on the window alone, increases monotonically as the window widens, and returns exactly to the global $C_P$ as $c\to\infty$. But in a four-dimensional box, $C_P^{\rm loc}$ is finite, of order the square of the side of the box, even for the massless free field. Merely satisfying "$C_P^{\rm loc}\le K\ell^2$" says nothing, so we ask that the ratio $K_{\rm rel}$ to the massless value on the same box be sufficiently smaller than 1.

The sector H. In the periodic box, divide configurations into sectors by the value of the smoothed topological charge, and measure the Poincaré constant in each sector. Splitting the variance into within sectors and between sectors, the global $C_P$ is sandwiched: above by the sum of "the worst constant within a sector" and "the quantity between sectors $C_{\rm trans}$", and below by $C_{\rm trans}$. Assuming the sector H, H↑ becomes the same as $C_{\rm trans}\lesssim\xi^2$. The obstruction does not vanish; it only moves into one quantity.

What is asked. Writing the ratio for the charge of a subregion with two coefficients ((7) of 11-01), the local H↑ needs the transport coefficient of the charge $d$ not to be too small, and the global H↑ needs the coefficient of breaking of conservation of the charge $h$ not to be too small. In four dimensions, the freezing of the topological charge acts on the $h$ side, so the local H is expected not to be affected by freezing Estimate. But the local H does not become a part of the chain that builds H↑; it is a separate sentence (the dependency graph of §06). That what the one-sided bridge needs is the side local in time (under the assumptions of Proposition B) is treated in the chapter on the bridge, 11-04.

Setting. We use the same setting as Proposition B of Part 10 ((17) of 10-01): a finite spatial lattice, the configuration space of one time $\mathcal Q$ (a finite-dimensional connected Riemannian manifold with metric $M$), $H=-\frac{\hbar^2}2\Delta_M+V$ with assumption (A1) (discrete spectrum, simple ground state $\psi_0\gt0$, $\Delta=E_1-E_0\gt0$), $\mu$ the stationary path measure with $T=\infty$ (one-time distribution $\psi_0^2$, transition $e^{-\tau A}$, $A=(H-E_0)/\hbar$), and $m=\Delta/\hbar$. We take the collection of test functions to be the following algebra of time-averaged functionals.

Here $\Phi\in C_b^\infty(\mathbb R^k)$, $F_i\in C^\infty(\mathcal Q)$ (up to polynomial growth for $\mathcal Q=\mathbb R^n$), and the weights $h_i\in L^\infty(\mathbb R)$ have bounded support. $d_{q(\tau)}f=\sum_i\partial_i\Phi\cdot h_i(\tau)\,dF_i(q(\tau))$, and the metric $G$ satisfies (A3) of Proposition B. The global Poincaré constant is defined as $C_P:=\sup_{f\in\mathcal D,\ \mathcal E(f)\gt0}\operatorname{Var}(f)/\mathcal E(f)$ over $\mathcal D$. The test functions of Proposition B, $f_T=\int_0^TF(q)\,d\tau$, belong to this $\mathcal D$.

$\ell$ is not fixed by the gap because, when a lower bound on the gap is derived from the local H in 11-04, the argument would be circular if the gap entered the length of the window. Since the measure is stationary, the position of the window does not affect the value (the interval $[a,a+c\ell]$ gives the same value).

Proved here Proof. (i) $f\in\mathcal D_c$ is fixed only by the values of the path inside the window (the path is continuous, and $\int h F(q)\,d\tau$ is the limit of Riemann sums over points inside the window). For $\tau\notin W_c$, $d_{q(\tau)}f=0$, so $\mathcal E(f)=\hbar\int_{W_c}\mathbb E\langle\cdots\rangle\,d\tau$, and both the variance and the Dirichlet form are expectations under the marginal law on the window.

(ii) The closure of $\mathcal D_c$ lies in $L^2(\mathcal F_{W_c})$, the space of functions measurable with respect to the $\sigma$-algebra $\mathcal F_{W_c}$ generated by the path inside the window. The value $F(q(\tau_0))$ at a time $\tau_0$ outside the window ($F$ non-constant) is not measurable with respect to $\mathcal F_{W_c}$: with $s\gt0$ the distance from the end of the window, the conditional expectation is of the form $\psi_0^{-1}e^{-sA}(F\psi_0)$, which under (A1) does not coincide with $F$ itself.

(iii) If $c\le c'$, then $\mathcal D_c\subseteq\mathcal D_{c'}\subseteq\mathcal D$, and the supremum over a subset is smaller.

(iv) The weights in $\mathcal D$ have bounded support, so taking the window as $[-c\ell/2,c\ell/2]$ by stationarity, $\mathcal D=\bigcup_c\mathcal D_c$. If $b\lt C_P$, there is $f\in\mathcal D$ whose ratio exceeds $b$, and that $f$ lies in $\mathcal D_c$ for some $c$, so $C_P^{\rm loc}(c)\gt b$. Together with the monotonicity of (iii), the limit is $C_P$. ∎

Lean LatticeGauge.LocalH.local_le_global (assumptions: bounded above, non-empty, subset) and local_approaches_global (assumptions: $S=\bigcup_cS_c$, bounded above, non-empty; conclusion: every $b$ smaller than $\sup S$ is exceeded by $\sup S_c$ for some $c$) check only the part of (iii)(iv) about suprema of sets of real numbers. The path measure, $\mathcal D$ and $\mathcal E$ are outside Lean.

(iv) holds without assumptions because $C_P$ was defined over $\mathcal D$. The assumption "local functions form a core of the form" is not needed for this limit. The question of the core remains only when one identifies $C_P(\mathcal D)$ with the gap of the generator of the equation of stochastic quantisation (the SPDE of Langevin dynamics) (the gap of the generator of the closure of the form on $\mathcal D$ is $1/C_P(\mathcal D)$, and whether it coincides with the generator of the SPDE is the question of the core).

The local H. On Definition 1, we write the local H at two levels.

(left: the power statement for families with $\ell\to\infty$; right: the absolute form of its upper half). Two things follow in one line from Proposition 1 Proved here. First, the global H↑ $C_P\le K\ell^2$ implies $\mathrm{H}{\uparrow}^{\rm loc}_c(K)$ with the same constant for every $c$ ((iii)). Second, if $\mathrm{H}{\uparrow}^{\rm loc}_c(K)$ holds for every $c$ with a $K$ uniform in $c$, it returns by (iv) to the global H↑. So the local H↑ has content different from the global H↑ only when stated with a single $c$ fixed.

A version local in space too. On an infinite spatial lattice (or a spatial lattice containing the box), take the product $B\times W_c$ of a spatial box $B$ and a window, and define in the same form the family $\mathcal D_{B,c}$, with $F_i$ restricted to single-time functions depending only on variables in $B$, and $C_P^{\rm loc}(B,c)$. (i)–(iv) of Proposition 1 hold with the same proof, and the marginal in (i) is the law of the field on $B\times W_c$. The local H on the four-dimensional Wilson measure copies this definition onto a spacetime box $B_c$ of side $c\ell$ ($\ell=\xi_{\rm RG}$), treating time and space alike, and the $\sigma$-algebra generated by the links in the box; the Dirichlet form is that of link Langevin, $\mathcal E(f)=\mathbb E\sum_\ell|\nabla_\ell f|^2$. What is presupposed at the copying stage is the existence and uniqueness of the infinite-volume Wilson measure ((S1) of Part 10) Assumption.

The absolute form is empty. The right-hand form of (3), $\mathrm{H}{\uparrow}^{\rm loc}_c(K)$, holds on a four-dimensional spacetime box, local in space too, even for the massless free field. Writing $C_P^{\rm loc,0}(R)$ for the local Poincaré constant of the infinite-volume massless free scalar field (one component, canonical graduation) on a spacetime box $B_R$ of side $R$,

Proved here. The gradient Poincaré constant of a Gaussian measure equals the operator norm of the covariance compressed to the subspace spanned by the family (a transcription to subspaces of Lemma 2.2 of Hairer–Stuart–Voss–Wiberg 2005) Cited, so $C_P^{\rm loc,0}(R)=\|P_B(-\Delta)^{-1}P_B\|$. In four dimensions $(-\Delta)^{-1}$ is finite (transient) and its kernel is continuous, of the type $1/(4\pi^2|z|^2)$, so the Schur estimate $\sup_x\sum_{y\in B}G(x-y)\le\int_{|z|\lt2R}d^4z/(4\pi^2|z|^2)=R^2$ gives the upper side, and the value with the Dirichlet ground state of the box as test function gives the lower side. Hence, reading $\ell=R/c$, $\mathrm{H}{\uparrow}^{\rm loc}_c(K)$ holds with $K=c^2$, and the form "$C_P^{\rm loc}(c)\le K(c)\ell^2$ for some $K(c)$" says nothing about the gap. This is the negative control of this definition (the massless free field). In numbers, $\Gamma(R):=C_P^{\rm loc,0}(R)/R^2$ is $0.0802,\ 0.0768,\ 0.0756,\ 0.0750,\ 0.0747,\ 0.0744$ at $R=4,6,8,10,12,16$; the correction is $1/R^2$ with no logarithmic correction, and $\Gamma_\infty=0.0740$ Reading (floating point, numerical integration of the heat kernel on the infinite lattice, extrapolation from $R\le16$; within relative $2\cdot10^{-3}$ of an independent route via periodic lattices). The massless field itself satisfies $K(c)=0.0740\,c^2$.

One line of contrast with lower dimensions. For massless fields in two spacetime dimensions the infrared diverges and $C_P^{\rm loc}=\infty$; localising only to a time window on a finite spatial lattice, the constant mode gives $C_P^{\rm loc}\approx T/(2m)$ ($mT\ll1$), which diverges as $m\to0$. A massless field has a finite local constant in spacetime dimension three or more when localised in space too, and the four-dimensional setting of Part 11 is exactly that case Proved here.

It corresponds to the constant of the absolute form by $K(c)=K_{\rm rel}\,\Gamma\,c^2$ Proved here. In the ratio form the constant $K$ becomes an absolute number independent of the box size, and "sufficiently smaller than 1" has meaning.

Why it is not required uniformly in $c$. For the free field of mass $m$ the ratio is a function of $mR$ alone, $r(mR)$, and is $0.932,\ 0.825,\ 0.623,\ 0.359,\ 0.0772$ at $mR=0.5,\ 1,\ 2,\ 4,\ 12$ Reading (agreeing within $0.3\%$ for boxes of side $4$ to $12$). When the box is smaller than the correlation length ($mR\to0$) the ratio tends to 1. An asymptotically free theory is close to a free field at short distance, so a sentence requiring $K\lt1$ uniformly down to $c\to0$ is expected to be false Estimate. For a field whose action is a convex perturbation $\tfrac12\phi(-\Delta)\phi+\sum U(\phi_x)$ ($U$ convex), the Brascamp–Lieb inequality gives $K_{\rm rel}\le1$ Assumption (the statement of the inequality in the original source has not been checked in this part). What the bridge needs is not that the ratio is below 1 but that it is below the threshold $K_*$, and the threshold moves with the choice of reference: $0.342\kappa$ (scalar), $0.502\kappa$ (gauge quotient, one-colour linearisation), $\approx0.62\kappa$ (colour singlet of $SU(2)$, leading term). The threshold, and that the band $[K_*,1)$ is undecided, are treated in 11-04.

The reference for gauge fields is taken on the gauge quotient. For four-dimensional gauge fields, the value of the reference changes with the test space in which it is measured. The target Wilson measure is gauge invariant, and the Poincaré constant of a non-gauge-invariant function is infinite along gauge orbits, so the numerator of the ratio is finite only on gauge-invariant functions. So the reference in the denominator is also measured on gauge-invariant functions of the same box.

If the box is contractible, $V_F=V_Q$ Proved here (the cubical complex is contractible with zero first homology, so cycles in the graph of the box are boundaries of 2-chains of plaquettes). In non-contractible regions (with holes), $V_F\subsetneq V_Q$. Gauge-fixed references (Feynman gauge and so on) depend on the fixing parameter $\alpha$, with $\Gamma_Q\le\Gamma_{\rm Landau}\le\Gamma_{\alpha=1}$, and moreover $\Gamma_\alpha\ge\alpha/(\lambda_D^{\rm int}R^2)$, so they become infinite as $\alpha\to\infty$ (removing the fixing) Proved here. The numbers are in Table 1.

The last row of Table 1 is (e) of Table 1 in 11-01. Gauge transformations at the boundary sites of the box also act on links inside the box, so gauge-invariant observables of the box (Wilson loops and so on) lie in the space divided at all sites. "Dividing by gauge transformations at all sites of the box" is part of the definition, and leaving it out moves the numbers by a factor of $2.2$ to $2.5$. Every reference is a number for a free field; this book has not measured a single ratio for the target Wilson measure.

Note on 0+1. In 0+1 dimensions there is no finite massless reference (for a free particle $C_P^{\rm loc}=\infty$). The reference always has a scale, and for the double well we take the harmonic oscillator with the curvature of the bottom (angular frequency $\omega_b$). Its ratio for the shortest window (a single time) is exactly $\omega_b/m$ (when $\psi_0$ is the Schrödinger ground state) Proved here:

($\varepsilon$ is the time step. The numerator is the Poincaré constant of the marginal $\psi_0^2$, which moves to the gap of $(H-E_0)$ by the ground-state transformation. This identity is the same as the formula for the vacuum measure in 11-05.) Lean LatticeGauge.LocalHDoubleWell.one_slice_ratio (assumptions: $0\lt\varepsilon,m,\omega$; only the algebra of the ratio). In the double well, tunnelling gives $m\ll\omega_b$, so the ratio does not become 1 even for short windows; on the lattice $\varepsilon=\tfrac12$ it is $4.61$ ($\hbar=0.3$) to $638$ ($\hbar=0.08$) Reading. Hence "the ratio tends to 1 at short distance" is not a fact that follows from the ratio form but an assumption depending on a property of the theory, asymptotic freedom (being close to a free field at short distance) Assumption. The three regimes of the ratio as the window is lengthened are treated in 11-04.

Setting. Take the periodic box $\Lambda_L=(\mathbb Z/L)^4$, the configuration space $\mathcal M=G^{\text{links}}$ (a compact connected manifold with a bi-invariant metric), the Wilson measure $\mu$ (smooth positive density), and $\mathcal E(f)=\int|\nabla f|^2d\mu$. For the topological charge we use the lattice charge $Q_s$ smoothed by the Wilson flow (flow time $s$) or by finitely many smoothings. The smoothing map is smooth and the charge density is a polynomial in matrix elements, so $Q_s\in C^\infty(\mathcal M)$.

Because the cut is at regular values, the closure of a sector is a compact manifold with smooth boundary, and the walls are smooth hypersurfaces of codimension 1. Cutting by a geometrically defined integer-valued charge (piecewise constant and discontinuous), the form can be defined as $H^1$ on open sets, but the regularity of the walls is unknown and (i) below cannot be said. This is why this chapter uses a smoothed charge.

Proved here Proof. (i) The closure of a sector is compact with smooth boundary and the density is positive and smooth, so the resolvent of the Neumann operator is compact (Rellich's theorem Assumption: the statement in the original source has not been checked in this part). The multiplicity of the zero eigenvalue equals the number of connected components, so a positive gap is the same as connectedness. (ii) The walls have measure 0, so the Dirichlet form splits into a sum over sectors, and the variance is the sum of the expectation of the conditional variance and the variance of the conditional expectation. (iii) The left is $\operatorname{Var}(\mathbb E[f|N])\le\operatorname{Var}f$; the right uses the Poincaré inequality in each sector on the first term of (9) and closes with $\sum_n\mu(I_n)\mathcal E_n=\mathcal E$. (iv) Functions on sectors may jump at the walls (they are not in $H^1(\mathcal M)$), so if conditioning creates a new bottleneck, $C_P^{\rm sec}(n)\gt C_P$ is possible. ∎ Lean LatticeGauge.LocalH.sector_split (assumptions: weights $\ge0$, the inequality in each sector, the two formulas of (9), $\operatorname{Var}(\mathbb E[f|N])\le C_{\rm trans}\mathcal E$; conclusion: the right of (10)), trans_le_total (assumptions: weights and sector variances $\ge0$, (9); conclusion: the inequality for the numerator on the left of (10)). The identity (9) itself, Rellich, and the measure are outside Lean.

The sector H and its corollary. The sector H is the sentence that, in a box with fixed physical side, the Poincaré constant within a sector is bounded by the square of the correlation length.

Proved here (from (10); $\lesssim$ is up to replacing constants). The sector H pushes the obstruction to H exactly into the quantity between sectors $C_{\rm trans}$. It does not remove the obstruction but only moves it into one quantity, the same form as type 7 in 10-10 (transfer). The physical reading of $C_{\rm trans}$ is the rate of dislocations, and statically $C_{\rm trans}\ge\operatorname{Var}(\mathbb E[Q_s|N])/\mathcal E(Q_s)$.

It does not connect to the bridge. What follows from (10) and the one-sided bridge ((17) of 10-01, with corrections for finite $T$ in periodic time) goes only as far as $\kappa/m^2\lesssim\sup_nC_P^{\rm sec}(n)+C_{\rm trans}$, and no gap comes out unless $C_{\rm trans}$ is bounded Proved here. The Langevin gap within a sector, $1/C_P^{\rm sec}(n)$, is a fifth quantity of time, and the conditioned measure $\mu_n$ has no transfer operator (the constraint on the total charge breaks the Markov property in time). A "physical gap within a sector" of the type of Proposition B does not stand by definition; to set it up one would need a treatment of superpositions of θ vacua, which has not been formulated Estimate.

Three weaknesses. (a) The walls of conditioning can create artificial bistability. In the core on the box side of Part 10, two local minima appeared on the walls of conditioning the interval to "exactly this interval", giving rise to a bistability that was not physical metastability (10-11 §06). The walls of sectors lie on dislocation configurations of small measure, so we expect the harm to be small, but this has not been checked Estimate. That it is unknown whether sectors are connected in (i) of Proposition 2 is the extreme form of this weakness. (b) When the physical volume is made large, the obstruction remains even with the sector fixed. With $\beta$ fixed and $L\to\infty$, the charge of a half box can move even within a sector $I_n$. With the coefficients of (5) in 11-01, the conditional variance is of order $\chi_tV/4$ and the Dirichlet form of order $hV/2+g\,|\text{face}|$, and the ratio tends to $\chi_t/(2h)$ as $V\to\infty$, the same order as the overall ratio of the periodic box $\chi_t/h$ Assumption (on the definitions of $h,g$ and the accounting for the conditional variance of a normal distribution with the sum of the two halves fixed). Fixing the sector stops only the one mode of the total charge; the modes of the charges of subregions remain. Hence the sector H is expected to have content different from H↑ only when the box is smaller than the crossing length (subregions within a sector are in the transport regime), that is, to have meaning only in the order in which the physical volume is fixed Estimate. (c) The path measure with $T=\infty$ has no sectors. The four-dimensional charge is a quantity integrated over time, and for $T=\infty$ it is not in $L^2$.

Correspondence with the splitting of the first step. (9) is the same splitting as the splitting of the variance in the first step (local Poincaré within a phase, and transport between phases, 10-05 §01), with phase ↔ sector and kink ↔ dislocation Estimate. It is one concrete destination of "the two-scale splitting is expected to lift" in 10-03 §06. But in the first step, the time for transport between phases (diffusion of kinks) is read to be of order $\xi^2S_0/\hbar$ ((14) of 10-04 Assumption), and fits into the power form of H. The four-dimensional quantity between sectors $C_{\rm trans}$ is exactly a quantity of the pattern of 11-01, and there is no prospect yet that it fits into the same form.

Local asks about the transport coefficient; global asks about the coefficient of breaking of conservation. The charge $Q_w$ in a smooth window of (7) in 11-01 belongs to the local family of a box of side $R=c\xi$. Putting (7) into the local H↑ (absolute form), the necessary condition is $\chi_tc^2\xi^2/(hc^2\xi^2+d\lambda_W)\le K\xi^2$, and in the regime where $h$ is small (the crossing length $R_*$ much longer than $c\xi$),

Proved here (algebra) Lean LatticeGauge.LocalH.transport_requirement (assumptions: $0\lt d$, $0\lt\lambda_W$, $0\lt\xi$). In the ratio form, $K$ is read as $K_{\rm rel}\Gamma c^2$. On the other hand, the necessary condition for the global H↑ is (2)(6) of 11-01 with $R\to\infty$,

Assumption (on top of (5) of 11-01). (12) and (13) bear on different coefficients. The local H asks about the transport coefficient of the charge $d$, and the global H about the coefficient of breaking of conservation of the charge $h$. The freezing of the four-dimensional topological charge can be read as the appearance of $h$ vanishing quickly in the continuum limit (dislocation configurations becoming rare), so what it affects is the side of (13). The prospect that the local H is unaffected by freezing comes from here Estimate. There are no four-dimensional lattice values for $d$ and $h$.

Closed boxes and open boxes. Proposition H3 of Part 10 (10-11 §02) is a reduction bounding the global $C_P$ by a box constant $C_{\rm box}$, and $C_{\rm box}$ was the constant of a conditioned measure with the boundary values fixed (with $\sup$ over the boundary values). We call this a closed box. The box of the local H is a marginal box without fixing the boundary, and we call it an open box. In four dimensions, if the boundary links are fixed, the integer part of the topological charge inside the box is expected to change only through dislocations inside the box (in the continuum, the charge of the box is the sum of the Chern–Simons value on the boundary and a winding number) Estimate. Hence for closed boxes of size $\chi_t(c\xi)^4\gtrsim1$, $C_{\rm box}$ is expected to inherit the time of the dislocation barrier Estimate. In open boxes the charge flows in and out through the boundary, and relaxation is of the transport type of (12) Assumption.

The difference between closed and open boxes has the same form as the difference between periodic and open boundaries. Open boundaries remove the topological obstruction only inside a finite box with fixed physical side (the autocorrelation time of $Q^2$ with open boundaries grows roughly in proportion to the length in the time direction, §5.3 of Lüscher–Schaefer 2011 Reading). This is the transport type of (12) restated in the language of boxes. A reading of the consequence Estimate: if, under the reading that extends the conjecture (§06 (a) of 11-01), H↑ reads as false in four dimensions, then even when the local H↑ holds, the four-dimensional version of Proposition H3 fails at the stage of closed boxes. The place where it breaks is not "relaxation inside the box" but "the passage between sectors inside a closed box".

The original source writes that the scaling of the autocorrelation of algorithms implementing Langevin ($1/a^2$ in lattice units) "is expected to hold on infinite lattices and on finite lattices with open boundaries", and excludes transitions changing topology with periodic boundaries from the analysis as lattice artefacts (the footnote of Lüscher–Schaefer, Non-renormalizability of the HMC algorithm, 2011) Cited. The second row of Table 2 points the same way as this wording, but the object of the original source is the autocorrelation of local quantities, not the local H itself.

We put the rows of this chapter into one table in the format of the five grades, with Estimate added. The symbols of the rows are the same as the nodes of the dependency graph, and one row of 11-01 (CL) is carried over.

Dependency graph

[01] ASM S1  infinite-volume Wilson measure exists, unique        β ≥ β₀
[02] ASM RL  Rellich's theorem                                    smooth boundary
[03] CIT GP  Gaussian C_P = norm of compressed covariance         Gaussian
[04] PRV L1  Proposition 1 (path measure, continuous time)        T = ∞
             Lean local_le_global, local_approaches_global (supremum part only)
[05] ASM W4  proof of Proposition 1 goes through in 4D boxes      four dims
[06] PRV LD  properties of the 4D local C_P ⟸ [01][04][05]
[07] PRV VC  emptiness of the absolute form (4) ⟸ [03]            4D spacetime
[08] RDG GM  reference numbers (Γ_∞, Γ_Q, λ_W^Q, freezing factor)  free field
[09] PRV GQ  V_F = V_Q in contractible boxes; fixing depends on α  contractible box
[10] PRV OS  ratio for the shortest window = ω_b/m                0+1
             Lean one_slice_ratio
[11] ASM AF  ratio → 1 at short distance (asymptotic freedom)     four dims
[12] ASM HR  local H↑ (ratio form, one c)                         β ≥ β₀
[13] PRV P2  sector constant finite ⟺ connected ⟸ [02]            regular-value cut
[14] PRV SW  sandwich C_trans ≤ C_P ≤ sup C^sec + C_trans         periodic box
             Lean sector_split, trans_le_total
[15] PRV CO  under the sector H, H↑ ⟺ C_trans ≲ ξ² ⟸ [14]
[16] PRV NB  bridge inequality from sectors ⟸ [14] (+ Prop. B)
[17] ASM CL  cluster decay (11-01)                                infinite volume
[18] PRV TR  local asks d, global asks h ⟸ [17]
             Lean transport_requirement
[19] ASM HB  ratio of half boxes in a sector → χ_t/(2h)
[20] ASM MX  (MIX4) max. correlation of closed-box families < 1   four dims
[21] ASM BX  (BOX4) C_box ≤ K′ξ_RG² for closed boxes
             ← the row the topological barrier is expected to enter
[22] ASM RD  4D multi-family box reduction (not written down)
[23] PRV HU  H↑ ⟸ [20][21][22]

How to check: on each "⟸" row, see by eye that the numbers on the right are smaller than the one on the left ([06] has [01][04][05], [07] has [03], [13] has [02], [15][16] have [14], [18] has [17], [23] has [20][21][22]). There are no circles. The "Assumption" leaves are the ten [01][02][05][11][12][17][19][20][21][22], the "Cited" leaf is [03], and the "Reading" leaf is [08].

Two things are read from this graph. First, the local H [12] enters nowhere in the chain that builds H↑ [23]. A reduction of the type of Proposition H3 uses closed boxes [21], whereas the boxes of the local H are open boxes. The local H is not a part of H↑ but a separate sentence. Where the local H is used is in the graph of the bridge in 11-04. Second, the place where H↑ is expected to break in four dimensions is concentrated in the single row [21]. The statement of [21] contains only the constant of closed boxes $C_{\rm box}$ and not the global $C_P$, so it is not of the form that puts the conclusion among the premises.

(a) The two forms of H are candidate definitions; at no $\beta$ is there an inequality shown for the target gauge theory. What has been shown are properties of the definitions (Propositions 1 and 2) and values of the free-field reference. Nor do we claim "the right way to correct" H.

(b) The local H on the four-dimensional Wilson measure is a copy of the definition on the path measure in continuous time, and the existence of the infinite-volume measure [01] and the treatment of domains [05] are premise rows.

(c) The accounting for the charge of a subregion (transport in open boxes, barrier in closed boxes) is an estimate on top of cluster decay and the formal coefficients $h,g,d$, with no four-dimensional lattice values.

(d) The four-dimensional multi-family box reduction [22] has not been written. The harm of sector walls, and how the sector H breaks when the physical volume is made large, are estimates.

(e) There is no evidence that the local H holds in four dimensions. There is only the accounting "freezing acts on the $h$ side, and the local H asks about $d$".

(f) Every number in Table 1 is a number for the free field of the reference (one-colour Gaussian, $\beta=1$, floating point, extrapolation); not a single ratio $K_{\rm rel}$ for the target measure has been measured. The threshold depends on the reference and on the family of test observables, and the bridge says nothing about fields in the band $[K_*,1)$ (11-04).

(g) "The ratio tends to 1 at short distance" is an assumption depending on asymptotic freedom; in the 0+1 double well this reading does not hold ((7)).

What this chapter can say

Sources and reproduction

Terms used in this chapter — the algebra of time-averaged functionals $\mathcal D$, window, the local Poincaré constant $C_P^{\rm loc}(c)$, the ratio form $K_{\rm rel}$, the massless reference and $\Gamma$, the gauge-quotient reference, sectors and the reflecting form, the quantity between sectors $C_{\rm trans}$, open boxes and closed boxes, the transport coefficient $d$ and the coefficient of breaking of conservation $h$. The notation is in 11-A1.

Next chapter to read: 11-04 Which H the one-sided bridge needs — the threshold $K_*$ at which a lower bound on the gap comes out of the local H↑ in ratio form, and the size of box at which the bridge sees the mass.

← 11-02 Choosing the ruler and the coordinates again — the length at which conservation breaksContents11-04 Which H the one-sided bridge needs →