2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerationsthe one-sided bridge, the local H on a window, the ratio form and the threshold · Kiichi, Shiori and Rin

Which H the one-sided bridge needs

In 10-01 §06 of Part 10, we showed the one-sided bridge $C_P\ge\kappa\hbar^2/\Delta^2$ (Proposition B) for the path measure on a lattice with continuous time. The test function is the single one $f_T=\int_0^TF(q(\tau))\,d\tau$, the time integral of a function of the configuration at one time, $F=\psi_1/\psi_0$. Setting this one function beside $C_P\le C\xi_{\rm RG}^2$ of H↑ gives a lower bound on the gap of the lattice in physical units, $m_{\rm phys}^2\ge\kappa/C$ ((19) of 10-01). The question of this chapter is which part of H↑ is used there.

The answer comes in three stages.

First stage. On the family $\mathcal F_1$ of "functions of one time averaged with time weights", the supremum of $\operatorname{Var}/\mathcal E$ is exactly $\xi^2=\hbar^2/\Delta^2$ in the canonical graduation Proved here. Hence H↑ restricted to this family is the same as "the gap of the lattice in physical units is at least $C^{-1/2}$". The content of H↑ beyond the gap lies only outside this family — in functions not local in time, such as winding numbers and the topological charge (Proposition C of 10-01, the pattern of 11-01). The bridge uses only the part of H↑ local in time.

Second stage. From the local H↑ with the time window limited to length $c\ell$ ($\ell$ is a ruler given from outside, independently of the gap; in four dimensions it is the bookkeeping length $\xi_{\rm RG}$), too, a lower bound on the gap in finite spatial volume follows, $m_{\rm phys}\ge\sqrt{1/c^2+\kappa/K}-1/c$ Proved here. As $c\to\infty$ it returns to the $\sqrt{\kappa/K}$ of Part 10.

Third stage. Going local in space too and on to infinite volume, a sentence of the form "the local Poincaré constant in the box is at most $K(c)\ell^2$" becomes empty, because the four-dimensional massless free field also satisfies it for some $K(c)$ Proved here. So we state it by the ratio $K_{\rm rel}$ to the value of the massless free field (the reference) on the same box with the same graduation. Under assumption $\text{(Loc-}\Delta_c)'$, if the ratio is below the threshold $K_*=\kappa/(\lambda_W\Gamma)$, a lower bound on the gap follows Proved here (algebra on top of the assumption). The threshold changes with the choice of reference: $0.342\kappa$ for the scalar, $0.502\kappa$ for the gauge quotient (one-colour linearisation), and about $0.62\kappa$ for the colour singlet of $SU(2)$ (leading term, second chaos; a test value, so a lower value of the usable threshold) Reading. Examining free fields with mass, the bridge sees the mass only when the side of the box is at least about 4.2 times (scalar) to 5.2 times (gauge quotient) the correlation length Reading. For fields whose ratio lies in the band $[K_*,1)$ between the threshold and 1, this bridge says nothing.

Finally we place a note on 0+1 dimensions. In the double well, even for the shortest window (a single time), the ratio is exactly $\omega_b/m$ and does not become 1 Proved here. "At short distance the ratio is close to 1" is not a fact that follows from the ratio form; in four dimensions it is an assumption depending on asymptotic freedom Assumption.

As a metaphor, the bridge looks only at "the time for the hand of the clock to go round once", not at the slowness of "the gear that counts how many times the hand has gone round". H↑ is a sentence about the whole, gear included, and what the bridge needs is only the part about the hand.

The setting is the same as Proposition B in 10-01 §06. Space is a finite lattice, the configuration space at one time is $Q$, the Hamiltonian is $H=-\frac{\hbar^2}2\Delta_M+V$, with assumptions (A1) (discrete spectrum, ground state $\psi_0\gt0$ non-degenerate and smooth, $\Delta=E_1-E_0\gt0$), (A2) (the stationary path measure with $T=\infty$), (A3) (comparison of the Langevin metric, $\kappa=1/\lambda_G$). $m=\Delta/\hbar$, $\xi=1/m$, $A=(H-E_0)/\hbar$.

We set the family local in time as follows.

The $f_T$ of Proposition B is the case $h=\mathbf 1_{[0,T]}$, $G_1=\psi_1/\psi_0$.

Proved here Proof. Let $\nu_\varphi$ be the spectral measure of $A$ for $\varphi=(G_1-\langle G_1\rangle)\psi_0$; $\nu_\varphi$ lies on $[m,\infty)$. From the formula for correlations and Plancherel's theorem,

(the right from the identity (18)). For $a\ge m$, $2a/(a^2+k^2)\le2a/m^2$, with equality only at $k=0$, so the ratio is smaller than $\xi^2$. That the supremum is $\xi^2$ is seen by taking $T\to\infty$ in the $f_T$ of Proposition B. ∎ For a general metric, (A3) and the reverse comparison $G\ge\lambda'_GM^{-1}$ sandwich the supremum between $\kappa\xi^2$ and $\xi^2/\lambda'_G$. LeanEuclideanPath.GapBridge.local_rayleigh_le (assumptions: $0\lt m$, $m\le a_k$ for each $k$, $0\le c_k$; conclusion: $m^2\sum_kc_k/a_k\le\sum_kc_ka_k$) and fourier_weight_le (assumption: $0\lt m\le a$; conclusion: $2a/(a^2+k^2)\le2a/m^2$) check only the finite sums and the pointwise inequality of this proof.

Consequence. The sentence of H↑ restricted to $\mathcal F_1$, "for every $f\in\mathcal F_1$, $\operatorname{Var}(f)\le C\xi_{\rm RG}^2\,\mathcal E(f)$", is by (2) the same as $\xi^2\le C\xi_{\rm RG}^2$, that is,

H↑ on $\mathcal F_1$ $\iff$ $m_{\rm phys}=m\,\xi_{\rm RG}\ge C^{-1/2}$ (canonical, each $\beta$)

Proved here. Right to left also follows from (2). Everything by which H↑ is stronger than the gap of the lattice in physical units (the conclusion of 10-01 §06) lies outside $\mathcal F_1$. For a free particle on a circle (Proposition C), the winding-number functional $\int w\,d\tilde x$ lies outside $\mathcal F_1$ and produces $C_P=\infty$, while the supremum over $\mathcal F_1$ remains $\xi^2$.

Correspondence with the paper. Proposition B, (2) and Proposition C are collected in the paper (list of papers, PDF) “The inverse squared spectral gap bounds the Langevin Poincaré constant of Euclidean path measures from below” (Kiichi, Shiori and Rin). The one sentence of its abstract is "for Euclidean path measures stationary over all time on a finite-dimensional configuration space, the Poincaré constant of the stochastic quantisation (Langevin) Dirichlet form on time-averaged smooth functionals satisfies $C_P\ge\kappa(\hbar/\Delta)^2$". Proposition B corresponds to Theorem 3.1 of the paper, (2) to Proposition 4.1, and Proposition C to Proposition 5.1 ($C_P=\infty$ even on the family of time-averaged smooth functionals $\mathcal D_0$). This book only cites the equation numbers, and leaves the proofs to 10-01 §06 and the paper. The paper has not been checked by outside mathematicians.

The definition of the local H is in (1)–(3) of 11-03 §02. We copy only the form used here. The ruler $\ell\gt0$ is a length given from outside, independently of the gap (in four dimensions, the bookkeeping length $\xi_{\rm RG}$). Using the $\xi$ fixed by the gap as the ruler would be circular when deriving a lower bound on the gap. For $c\gt0$, write $\mathcal D_c$ for the family of time-averaged functionals whose weights are supported in a window of length $c\ell$, and put

$C_P^{\rm loc}(c)$ is fixed only by the marginal law of the path on the window, increases monotonically in $c$, and returns to the global $C_P$ as $c\to\infty$ (11-03).

The window factor, and its monotonicity

Write $\varphi(x)$ for the bracket on the right of (17) of Proposition B (the $\varphi$ without argument is the vector of §02, a different quantity; the notation follows 11-A1).

The ratio for a window $T$ is $\kappa\xi^2\varphi(T/\xi)=\kappa T^2g(mT)$. $\varphi(x)=x/2+O(x^2)$, $\varphi(x)\to1$ ($x\to\infty$), and the values are $\varphi(1)=0.368$, $\varphi(4)=0.755$, $\varphi(10)=0.900$ Proved here (values of the closed formula).

Proved here Proof. Since $x-1+e^{-x}=\int_0^x\!\int_0^ue^{-v}\,dv\,du=x^2\int_0^1(1-t)e^{-xt}\,dt$, $g(x)=\frac1x\int_0^1(1-t)e^{-xt}\,dt$. It is the product of the positive strictly decreasing function $1/x$ and the positive strictly decreasing function $\int_0^1(1-t)e^{-xt}dt$, so it strictly decreases. ∎ LeanLatticeGauge.LocalH.wr_strictAntiOn (no assumptions; conclusion: $g$ is strictly decreasing on $(0,\infty)$. In Lean it is closed via the sign of the derivative $g'(x)=h(x)/x^4$, $h(x)=3-2x-(3+x)e^{-x}\lt0$).

Proposition G has two uses. First, in the family where the weight is fixed to the indicator of the window $\mathbf 1_{[0,T]}$ and only the single-time function varies, the supremum is attained exactly at the lowest excitation $a=m$ and equals $T^2g(mT)=\xi^2\varphi(T/\xi)$ Proved here (the ratio of weighted averages over the spectral measure does not exceed the maximum of the ratios per component; that the maximum is attained at $a=m$ is exactly the monotonicity of $g$). Second, it turns the window-limited H↑ $\kappa T^2g(c\,m_{\rm phys})\le K\ell^2$ ($T=c\ell$) into an inequality on one number. $g$ is a continuous strictly decreasing bijection from $(0,\infty)$ to $(0,\infty)$, so

From $e^{-x}\ge1-x+x^2/2-x^3/6$ ($x\ge0$) comes $g(x)\ge1/(2x)-1/6$, and the explicit form

is obtained Proved here. LeanLatticeGauge.LocalH.gap_lower_of_window (assumptions: $0\lt\kappa,c,K,\ell,m$, $\kappa(c\ell)^2g(c\ell m)\le C_P$, $C_P\le K\ell^2$; conclusion: $3\kappa c/(6K+\kappa c^2)\le m\ell$). (7) is sharp for $c\,m_{\rm phys}\lesssim1$, and for $c\,m_{\rm phys}\gtrsim3$ the approximation on the right becomes weak, so for large $c$ we use the next form.

Letting the weight inside the window vary too — Proposition W

Proved here Proof [Res]. Splitting $\varphi$ into eigencomponents $\varphi_k$ of $A$ (eigenvalues $a_k$), $\operatorname{Var}=\sum_k\|\varphi_k\|^2\langle h,e^{-a_k|s-t|}h\rangle$ and $\mathcal E=2\|h\|^2\sum_ka_k\|\varphi_k\|^2$ (the identity (18)). The ratio of weighted averages does not exceed the maximum of the ratios per component, so $\operatorname{Var}/\mathcal E\le\max_k\langle h,K_{a_k}h\rangle/\|h\|^2$, where $K_a$ is the integral operator with kernel $e^{-a|s-t|}/(2a)$ restricted to $[0,T]$, the compression of $(a^2-\partial^2)^{-1}$ on the real line. $a\mapsto(a^2-\partial^2)^{-1}$ is decreasing as an operator, so the maximum is at $a=m$. The largest eigenvalue of $K_m$ is fixed, since the eigenfunction satisfies $u''=-\omega^2u$ with the boundary conditions $u'(0)=m\,u(0)$ and $u'(T)=-m\,u(T)$, by $\omega\tan(\omega T/2)=m$ for the first symmetric solution $u=\cos(\omega(t-T/2))$, and its value is $1/(m^2+\omega^2)$. Equality is attained at $G_1=\psi_1/\psi_0$, $h=\cos(\omega_0(t-T/2))$. ∎

The root $\omega_0$ satisfies $\omega_0^2\le2m/T$ by $\tan z\ge z$, and also $\omega_0\lt\pi/T$, so

Proved here. The value for the indicator weight, $T^2g(mT)$, is strictly smaller than $\Psi_T(m)$ ($\cos$ is not constant); as $T\to0$ both are $\approx T/(2m)$, and as $T\to\infty$, $\Psi_T=1/m^2-O(T^{-2})$ and $T^2g=1/m^2-O(T^{-1})$.

Lower bound in finite spatial volume

Combine the local H↑ (4) with Proposition W. From $\kappa\Psi_{c\ell}(m)\le C_P^{\rm loc}(c)\le K\ell^2$ and $\Psi_{c\ell}(m)\ge1/(m^2+2m/(c\ell))$ of (9), $\mu=m\ell=m_{\rm phys}$ satisfies $\mu^2+2\mu/c\ge\kappa/K$, and

Proved here (under the assumptions of Proposition B). LeanLatticeGauge.LocalH.gap_lower_of_local_window (assumptions: $0\lt\kappa,c,K,\ell,m$, $\kappa/(m^2+w^2)\le C_P$, $w^2\le2m/(c\ell)$, $C_P\le K\ell^2$; conclusion: (10)), local_bound_tendsto (no assumptions; conclusion: the right-hand side of (10) tends to $\sqrt{\kappa/K}$ as $c\to\infty$). As $c\to\infty$ it returns to (19) of 10-01, and as $c\to0$ it is $\approx c\kappa/(2K)$. (10) is stronger than (7) for every $c$.

For the free field, both directions. For a Gaussian path measure the Poincaré constant equals the norm of the covariance operator Cited (Hairer–Stuart–Voss–Wiberg 2005, Lemma 2.2), and $C_P^{\rm loc}(c)=\Psi_{c\ell}(m)$ holds exactly even over the whole window family $\mathcal D_c$ Proved here (the largest eigenvalue of the covariance compressed to the window is the value of Proposition W). Hence for the free field, besides "$\mathrm H{\uparrow}^{\rm loc}_c(K)$ ⇒ (10)", also "$m_{\rm phys}\ge K^{-1/2}$ ⇒ $\mathrm H{\uparrow}^{\rm loc}_c(K)$ for every $c$" holds ($\Psi\lt1/m^2$). One also sees that localisation is not trivial: $C_P^{\rm loc}(c)\lt C_P$ holds strictly for every finite $c$, and as $c\to0$ it is $\approx c\ell/(2m)\to0$.

Conditions of this section. (4)–(10) are sentences on a lattice with continuous time, a finite spatial lattice, $T=\infty$, and (A1)–(A3). On Wilson's lattice, which also discretises time, assumption (D) ((20) of 10-01) remains, and the window version has not yet been written on top of it.

Now go local in space too. On an infinite spatial lattice, write $C_P^{\rm loc}(B_R)$ for the local Poincaré constant with only functionals supported in a spacetime box $B_R$ of side $R=c\ell$ allowed to vary (the version of 11-03 §03 that is local in space too; (11) below is the same inequality as (4) of 11-03). The plain local H↑ is "$C_P^{\rm loc}(B_{c\ell})\le K(c)\,\ell^2$". But in four dimensions this form says nothing about the gap.

Proved here Proof. The Gaussian Poincaré constant is the norm of the covariance compressed to the box Cited (previous section). The covariance $(-\Delta)^{-1}$ is transient in four dimensions and hence finite, with continuum kernel $1/(4\pi^2|z|^2)$. Above, by the Schur estimate, the norm is at most $\sup_x\sum_{y\in B_R}G(x-y)\le\int_{|z|\lt2R}\frac{d^4z}{4\pi^2|z|^2}=R^2$ (the asymptotics of the lattice kernel replaced by the continuum kernel; this step is Assumption). Below, take the Dirichlet ground state $w$ of the box as test direction; from $\langle w,Gw\rangle\langle w,-\Delta w\rangle\ge\|w\|^4$ (Cauchy–Schwarz), $C_P^{\rm loc}\ge1/\lambda_D(B_R)\approx R^2/(4\pi^2)$ ($\pi^2/R^2$ for each of the four directions of the spacetime box). ∎

In numbers, the ratio $\Gamma(R):=C_P^{\rm loc,0}(B_R)/R^2$ decreases as $0.0802,\ 0.0768,\ 0.0756,\ 0.0750,\ 0.0747,\ 0.0744$ at $R=4,6,8,10,12,16$; the correction is $1/R^2$ with no logarithmic correction, and by extrapolation $\Gamma_\infty=0.0740$ Reading (floating point; numerical integration of the lattice heat kernel agreeing with an independent route via periodic lattices within relative $2\times10^{-3}$; extrapolation from $R\le16$; one component). This is inside the paper sandwich $[1/(4\pi^2),\,0.1125]$, 2.92 times the Dirichlet value $1/(4\pi^2)=0.0253$. Since the continuum kernel is homogeneous of degree $-2$, the compression being exactly proportional to $R^2$ is as expected.

One line of contrast with lower dimensions as well. For a massless field in two spacetime dimensions the infrared diverges and $C_P^{\rm loc}=\infty$, and even in finite spatial volume with only the time window localised, the constant mode grows as $\approx T/(2m)$. A massless field has a finite local constant only 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.

The ratio form

So we state the local H as a ratio to the massless free field (the reference) on the same lattice, the same box and the same graduation.

(uniformly in $\beta\ge\beta_0$, $\ell=\xi_{\rm RG}(\beta)$). The constant $K$ becomes an absolute number. We set it as a sentence at a single $c$, and do not require it uniformly in $c$. The ratio $r(mR)$ for the free field of mass $m$ tends to 1 as $mR\to0$, so a sentence requiring $K\lt1$ uniformly down to short distances becomes false for theories close to free fields Estimate. Requiring a $K$ uniform in $c$ returns to the global H↑ as $c\to\infty$ (11-03).

For a field with a convex interaction added (action $\frac12\langle\Phi,-\Delta\Phi\rangle+\sum_xU(\Phi_x)$, $U$ convex), the Hessian is at least $-\Delta$, so the Brascamp–Lieb inequality gives $K_{\rm rel}\le1$ Assumption (the bibliographic data of the inequality have not been checked). Convex fields lie in the band $[0,1]$. What the bridge needs, as the next section shows, is the part of it below the threshold.

In infinite volume, the single-time function $F=\psi_1/\psi_0$ is neither local nor explicit. Instead we assume that local observables in a box produce low-energy excitations.

Assumption It has not been shown for any interacting model. Assumption (13) contains the quantity of the conclusion $m_\infty$, but it appears only in a comparison from above and does not imply $m_\infty\gt0$. It is a sentence that has meaning even for $m_\infty=0$ (a property of approximation, that low-energy excitations can be approached by local observables), and the massless free field satisfies it with $\gamma=d\pi^2$.

Proved here It holds for the free field. On one-particle states $\nu_\varphi=\sum_p|\hat w_p|^2/(2\omega_p)\,\delta_{\omega_p}$, $M_3/M_1=m_0^2+\langle w,-\Delta w\rangle/\|w\|^2\le m_0^2+d\pi^2/(c\ell)^2$ ($w$ is the Dirichlet ground state of the spatial box, $d$ the spatial dimension).

By Jensen's inequality, the ratio for a window $T$ is bounded as $\ge\kappa/(M_3/M_1+\pi^2/T^2)$. Treating the spacetime box isotropically and putting $\lambda_W:=\gamma+\pi^2$,

and from this and the local H↑ $C_P^{\rm loc}\le K\ell^2$, $m_{\rm phys}^2\ge\kappa/K-\lambda_W/c^2$ Proved here (on top of (13) and the identity of the ground-state transformation in infinite volume). LeanLatticeGauge.LocalH.gap_lower_dirichlet (assumptions: $0\lt\kappa,c,K,\ell$, $0\lt m^2+\lambda$, $\kappa/(m^2+\lambda)\le C_P$, $\lambda\le\gamma/(c\ell)^2$, $C_P\le K\ell^2$; conclusion: $\kappa/K-\gamma/c^2\le(m\ell)^2$).

Transferring to the ratio form (12), from $C_P^{\rm loc}\le K_{\rm rel}\Gamma R^2$ and (14),

Proved here (algebra on top of (13) and the infinite-volume identity). LeanLatticeGauge.LocalHReference.ratio_bridge (assumptions: $0\lt\lambda_W,\Gamma,R,K_{\rm rel}$, $\kappa/(m^2+\lambda_W/R^2)\le C$, $C\le K_{\rm rel}\Gamma R^2$; conclusion: (15)), kstar_correspondence (assumptions: $0\lt\lambda_W,\Gamma,c$; conclusion: when $K(c)=K_{\rm rel}\Gamma c^2$, $K(c)\lt\kappa c^2/\lambda_W\iff K_{\rm rel}\lt K_*$), ratio_bridge_vacuous_of_ge (assumption: $K_*\le K_{\rm rel}$; conclusion: the right-hand side of (15) $\le0$).

How to read it. If $K_{\rm rel}\lt K_*$, the right-hand side is positive, and the lower bound on the mass

follows. If $K_{\rm rel}\ge K_*$, (15) is empty. A reference satisfies its own Dirichlet lower bound ($\Gamma\ge1/\lambda_W$, the same Cauchy–Schwarz as on the left of (11)), so for every reference $K_*\le\kappa$ Proved here. LeanLatticeGauge.LocalHReference.kstar_le_kappa (assumptions: $0\le\kappa$, $0\lt\lambda_W,\Gamma$, $1\le\lambda_W\Gamma$; conclusion: $K_*\le\kappa$). For a field with ratio 1 ($K_{\rm rel}=1$) the bridge does not stand (rel_one_no_bridge, negative control).

The $\lambda_W$ in (15) needs care. $\lambda_W$ is "the value of $M_3/M_1$ attainable within the allowed family of local observables", a number fixed once the family of test observables is chosen. The reference $\Gamma$ in the denominator and the $\lambda_W$ of the lower bound in the numerator must be taken over the same family. In the next section, this point moves the threshold.

Table 2 sets out the values for each choice of reference. Every row is a number for a free-field reference; not a single ratio $K_{\rm rel}$ for the target Wilson measure has been measured.

LeanLatticeGauge.LocalHReference.kstar_measured (assumption: measured $\Gamma\in[0.0739,0.0741]$; conclusion: $0.34\lt K_*\lt0.344$), LatticeGauge.GaugeReference.kstar_gauge_measured (assumptions: measured $\Gamma_Q\in[0.02762,0.02772]$, $\lambda_W^Q\in[71.8,72.1]$; conclusion: $0.5\lt K_*^Q\lt0.505$), LatticeGauge.WilsonLoopReference.kstar2_measured (assumptions: measured $\lambda^{(2)}\in[115,118]$, $\Gamma_Q\in[0.02762,0.02772]$; conclusion: $0.61\lt K_*^{(2)}\lt0.63$). All three are arithmetic with measured numbers as assumptions; the numbers themselves are outside Lean.

Why the threshold changes with the reference

Gauge-fixed values are not references. The value for the gauge-fixed free gauge field (Feynman gauge) is $\Gamma=0.0740$, the same as the scalar, but it changes when the gauge-fixing parameter $\alpha$ is changed, and when the fixing is removed ($\alpha\to\infty$) it becomes infinite in the directions of gauge transformations within the box Proved here (with test functions in the directions of the image of the gauge, $\Gamma_a(\alpha)\ge\alpha/(\lambda_D^{\rm int}R^2)$). The target Wilson measure is not gauge-fixed, so the numerator of the ratio is finite only on gauge-invariant functions. The denominator is measured on the same space. LeanLatticeGauge.GaugeReference.gauge_param_unbounded (assumptions: $0\lt\lambda_D,R$, $\alpha/(\lambda_DR^2)\le\Gamma_a(\alpha)$; conclusion: $\Gamma_a$ is unbounded in $\alpha$).

Gauge-invariant observables can only have high momentum. The test space of the gauge quotient is the divergence-free fields (transverse waves) on the graph of the box. If the box is contractible, this coincides exactly with the space of functions of the plaquette angles of the box Proved here (the first homology of the cubical complex is 0, so closed paths are boundaries of sums of plaquettes). Under the divergence-free constraint one cannot build a positive field pointing the same way over the whole box (the optimal test direction for the scalar). The optimal direction becomes a transverse wave of large momentum: $\lambda_W^Q=72.0$ is 1.82 times the scalar $4\pi^2$, and $\Gamma_Q$ is 0.374 times the scalar Reading. The threshold of the ratio relaxes, $0.342\to0.502$, but this is because $\Gamma$ shrinks by more than $\lambda_W$ grows ($0.502/0.342=1/(0.374\times1.82)$); the condition rewritten for the local Poincaré constant itself, $C_P^{\rm loc}\lt\kappa R^2/\lambda_W$, tightens from $0.0253\kappa R^2$ to $0.0139\kappa R^2$ Reading.

Divide by the gauge transformations at the boundary of the box too. If one divides only by gauge transformations at inner sites of the box and freezes the gauge at the boundary sites, flux passing uniformly through the box enters the test space, and the value returns roughly to the gauge-fixed value, 2.2–2.5 times $\Gamma_Q$ Reading. Gauge-invariant observables of the box (Wilson loops) are invariant under transformations at all sites, so "dividing by gauge transformations at all sites" is part of the definition of the reference (the row "flux through a box" in 11-01).

What the one-colour approximation tacitly assumed. Linearising to one colour component, linear combinations of plaquette angles inside the box alone span the whole transverse test space, and neither large loops nor plaquettes reaching outside the box are needed Proved here (the angle of a rectangular loop is the sum of the angles of the plaquettes inside). But the gauge invariants of $SU(2)$ (colour singlets) have no linear term in the expansion around the identity Proved here. A linear term $\sum c_\ell^aA_\ell^a$ must also be invariant under constant gauge transformations (global colour rotations $SO(3)$), and $SO(3)$ has no fixed vector in $\mathbb R^3$, so $c=0$. The trace of a loop $\tfrac12\operatorname{tr}U_C=\cos(|\theta_C|/2)=1-|\theta_C|^2/8+\cdots$ starts at second order in the angles. LeanLatticeGauge.WilsonLoopReference.no_linear_singlet (assumption: a linear form invariant under the two rotations $\mathrm{diag}(-1,-1,1)$, $\mathrm{diag}(1,-1,-1)$; conclusion: the coefficients are 0).

The reference is exactly half. In the leading term ($\beta\to\infty$, independent Gaussians for three colours), functions that are colour singlets lie in the second chaos and above. The eigenvalues of the Ornstein–Uhlenbeck generator of a Gaussian measure are sums of one-particle eigenvalues in the eigencoordinates of the covariance, so the minimum on the second chaos is twice the one-particle minimum, and the Poincaré constant is $\Gamma_Q/2$ Proved here (equality at $f_w=\sum_a(\langle w,A^a\rangle^2-\mathbb E)$, with $w$ the optimal transverse direction). The value of the first-order test is $\lambda_W(f_w)=\lambda(w)+1/C(w)$, and by Cauchy–Schwarz $\lambda(w)\lt\lambda_W(f_w)\le2\lambda(w)$ Proved here. In the four-dimensional box $\lambda^{(2)}R^2\to115$ to $117$ (about 1.6 times $\lambda_W^Q$), and the threshold is $K_*^{(2)}=2/(\lambda^{(2)}\Gamma_Q)\approx0.62\kappa$ Reading. $\lambda^{(2)}$ is a test value (a value from above), so the threshold obtained is a lower value of the usable threshold. Leanchaos2_gamma, chaos2_lambda (assumptions: $0\lt n,g$ and the Wick moments as formulas; conclusion: algebra of the ratio), chaos2_bracket (assumptions: $0\lt C$, $1\le\lambda C$; conclusion: $\lambda\lt\lambda+1/C\le2\lambda$).

Large loops are needed. To get this value, loops the size of the span of the box are needed. With the family of plaquettes only, $\{\operatorname{tr}U_p\}$, the reference in the leading term vanishes as $1/R^2$ (in two dimensions, from 0.29 times to 0.035 times $\Gamma_Q/2$ for $R=4\to12$), and even adding loops up to $2\times2$ it is 0.12 times ($R=12$), so the bridge speaks only of the lattice scale Reading. Including all rectangles spanning the box, it agrees with $\Gamma_Q/2$ to all digits in two dimensions, while flat rectangles alone give 0.73 times in three dimensions Reading. There are no values for large loops in four dimensions (including non-flat ones and products of lassos).

The box at which the bridge sees the mass

The ratio for the free field of mass $m$, $r(mR)=C_P^{\rm loc}(R;m)/C_P^{\rm loc}(R;0)$, collapses within 0.3% for $R=4$ to $12$ onto a function of $mR$ alone Reading. For the scalar $r=0.932,\ 0.825,\ 0.623,\ 0.359,\ 0.0772$ ($mR=0.5,1,2,4,12$); for the transverse waves of the gauge quotient (Proca) $r_Q=0.950,\ 0.842,\ 0.721,\ 0.610,\ 0.514,\ 0.434$ ($mR=1,\dots,6$). The ratio falls to the threshold at $mR\approx4.2$ for the scalar and $mR\approx5.2$ for transverse waves (both by linear interpolation) Reading. The bridge sees the mass only when the side of the box is at least 4–5 times the correlation length. Transverse waves have large momentum, so they notice the mass later.

Put the other way, for fields whose ratio lies in the band $[K_*,1)$, this bridge says nothing. Free fields of mass $mR\lt4.2$ (with the scalar reference) also fall in this band, so the bridge misses fields that do have mass. Whether there is another argument (outside Jensen's inequality and Dirichlet test functions) that sees the mass for fields in the band is not settled.

We try the ratio form on the 0+1-dimensional double well $V(x)=(x^2-1)^2/8$ ($V''(\pm1)=1=\omega_b^2$, $S_0=2/3$). In 0+1 dimensions there is no finite massless reference (a free particle has no stationary measure and $C_P^{\rm loc}=\infty$). There is no choice but to take as reference the harmonic oscillator $\omega_b$ with the curvature of the bottom, and the ratio sentence becomes "if $K_{\rm rel}(T)\lt\kappa$ then $m\gt\omega_b$" Proved here (from the lower bound of Proposition W and the decrease of $\Psi_T$ in $m$. The threshold is $\kappa$, and its difference from the four-dimensional $0.342\kappa$ is exactly the Dirichlet loss of the spatial box). LeanLatticeGauge.LocalHReference.gap_gt_ref_of_ratio (assumptions: $\Psi$ is monotone non-increasing, $0\lt\kappa$, $0\lt\Psi(\omega_b)$, $\kappa\Psi(m)\le C\le K\Psi(\omega_b)$, $K\lt\kappa$; conclusion: $\omega_b\lt m$).

The window of a single time can be computed in closed form. On a lattice with time step $\varepsilon$, the Langevin generator for the single-time marginal $\psi_0^2$ and $\mathcal E_1(f)=(\hbar/\varepsilon)\int\psi_0^2|f'|^2$ is, by the ground-state transformation,

Proved here (when $\psi_0$ is the Schrödinger ground state; the reference is $\varepsilon/(2\omega_b)$). This is exactly the identity of 11-05. LeanLatticeGauge.LocalHDoubleWell.one_slice_ratio (assumptions: $0\lt\varepsilon,m,\omega$; conclusion: $(\varepsilon/(2m))/(\varepsilon/(2\omega))=\omega/m$).

The behaviour as the window is lengthened divides, on top of Proposition W, into three regimes Proved here.

The crossover is the window at which $\Psi_T(m)$ becomes half of $1/m^2$: $\omega_0=m$, that is $\tan(mT/2)=1$, $T_\times=\pi/(2m)$ Proved here. LeanLatticeGauge.LocalHDoubleWell.crossover_root, crossover_half (assumption: $0\lt m$; conclusion: at $T=\pi/(2m)$, $m\tan(mT/2)=m$ and the value is half. Uniqueness of the root is outside Lean).

The lattice values (time step $\varepsilon=\tfrac12$, transfer kernel as Trotter's symmetric product, $\hbar=0.3,\ 0.15,\ 0.08$) are $K_{\rm rel}(1)=4.61,\ 21.6,\ 638$ and $K_{\rm rel}(1)\cdot m=1.04,\ 1.04,\ 1.05$ for the window of a single time Reading (the deviation from (17) comes from $\psi_0$ of the time-discretised transfer and the discretisation of the reference). The crossover on the lattice is $k_\times m\varepsilon=1.571$ to $1.574$ ($\pi/2=1.5708$) Reading. Extending the functionals in the window from sums of one-point functions to multi-point functions raises the ratio further: the direct value for a window of two times is 1.02, 1.09, 1.34 times the $\mathcal F_1$ value, and the lower bound for a window of eight times with a family up to three-point products is at least 1.07, 1.27, 2.12 times Reading (lower bounds; the smaller $\hbar$, the more they rise with each widening of the family, and they have not saturated).

The ratio sentence "if $K_{\rm rel}\lt\kappa$ then $m\gt\omega_b$" correctly fails for the double well, whose gap is small because of tunnelling ($K_{\rm rel}\ge4.6\gt\kappa=1$ for every window). 0+1 is not an analogue of the four-dimensional ratio but a negative control showing that the ratio form responds correctly to non-convexity.