2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircaseProposition H3, Proposition H4, Proposition CORE3, the set of ends · Kiichi, Shiori and Rin

(H3)(H4) and the constant of the core

(H3) bounds the Poincaré constant of a long path by the Poincaré constant of a box of length $C\xi$, and (H4) bounds the gap inside that box at the order of $D/(C\xi)^2$.

1. Proposition H3. Cover the line with overlapping boxes and split them into even boxes and odd boxes; then $C_P\le(1+1/n)\,C_{\rm box}/(1-c_{\rm alt})$. It uses only the lemma of two projections, the product inequality for boxes and averaging over shifts, and does not use the logarithmic Sobolev inequality. The remaining assumption rows are that the maximal correlation between even and odd, $c_{\rm alt}\lt1$, is uniform, and the $\sup$ over the boundary values. The correlation length is read as $\xi\propto\hbar^{1/2}e^{S_0/\hbar}$, and a box does not cut the plateau at the top.
2. Proposition H4. Splitting the inside of the box by the configuration of kinks $Y$ gives $C_{\rm box}\le C_{\rm loc}+(1+\chi)C_{\rm kg}$. If the reduced kink gas is put in the same type as the Glauber dynamics of the one-dimensional Ising model, the gap is the sum of "creation and annihilation of pairs" and "diffusion of kinks", $(C^2+\pi^2)D/(C\xi)^2$. The remaining rows are the cross term (X), the type of the gas (KG), (H1) for scraps, and the constant of the core.
3. Three stages for the constant of the core. Core length $k\le7$ is bounded by convexity, $\le9.18$ (independent of $\hbar$ and of the end values); $8\le k\le14$ by the lemma of two blocks; and $k\ge15$ by applying the reduction to the box to the core. Taking the end values as samples in $[0,x_c]$ gives $\sup_kC_P(\text{core})\le22.1／30.5／66.4$ Reading.
4. The set of ends. In the core of the sector framework, the ends are only on the side of the well, and there is no hole. In the core on the box side, the ends can lie beyond the top, and there the wall of conditioning creates bistability; but its barrier $\Delta S_*\approx10^{-2}$ is about 95 times smaller than $2S_0$, and the local term of Proposition H4 vanishes as $\hbar\to0$ (on the reading at $\hbar=0$).

Setting. Take the action $S=\sum_i(x_{i+1}-x_i)^2/(2\varepsilon)+\varepsilon\sum_iV(x_i)$ on the lattice line $\mathbb Z$ (spacing $\varepsilon=\tfrac12$) at $T=\infty$, the infinite-volume measure $\mu$ of $e^{-S/\hbar}$, and the Dirichlet form $\mathcal E(f)=(\hbar/\varepsilon)\sum_i\mathbb E_\mu|\partial_if|^2$ (the uniqueness of $\mu$ and the nearest-neighbour Markov property are the premise row P0, 10-05 Step 0). A box $B$ is an interval of $L$ sites, the boundary values $(a,b)$ are the values at the neighbouring sites to the left and right of $B$, $\mu_B^{a,b}$ is the conditional measure with these fixed, and $\mathcal E_B$ is the sum over the sites of $B$ only. We put

The maximal correlation of two $\sigma$-algebras is $c(\mathcal G_1,\mathcal G_2):=\sup\{\operatorname{corr}_\mu(F,G):F\in L^2(\mathcal G_1),\,G\in L^2(\mathcal G_2)\}$.

Lemma TP (two projections). If $c=c(\mathcal G_1,\mathcal G_2)\lt1$, then for every $f\in L^2(\mu)$

Proved here (proof in 10-05 Step 1) Leantwo_proj_sum_le (assumptions: Cauchy–Schwarz and the inequality of maximal correlation), two_proj_var (assumption: $A+B\le(1+c)V$).

The family of boxes. Line up boxes $B_m$ of $L$ sites so that neighbours overlap in $l$ sites ($l\le L/3$), and let $E$ be the family of even-numbered boxes and $O$ the family of odd-numbered ones. $\mathcal G_E$ is the $\sigma$-algebra generated by the values at the sites outside $E$, $\mathcal G_O$ likewise, and $c_{\rm alt}(L,l):=c(\mathcal G_E,\mathcal G_O)$ (independent of translation).

Figure 1  Even and odd boxes, and the shift (example with L = 8, l = 2; each character in a column is 1 site)
sites       ....|....|....|....|....|....|....|....
even E      [ B0   ]       [ B2   ]       [ B4   ]
odd O             [ B1   ]       [ B3   ]
overlap           ^^     ^^      ^^     ^^           ← l sites each
shift s=1   move the whole row of boxes l sites right (the overlaps move to other sites)
with n = ⌊(L−l)/l⌋ shifts, the overlap sites are mutually disjoint

The product over boxes. Even boxes are $L-2l\ge1$ sites apart from each other, and the following inequality holds (proof in §03).

Proposition H3. $n:=\lfloor(L-l)/l\rfloor$. If $c_{\rm alt}(L,l)\lt1$, then

Proved here Proof. Fix one shift $s$ and let its even and odd families be $E^s,O^s$. From (1) and (2) for the two families, $(1-c_{\rm alt})\operatorname{Var}f\le C_{\rm box}\bigl(\sum_{B\in E^s}\mathcal E_B+\sum_{B\in O^s}\mathcal E_B\bigr)$. The boxes cover the line, each site lies in the even family at most once and in the odd family at most once, and only the overlap sites lie in both, so the bracket is $\mathcal E(f)+\mathcal E_{{\rm ov}^s}(f)$ (the sum over the overlap sites only). Averaging over the shifts $s=0,\dots,n-1$, the overlap sites are mutually disjoint across shifts, so $\sum_s\mathcal E_{{\rm ov}^s}\le\mathcal E$, hence $(1-c_{\rm alt})\operatorname{Var}f\le C_{\rm box}(1+1/n)\mathcal E(f)$. ∎ Leanbox_two_family (assumptions: Lemma TP, (2) for each family, $\mathcal E_E+\mathcal E_O\le\mathcal E+\mathcal E_{\rm ov}$, $C_{\rm box}\ge0$), box_shift_average (assumptions: the two-family inequality for each shift, the sum over overlaps is at most $\mathcal E$).

(3) is built only from Poincaré inequalities and does not use the logarithmic Sobolev inequality. Conditions (H1)(H2) enter only through $C_{\rm box}$ (§04). Putting in $C_{\rm box}\le K_4(C\xi)^2/D$ of (H4) together with $L\varepsilon=C\xi$ and $D=\hbar/S_0$ gives $C_P\le(1+1/n)K_4C^2/(1-c_{\rm alt})\cdot\xi^2S_0/\hbar$, the form of Proposition 4.5 LeanH3_with_H4, H3_H4_compose (assumptions: (3), (H4), $D=\hbar/S_0$, $n\gt0$, $c_{\rm alt}\lt1$). It is the same type as the box dynamics of the Glauber dynamics of discrete spins (Propositions 3.4 and 3.5 of Martinelli's lectures) Cited; here we rewrite it in one dimension, in $L^2$, and in the form of maximal correlation.

The remaining assumption rows, and the numbers

• (MIX_alt): $c_{\rm alt}\lt1$ uniformly in $\hbar$. There is no paper proof Assumption. As values, on the reading that the slow part is a two-state chain of the sign (the telegraph chain), boxes with $\theta=l/L=\tfrac14$ give $c_{\rm alt}=0.731／0.410／0.138$ for $C=2／4／8$ Reading (values of a family that copies the values of the closed formula of the two-state chain). These values agree with $\varphi(s,t)$ of Proposition G in §05 with $s=e^{-C\theta}$ and $t=e^{-C(1-2\theta)}$ (at $C=2$, $\varphi=0.7308$).
• The $\sup$ over boundary values. $C_{\rm box}$ takes the $\sup$ over all real boundary values. We see in §04 that boundary values near the top are the worst.
• The size of $\xi$. Computing the difference of the even and odd eigenvalues of the transfer operator in the logarithmic domain gives $\ln\xi=218.08／661.04／2212.72$ and $\ln\xi-S_0^{\rm lat}/\hbar-\tfrac12\ln\hbar\to-0.778$ Reading (two independent computational routes agree to six digits at intermediate $\hbar$). That is, $\xi\propto\hbar^{1/2}e^{S_0/\hbar}$, and its ratio to the length of the plateau at the top (the crossing length $\ell_\times\approx0.34/\hbar$, 10-10 type 5) is $\xi/\ell_\times\ge e^{213}$ Leanxi_over_ellx (assumption: the measured $\ln\xi-\ln\ell_\times\ge213$). A box of length $C\xi$ does not cut the plateau; it contains it whole. In exchange, the constant of the core inside the box has to be uniform in the length of the core (§05).

In the Glauber dynamics examples, even when the gap of the ring (the lower edge of the band) was only about a sixth of the gap of the box, $1-c_{\rm alt}$ carried exactly that difference. The band structure of the kink gas does not break the form of the box.

We give the proof of (7) of 10-05 ((2) of this chapter). Row ES was graded "Proved here" in the table of the five grades in 10-05 because of the proof in this section.

Step (a) Conditional independence. The action $S$ is the sum of terms between neighbours $(x_{i+1}-x_i)^2/(2\varepsilon)$ and one-site terms $\varepsilon V(x_i)$. If all values outside a box $B$ are fixed, the conditional density of $x_B$ is determined only by the terms of $e^{-S/\hbar}$ that contain $x_B$, which are the terms inside $B$ and the two terms joining the two ends of $B$ to the two neighbours $a_B,b_B$. So the conditional law is $\mu_B^{a_B,b_B}$, and it depends on the values outside $B$ only through the two neighbours (the nearest-neighbour Markov property). Between even boxes there is at least one site fixed by $\mathcal G_E$, so given $\mathcal G_E$, the conditional density of the even boxes splits into a product of factors, one per box, $$\prod_{B\in E}\frac{1}{Z_B}\exp\Bigl(-\frac{S_B(x_B;a_B,b_B)}{\hbar}\Bigr)$$ and the values of the even boxes $(x_B)_{B\in E}$ are mutually independent with the product law $\bigotimes_{B\in E}\mu_B^{a_B,b_B}$ Proved here (one line on the density splitting into a product in finite volume; that the measure at $T=\infty$ has the same conditional law is the premise row P0).

Step (b) The Efron–Stein inequality for product measures. Let $\nu=\nu_1\otimes\cdots\otimes\nu_m$ be a product measure on $X_1\times\cdots\times X_m$, $F\in L^2(\nu)$, and $\operatorname{Var}_jF$ the variance obtained by moving only the $j$-th coordinate (the other coordinates fixed). Then

Proved here Proof. Put $F_j:=\mathbb E[F\mid X_1,\dots,X_j]$ ($F_0=\mathbb EF$, $F_m=F$) and $\Delta_j:=F_j-F_{j-1}$. The $\Delta_j$ are mutually orthogonal, so $\operatorname{Var}F=\sum_j\mathbb E\Delta_j^2$. Since the coordinates are independent, $F_{j-1}$ is $F_j$ integrated over the $j$-th coordinate with $\nu_j$, and $\mathbb E\Delta_j^2=\mathbb E[\operatorname{Var}_jF_j]$. Moreover $F_j$ is $F$ averaged over the later coordinates $X_{j+1},\dots,X_m$, and $G\mapsto\operatorname{Var}_jG=\|G-\int G\,d\nu_j\|^2_{L^2(\nu_j)}$ is convex (the square of a seminorm), so Jensen's inequality gives $\operatorname{Var}_jF_j\le\int\operatorname{Var}_jF\,d\nu_{j+1}\cdots d\nu_m$. Taking expectations, $\mathbb E\Delta_j^2\le\mathbb E\operatorname{Var}_jF$, and summing over $j$ gives (4). ∎

Step (c) Poincaré for the box. Apply (4) to the conditional product of Step (a) (with the coordinate $X_j$ the value $x_B$ of the box $B$). The variance obtained by moving only the coordinates of a box $B$ is the variance of the measure $\mu_B^{a_B,b_B}$, so

by the $\sup$ in the definition of $C_{\rm box}$.

Step (d) Putting it together. From (4) and Step (c), $\operatorname{Var}(f\mid\mathcal G_E)\le C_{\rm box}\sum_{B\in E}\mathbb E\bigl[(\hbar/\varepsilon)\sum_{i\in B}|\partial_if|^2\bigm|\mathcal G_E\bigr]$. Taking expectations, by the tower property of conditional expectation each term on the right becomes $\mathcal E_B(f)$, which is (2). Since $f$ depends on finitely many sites only, finitely many boxes are involved, and the terms of boxes that do not touch the support of $f$ are $0$. ∎

The extension to general $f$ in the domain of $\mathcal E$ (approximation by functions depending on finitely many sites) is included in the premise row P0. The odd family is the same. (4) of Step (b) is the Efron–Stein inequality (Efron–Stein 1981); since we give the proof here, the grade is "Proved here".

Fix a box $B$ ($L\varepsilon=C\xi$) and boundary values $(a,b)\in\mathbb R^2$, and let $\mu=\mu_B^{a,b}$ and $D=\hbar/S_0$. The expected number of kinks in the box is $C/2$ from the kink density $1/(2\xi)$, which is $O(1)$ Proved here Leanpair_fugacity (assumption: $\xi\ne0$).

The naive split goes round in a circle. Splitting by a rare set $E$ (for example, configurations with a long core), with $p=\mu(E)$, gives $\operatorname{Var}f=p\operatorname{Var}_Ef+(1-p)\operatorname{Var}_{E^c}f+p(1-p)(m_E-m_{E^c})^2$ Proved here (an identity). The second term can be paid with a constant on $E^c$, but the first term is as large as $\operatorname{Var}f$ if $f$ is concentrated on $E$, and the smallness of $p$ does not come out. What can be paid is up to $p\cdot\mathrm{osc}(f)^2/4$, which is not a gap but a weak form of the Poincaré inequality Leannaive_split_weak (algebra). It is an example of 10-10 type 4: rarity cannot be paid with variance. So we take the split not by a rare set but by a slow variable.

Splitting by the slow variable. Let $Y$ be the configuration of kinks in the box (the number of kinks, and the position of each kink defined by projection onto the manifold of kinks), and take the identity for conditional expectation Proved here

as the starting point.

• The first term (the fast side). By the nearest-neighbour Markov property, the conditional measure with $Y$ fixed splits into a product of scraps of the phase without kinks (length $\le C\xi$, with values at both ends), the tubes around the kinks, and the cores inside them. The Poincaré constant of a product measure is at most the maximum of the constants of the components ((4) and Poincaré for each component), so $\mathbb E\operatorname{Var}(f\mid Y)\le C_{\rm loc}\,\mathcal E(f)$ with $C_{\rm loc}=\max(C_{\rm (H1^{seg})},C_{\rm (H2^{tube})},C_{\rm (T0'_P)})$ Proved here (the algebra of products). The three constants are assumption rows Assumption, and the constant $C_{\rm (T0'_P)}$ of the core is bounded in §05 and §06.
• The second term (the slow side). $\mathbb E[f\mid Y]$ is a function of $Y$ alone and relaxes by diffusion of the kink positions (mobility $D$, row (Mob) Cited) and by the creation and annihilation of kink pairs. Row (KG): the Poincaré constant of the box of the reduced kink gas, $C_{\rm kg}\le(C\xi)^2/(c_{\rm kg}(1-\eta_{\rm KG})D)$ Assumption. Row (X): the cross term $\mathcal E_Y(\mathbb E[f\mid Y])\le(1+\chi)\mathcal E(f)$ Assumption (of the type of the cross constant of a two-scale inequality; a statement of this form has not been drawn from a known theorem).

Proposition H4. Under the rows above, for every $(a,b)$

Proved here (the sum of the two terms above) LeanH4_two_scale (assumptions: the decomposition inequality, first term $\le C_{\rm loc}\mathcal E$, (KG), (X), $C_{\rm kg}\ge0$), H4_gap_form (assumptions: $D\gt0$, $C\xi\ne0$, $c_{\rm kg}\gt0$, $\eta_{\rm KG}\lt1$, $1+\chi\ge0$). (5) is the same formula as (12) of 10-05.

Comparison with the Glauber dynamics of the one-dimensional Ising model

The constant $c_{\rm kg}$ of row (KG) is fixed by the assumption that puts the reduced kink gas in the same type as the heat-bath Glauber dynamics of the one-dimensional Ising model. The gap of a box of $L$ sites with the spins at both ends fixed has the closed formula

Assumption (the original source is bibliographic only; diagonalisation on the lattice agrees to the 6 digits shown, and it is the same whether the boundary values are $(+,+)$ or $(+,-)$ Reading) Leanglauber_split (the right-hand equality, no assumptions). In the limit of long correlation length, the first term $1-\gamma\approx1/(2\xi^2)$ is the infinite-volume gap = creation and annihilation of pairs, and the second term $\approx\pi^2/(2L^2)$ is the diffusion of kinks in an interval of length $L$. Written in the Glauber unit $D=\tfrac12$, it is $D/\xi^2+\pi^2D/L^2=(C^2+\pi^2)D/(C\xi)^2$, that is, $c_{\rm kg}=C^2+\pi^2$ Proved here (algebra) Leankg_constant (assumption: $C\xi\ne0$). The pair term is not a small perturbation; it enters additively as one of the main terms of the gap. The local term of the part without kinks does not appear at the lower edge of the gap.

When the boundary values are near the top, a half kink can sit at the end of the box. This corresponds to a free end of the Ising model, and there is no closed formula. Read by diagonalisation on the lattice, the gap does not fall below $1-\gamma$ for any type of boundary (ratio $\ge1.17$), but a free end is slower than a fixed end.

$K=(1+1/n)C^2/(c_{\rm kg}(1-c_{\rm alt}))$ LeanK_fixed_C2, K_ff_C2, K_ff_C4 (assumptions: the measured $c_{\rm ff}$ and $c_{\rm alt}$; arithmetic only). Since $K_4$ decreases with $C$, with fixed ends only $K$ hardly depends on $C$, and including boundary values at the top, the larger $C$ is taken, the better.

The rows remaining in (H4). (X), (KG) (the Glauber type and free ends), (H1^seg) (Proposition H1 of 10-06 is a statement on the ring, and there is no version for scraps with end values), (H2^tube), and the row of the core. The second term $C_{\rm loc}D/(C\xi)^2$ of $K_4$ vanishes if $C_{\rm loc}$ is sufficiently smaller than $e^{2S_0/\hbar}$, because $D/\xi^2$ is of size $e^{-441.6}／e^{-1328.6}／e^{-4433.1}$ Reading. We use this margin for the core in §06.

Setting. A core is a maximal interval of sites at which the path stays between the top and the well, $(0,x_c)$ ($x_c=0.89$). In this chapter we reverse the direction by symmetry and write the core as $(0,x_c)^k$ (the mirror image of $Q_k=(-x_c,0)^k$ in 10-09). The measure of the core $\mu_k^{\alpha,\beta}$ is $e^{-S/\hbar}$ with the outside values at both ends $\alpha,\beta$ fixed, restricted to $(0,x_c)^k$ with Neumann boundary, and the Dirichlet form is $\mathcal E=(\hbar/\varepsilon)\sum_i\mathbb E|\partial_if|^2$ (converted from the convention $\hbar\int|\nabla g|^2$ of 10-09 by $C^{(10\text{-}09)}=C/\varepsilon=2C$). What the fast side of condition (H4) needs is that $C_P(\mu_k^{\alpha,\beta})$ is bounded uniformly in the length of the core $k$ and in the end values (row (T0′_P)).

Stage 1 $k\le7$: convexity (Lemma BV)

The Hessian of the action of the core is

, and the end values $\alpha,\beta$ enter $S$ only in first-order terms, so they do not appear in the Hessian. From $\lambda_{\min}(T_k)=4\sin^2\frac{\pi}{2(k+1)}$ Proved here (the closed formula for the eigenvalues of a tridiagonal matrix) and $\inf V''=V''(0)=-\tfrac12$, at $\varepsilon=\tfrac12$

. This is the same formula that appeared in 10-09 §06 as the smallest eigenvalue on the surface of the top; it is positive for $k\le7$ and negative at $k=8$ ($8\sin^2(\pi/16)\approx0.304\gt\tfrac14$, $8\sin^2(\pi/18)\approx0.241\lt\tfrac14$) Leankappa7_pos, kappa8_neg (no assumptions).

Lemma BV. If $\kappa(k)\gt0$, then for all end values $\alpha,\beta\in\mathbb R$ and all $\hbar\gt0$, $C_P(\mu_k^{\alpha,\beta})\le\varepsilon/\kappa(k)$.

Proved here Proof. By (7), $S/\hbar$ is uniformly convex with $\kappa/\hbar$. A uniformly log-concave measure restricted with Neumann boundary to a convex domain (the box $(0,x_c)^k$) satisfies $\operatorname{Var}f\le(\hbar/\kappa)\int|\nabla f|^2$ Cited (Bakry–Émery on convex domains, checked in Theorem 1.2 (1) of Kolesnikov–Milman), and rewritten in $\mathcal E=(\hbar/\varepsilon)\int|\nabla f|^2$ this is $\le(\varepsilon/\kappa)\mathcal E$. ∎ Leanlemma_BV (assumptions: the lower bound of the Hessian is $\ge\kappa\gt0$ independently of the end values, and the pointwise Bakry–Émery inequality).

The values are $\varepsilon/\kappa(k)=0.133／0.286／0.543／0.973／1.749／3.42／9.18$ ($k=1,\dots,7$), independent of both $\hbar$ and the end values. The $\sup$ measured at $k=1,2$ ($0.128$ and $0.26$) lie within these Reading. If the blocks are cut to 7 sites or fewer, the difficulty of the $\sup$ over end values disappears by convexity. The price moves to the side of the correlation at the seams of the cut blocks.

Stage 2 $8\le k\le14$: two blocks (Lemma TB)

Lemma MR (reduction to the ends). If the chain $x_1,\dots,x_k$ is Markov, then for $a\lt b$, $c\bigl(\sigma(x_1,\dots,x_a),\sigma(x_b,\dots,x_k)\bigr)=c\bigl(\sigma(x_a),\sigma(x_b)\bigr)$.

Proved here Proof. Right-hand side $\le$ left-hand side follows from the inclusion of $\sigma$-algebras. Conversely, let $F$ be a centred function of the front piece and $G$ one of the back piece. By the Markov property $\mathbb E[G\mid x_1,\dots,x_a]=\mathbb E[G\mid x_a]=\mathbb E\bigl[\mathbb E[G\mid x_b]\bigm|x_a\bigr]$, so putting $\hat F=\mathbb E[F\mid x_a]$ and $\hat G=\mathbb E[G\mid x_b]$ gives $\mathbb E[FG]=\mathbb E[\hat F\,\mathbb E[\hat G\mid x_a]]=\mathbb E[\hat F\hat G]\le c(\sigma(x_a),\sigma(x_b))\|\hat F\|\|\hat G\|\le c(\sigma(x_a),\sigma(x_b))\|F\|\|G\|$. ∎

Lemma TB. Let $1\le a\lt b\le k$, $k-a\le7$, $b-1\le7$. Then

Proved here Proof. Apply Lemma TP to $\mathcal G_1=\sigma(x_1,\dots,x_a)$ and $\mathcal G_2=\sigma(x_b,\dots,x_k)$. By Lemma MR, $c(\mathcal G_1,\mathcal G_2)=c(x_a,x_b)$. The conditional law given $\mathcal G_1$ is the chain $x_{a+1},\dots,x_k$ of length $k-a\le7$ (with ends $x_a$ and $\beta$) restricted to the box, and Lemma BV gives $\operatorname{Var}(f\mid\mathcal G_1)\le(\varepsilon/\kappa(k-a))(\hbar/\varepsilon)\sum_{i\gt a}\mathbb E[|\partial_if|^2\mid\mathcal G_1]$. $\mathcal G_2$ likewise gives $\sum_{i\lt b}$ and $\varepsilon/\kappa(b-1)$. Taking expectations and putting them into (1), the coefficient of site $i$ is $[i\gt a]\varepsilon/\kappa(k-a)+[i\lt b]\varepsilon/\kappa(b-1)$; both are switched on only when $a\lt i\lt b$, which happens only when $b\ge a+2$. ∎ Leantb_far, tb_adjacent (assumptions: the conclusion of Lemma TP, BV for each block, $\mathcal E_i\le\mathcal E$, $c\lt1$).

From the conditions $k-a\le7$, $b-1\le7$, $a\lt b$ we get $k\le14$, and at $k=15$ there is no $(a,b)$ Leantb_range, tb_none_at_15 (arithmetic of natural numbers). The input is a single two-point maximal correlation $c(x_a,x_b)$, which is the second singular value of the normalised matrix of the joint distribution of two sites Cited (the singular-value representation of Witsenhausen and Rényi, checked in a restatement). For the measure on the lattice, this value can be computed from products of transfer matrices.

Values. With the ends sampled in $[0,x_c]$, the $\sup$ of the upper bound over $8\le k\le14$ is attained at $k=8$ with both ends $0$ and is $15.5／30.5／66.4$, and the lower bound from polynomial test functions (Rayleigh–Ritz) is $8.39／17.3／38.5$ Reading (at $3\cdot10^{-3}$, the $\sup$ from Lemma TB alone is $22.1$ at $k=13$). $k=8$ is just outside the threshold of convexity; on the reading at $\hbar=0$, the smallest eigenvalue of the Hessian at the constrained minimiser is $0.0175$, and $C_P\to\varepsilon/\lambda_{\min}=28.55$ is approached. There is a peak at $\hbar\approx10^{-4}$ (lower bound 57, upper bound 99) Reading, and it is not monotone in $\hbar$.

Why the form of Stage 3 is not used here. In a core with $k=8$ and both ends at the top, the alternating maximal correlation of subboxes of length 5 rises to $0.879／0.938／0.971$, and the coefficient of Stage 3 becomes $76$ or more Reading Leancoeff_k8 (assumption: the measured $\bar c\ge0.9713$). This is because the soft direction ($\lambda_{\min}=0.0175$) pushes the correlation at the seams towards 1; the form of two blocks avoids the alternating correlation and can be paid with the two-point correlation alone.

Stage 3 $k\ge15$: applying the reduction to the box to the core (form (β))

The measure of the core is also one-dimensional and nearest-neighbour Markov (restriction to a product box preserves the Markov property), so the construction of §02 and §03 can be used literally inside the core. Taking subboxes of length $L_c=5$ with overlap $l_c=1$ ($n_c=4$), the constant of the subboxes (all of 5 sites or fewer, including short ones cut at the ends) is $\le\varepsilon/\kappa(5)=1.749$ uniformly in the end values by Lemma BV, so for every $k$ and all end values

Proved here (the proof of Proposition H3 transferred to the core) Leancore_uniform_L5 (assumptions: the conclusion of Proposition H3, $C_{\rm box}\le7/4$, the measured $\bar c_{\rm core}$). $\bar c_{\rm core}$ is the alternating maximal correlation between the even and odd subboxes inside the core; the values measured at $k\ge15$ are $\le0.535$ over end values (including negative ends beyond the top), and at $k=20〜30$ they are saturated to 7 digits Reading (a measurement of a lower bound of the maximal correlation by additive polynomial functions and two points at the seams). Putting this in gives $C_P\le4.70$.

The form of the alternating maximal correlation (Proposition G). In a stationary Gaussian chain $\dots,L_j,R_j,L_{j+1},R_{j+1},\dots$ with correlation $t$ inside a piece and correlation $s$ at the seams ($\operatorname{corr}(L_j,R_j)=t$, $\operatorname{corr}(R_j,L_{j+1})=s$, $|s|,|t|\lt1$; the chain is taken to continue infinitely on both sides), the maximal correlation between the even pieces and the odd pieces is $\varphi(|s|,|t|)$, where

. Derivation: for the uniform type $F=\sum_{j\text{ even}}(L_j+R_j)$, $G=\sum_{j\text{ odd}}(L_j+R_j)$, the covariance per period is $S_{AB}=2s(1+t)^2/(1-s^2t^2)$, because by the Markov property a two-point correlation is the product of the correlations of the links in between; the variance is $S_{AA}=2(1+t)+2(1+t)^2s^2t/(1-s^2t^2)$, and the ratio is $\varphi$ Proved here (sum of a geometric series) Leangauss_mode0, one_sub_phi. This ratio falls within the framework of the convergence rate of two-block Gibbs sampling for a Gaussian target distribution Cited. The rate of Gibbs sampling that alternately redraws all the even pieces and all the odd pieces is the square of the largest eigenvalue of the block Jacobi matrix (Theorem 5 and §2.4 of Roberts–Sahu 1997), and that largest eigenvalue is the maximal correlation $\varphi$ of the two pieces. The case of pieces of length 1 ($L_j=R_j$; the value $2s/(1+s^2)$ obtained by putting $t=1$ in (10)) is in Theorem 1 of Pitt–Shephard 1999 and Corollary 2 of Mider–Jenkins–Pollock–Roberts 2022, its square being the rate in the limit of infinite chain length Cited. The value of (10) for general piece length is the same quantity as the convergence radius of line Jacobi iteration in numerical linear algebra (the bibliography on that side has not been cross-checked). That the maximal correlation of a Gaussian chain equals the linear canonical correlation is Cited (Kolmogorov–Rozanov, checked in a restatement); that the uniform type is the maximum within it has only been confirmed by computation Reading. Since $\varphi\ge s\ge st$, the naive product $c\le s\cdot t$ is false Leannaive_product_false (assumption: $0\lt s,t\lt1$). The $c_{\rm alt}=0.7308$ of the two-state chain in §02 is the $\varphi$ at $s=e^{-1/2}$, $t=e^{-1}$.

It does not reduce to two-point quantities. In the core, the Gaussian proxy $\varphi_G$ built from the two-point maximal correlation agrees with the measured value at the order of $10^{-4}$ Reading. Even so, for a general Markov chain there is no inequality that bounds the alternating maximal correlation by two-point quantities alone. A chain of three variables $X_1–X_2–X_3$: $X_2\sim\mathrm{Bern}(p)$; if $X_2=1$ then $X_1=X_3=1$; if $X_2=0$ then $X_1,X_3$ are independently 1 with probability $e$; $p=10^{-4}$, $e=10^{-3}$. A function of a binary variable is affine, so the maximal correlation between binary variables is $|\operatorname{corr}|$, and $\operatorname{corr}(X_1,X_2)^2=p(1-p)(1-e)^2/(m_1(1-m_1))\le1/10$ ($m_1=p+(1-p)e$). But $Z=X_1X_3$ has $\operatorname{corr}(Z,X_2)^2=p(1-p)(1-e^2)^2/(m_Z(1-m_Z))\ge98/100$ ($m_Z=p+(1-p)e^2$) Proved here (computation of covariances) LeanrareR_pair, rareR_joint (arithmetic of rational numbers). It is an example in which a non-linear function that holds a rare state from both sides is effective; the paper for row (MIX_core) needs a comparison theorem that uses the fact that the conditional law of the core is close to log-concave.

Proposition CORE3 and the table of the three stages

Proposition CORE3. For a set of end values $\mathfrak E$, let $B_2(\hbar;\mathfrak E):=\sup_{8\le k\le14}\sup_{(\alpha,\beta)\in\mathfrak E^2}\min_{(a,b)}M(a,b)/(1-c_{k,\alpha,\beta}(x_a,x_b))$ and $B_3(\hbar;\mathfrak E):=2.186/(1-\bar c_{\ge15}(\mathfrak E))$. Then for every $k\ge1$ and $(\alpha,\beta)\in\mathfrak E^2$

Proved here (split into cases by $k$ and apply Stages 1–3) Leanthree_stage (assumption: the upper bound of each stage).

From Table 3, $\sup_kC_P(\text{core})\le22.1／30.5／66.4$ Reading Leanthree_stage_3em4 (assumption: the measured values of Stages 2 and 3). The ratios of upper to lower bound are $2.6／1.76／1.72$. The input of Stage 2 (the two-point maximal correlation) is a product of matrices and a second singular value, so it is of a type that can be put on interval arithmetic; but it needs a paper estimate of the difference between the lattice measure and the continuum measure, and the cost of running it is estimated at the order of $10^3$ CPU hours (not yet). In Stage 3, what is measured is a lower bound, and there is no paper for the upper bound.

Table 3 took the end values as samples in $[0,x_c]$. A core is a maximal interval of sites in $(0,x_c)$, so the neighbouring sites outside it actually take values in $(-\infty,0]\cup[x_c,\infty)$. We separate which ends are needed in which construction.

The core of the sector framework — ends only on the side of the well

The core of the sector framework (10-09) is defined as having both end values on the side of the well ($[x_c,\infty)$ in the direction of this chapter). So ends beyond the top do not enter there Proved here (by definition) Leanp5_end_not_negative (assumption: $x_c=0.89$), and the set of ends has no hole. In the stationary chain, a fraction $1-e^{-0.63/\hbar}$ of the mass of runs of length $k$ is of this type, and the minimiser on the reading at $\hbar=0$ is a single shape with every site pressed against the wall $x_c$ Reading.

The core on the box side — ends beyond the top, and the wall of conditioning

In the core on the fast side of Proposition H4 (the core after splitting by the configuration of kinks), the outside neighbour can take negative values beyond the top. There, two local minimisers at $\hbar=0$ appear in cores with $k=9,10$, the barrier is $\Delta S\in[1.1\cdot10^{-4},2.5\cdot10^{-3}]$ (depending on the end values), and the lower bound of $C_P$ for $k=9$ with ends $(-0.035,-0.035)$ is $13.9／44.6／526$, growing in the manner of $e^{\Delta S/\hbar}$ Reading.

• Both minimisers sit on the wall of conditioning. One is a bump spanning the core interval as it is; the other has the zeros at both ends moved inwards by one site each. Because we conditioned the core interval to be "exactly this interval", a wall stands in the direction in which the zeros want to move, and the two types compete. The Peierls–Nabarro barrier of a discrete kink on this lattice is $10^{-12}$ or less, a different order from the bistability barrier $0.5〜2.0\cdot10^{-3}$ Reading Leanpn_vs_bistab (assumption: measured values). The bistability is not physical metastability; it is made by the choice of conditioning.
• Cutting by the binary of the bump brings the fibres back. Splitting by the binary $Y_3$ of whether a site $x_m$ near the top is at or above the threshold $\theta^*$ (whether there is a bump), the $C_P$ of each fibre returns, by Lemma TB (the same proof applies to products of intervals), to upper bound $25.8／46.2／101$ and lower bound $8.9／17.4／32.8$, the same order as Stage 2 of Table 3 Reading.
• $e^{\Delta S/\hbar}$ only moves. In exchange, the Poincaré constant of the marginal of the binary $Y_3$ takes on $e^{\Delta S/\hbar}$, with $\operatorname{Var}(\mathbb E[f\mid Y_3])\ge470\,\mathcal E(f)$ ($3\cdot10^{-4}$) Reading Leany_side_k9_3em4 (assumption: measured values). Cutting does not remove the barrier; it takes the form of transfer of 10-10 type 7. Extending the constant of the core to the actual set of ends while keeping it $O(1)$ cannot be done by this split.