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 4.5, conditions (H1)–(H4), dependency graph · Kiichi, Shiori and Rin

The statement of the first step — four conditions and Proposition 4.5

The setting is as in 10-04 §01. The lattice is $\varepsilon=\tfrac12$, the action $S_{\rm lat}$, the measure $\mu\propto e^{-S_{\rm lat}/\hbar}$, the Dirichlet form $\mathcal E(f)=(\hbar/\varepsilon)\sum_i\mathbb E|\partial_if|^2$, $C_P=\sup_f\operatorname{Var}(f)/\mathcal E(f)$, the diffusion coefficient of kinks $D=\hbar/S_0$, and $S_0=2/3$. We take $T\to\infty$ first and then $\hbar\to0$.

The first tool for building the upper side is the algebra of splitting the measure into two parts. Split the space of configurations in two (for example by the sign of the smoothed value of the path) and write $\mu=Z_+\mu_++Z_-\mu_-$ ($Z_\pm\ge0$, $Z_++Z_-=1$, and $\mu_\pm$ the measures restricted to each part and normalised). Then for every $f$

holds Proved here (the identity for the variance of a mixture. Put $m_\pm=\mathbb E_{\mu_\pm}f$ and $m=Z_+m_++Z_-m_-$, split $\mathbb E_\mu(f-m)^2$ on each part into $f-m_\pm$ and $m_\pm-m$, and use $Z_+(m_+-m)^2+Z_-(m_--m)^2=Z_+Z_-(m_+-m_-)^2$).

Suppose that each of the three terms on the right-hand side has the following inequality.

The first is local Poincaré inside a valley, the second is the "transport of the mean difference", which controls the difference of the means of the two valleys by the Dirichlet form, and the third is the additivity of the form. Since $Z_+Z_-\le\tfrac14$,

holds Proved here Leansplit_poincare_le (assumptions: $Z_++Z_-=1$, $Z_\pm\ge0$, the splitting formula (1), the two local Poincaré inequalities, the mean difference, additivity of the form, non-negative constants). The splitting formula (1) has the same form as Lemma 2.4 of Menz–Schlichting Cited (the skeleton of the Poincaré version of finite-dimensional Eyring–Kramers).

Of the two constants in (3), $C_{\rm loc}$ later becomes condition (H1) and $C_{\rm md}$ becomes condition (H2). But (3) cannot be applied to long paths as it stands.

For long paths, one coarse variable is not enough. (1) is a split of the form "take one slow variable (which valley the path is in) and treat the rest as fast". When the path is much longer than the correlation length ($T\gg\xi$), long-wave fluctuations of the density of kinks form a continuous band, and the Poincaré constant of the conditional measure with one valley fixed stays of the same size as $\xi^2/D$ (the Glauber band of 10-04 §05). That is, $C_{\rm loc}$ does not become $O(1)$. So instead of taking a number of slow variables proportional to the volume, we cut the problem into boxes of length $C\xi$. This is condition (H3) (reduction to the box) and (H4) (the gap inside the box), and (H1)(H2) are used inside the box (§04).

First we list the parts of the construction of the upper side that can be said without assumption rows.

(a) The algebra of the splitting. (1)–(3) are as in §01 Proved here.

(b) The convex core. For the measure restricted to the convex region deep in a valley, $\{x_i\ge a\ \forall i\}$ ($a\gt1/\sqrt3$), the Hessian of the action is at least $\varepsilon\kappa_a$ ($\kappa_a=(3a^2-1)/2$), so Bakry–Émery on a convex region gives $C_P\le2/(3a^2-1)$ uniformly in the length, $\hbar$ and the boundary Proved here (the lower bound on the curvature) Cited (Bakry–Émery on a convex region, Theorem 1.2 of Kolesnikov–Milman). The proof and the Lean statement are in 10-04 §06. But excursions out of the core have smaller action than a kink, and exponentially many of them fit in an interval of length $\xi$, so the convex core alone does not give (H1).

(c) The two-state value of the reduced plateau. On a short ring ($T\ll\xi$) the path that moves between valleys is one pair of kinks, and the separation $\ell$ of the pair becomes the one slow coordinate. In the one-dimensional model that keeps only $\ell$ ($\ell$ moves with diffusion coefficient $2D$, with an atom of a valley at each end and the plateau weight $\pi(\ell)$ in between), we control the mean difference by the Cauchy–Schwarz inequality, write the difference of the values at the two ends of $\ell$ as $\int f'(\ell)\,d\ell$, and

gives a value of the form $C_{\rm md}/4=J/(4D)$, with equality nearly attained by the sign test function Proved here ((4) is the Cauchy–Schwarz inequality) Leanreduced_two_state_value (assumptions: $0\lt T$, $0\lt D$, $J=4\xi^2I/T$, with $I$ the integral of the plateau weight). The reduced model itself (that everything other than $\ell$ may be thrown away) is an assumption of the model, and this value is "a value on the reduced model" Assumption.

(d) The diffusion coefficient of kinks. The position of a single kink moves with diffusion coefficient $D=\hbar/S_0$. At the level of the process of a single interface, this is a copy of the coefficient of a known theorem Cited ((9) of 10-04 §03). It is not a statement about the gap of the whole path.

These four are all. The remaining parts, namely that local Poincaré inside a valley is uniform in the length and in $\hbar$, that transport along the family of flat saddles is uniform in the dimension, the reduction of long paths to boxes, and the gap inside a box, lie outside the assumptions of known theorems (§05). We name them as four conditions.

We write the four conditions making three things explicit: the measure, the constant and the uniformity. All are statements about the 0+1-dimensional double well $V=(x^2-1)^2/8$, the lattice $\varepsilon=\tfrac12$ and the Dirichlet form $\mathcal E$. "Uniform" means that for some $\hbar_0\gt0$ with $\hbar\in(0,\hbar_0]$, the constant does not depend on $\hbar$ (does not grow as $\hbar\to0$).

The four are not conditions of the same weight. (H3) is the step that "moves the problem of long paths to a problem of boxes", (H4) is the step that "splits the inside of a box into slow variables (the arrangement of kinks) and fast variables", and (H1)(H2) are statements on the side of those fast variables. The next section puts into one line the place where (H1)(H2) enter (H4).

To be precise, the conditions named (H1)–(H4) are different from Assumption H (the power statement, 10-01 §05). Assumption H is a statement on the side of the conclusion, and (H1)–(H4) are conditions put in place to derive it in the first step (10-03 §05).

(5) is the same formula as (15) of 10-04 §05 and (3) of 10-03 §05. Below, each step is given with "what it uses" and its grade.

Step 0 The premise rows

That the measure $\mu$ with $T=\infty$ is uniquely determined (simplicity of the principal eigenvalue, from the positivity of the integral kernel of the transfer operator) Assumption (the original source for the version with a positive kernel on $\mathbb R$ has not been checked; the same row as in 10-04 §01). The relation between $C_P(T)$ of the finite ring and $C_P$ of the measure with $T=\infty$ (the step of the limit), and the regularity of the functions used in the steps below (conditional measures, smoothness of the kink coordinates) Assumption. These three are premises placed outside Proposition 4.5, and they are counted as assumption rows in the integrated table (10-12) too.

Step 1 Reduction to the box — condition (H3) in the form of Proposition H3

Cover the line $\mathbb Z$ by boxes of length $L$ sites. Neighbouring boxes overlap in $l$ sites ($l\le L/3$), and the boxes are divided alternately into an even family $E$ and an odd family $O$. Let $\mathcal G_E$ be the $\sigma$-algebra generated by the values at the sites outside the even family, define $\mathcal G_O$ in the same way, and let the maximal correlation of the two $\sigma$-algebras be

.

Lemma TP (two projections). If $c=c(\mathcal G_1,\mathcal G_2)\lt1$, then for all $f\in L^2(\mu)$, $\operatorname{Var}f\le(1-c)^{-1}\bigl(\mathbb E\operatorname{Var}(f\mid\mathcal G_1)+\mathbb E\operatorname{Var}(f\mid\mathcal G_2)\bigr)$.

Proved here Proof. Put $\tilde f=f-\mathbb Ef$ and $a_i=\mathbb E[f\mid\mathcal G_i]-\mathbb Ef$. Then $s:=\|a_1\|^2+\|a_2\|^2=\langle\tilde f,a_1+a_2\rangle\le\|\tilde f\|\,\|a_1+a_2\|$, while $\|a_1+a_2\|^2\le s+2c\|a_1\|\|a_2\|\le(1+c)s$. Hence $s\le(1+c)\|\tilde f\|^2$, and adding $\mathbb E\operatorname{Var}(f\mid\mathcal G_i)=\|\tilde f\|^2-\|a_i\|^2$ gives at least $(1-c)\|\tilde f\|^2$. ∎ Leantwo_proj_var (assumption: $A+B\le(1+c)V$. The geometry of the projections is outside Lean).

The product over boxes. Even boxes are $L-2l\ge1$ sites apart, so by the Markov property of the nearest-neighbour interaction they are mutually independent under $\mathcal G_E$. The variance of a function of independent components is at most the sum of the conditional variances component by component (of the Efron–Stein type), so

holds Proved here (the Markov property and the variance inequality for independent products; the proof is placed in 10-11). The same holds for the odd family. Each site lies in the even family at most once and in the odd family at most once, and only the overlaps lie in both.

Averaging over shifts. Shift the arrangement of boxes by $l$ at a time, in $n=\lfloor(L-l)/l\rfloor$ ways. Every shift has the same $c_{\rm alt}$, and the overlap sites are disjoint between shifts. Applying Lemma TP and (7) to each shift and averaging gives

Proved here Leanbox_two_family, box_shift_average (assumptions: the two-family inequality for each shift, $C_{\rm box}\ge0$, the sum of the forms on the overlaps at most $\mathcal E$). This is Proposition H3, and $c_M$ of (H3) corresponds to $(1-c_{\rm alt})/(1+1/n)$. This step uses only Poincaré inequalities, not the logarithmic Sobolev inequality.

One assumption row remains in (8): that the alternating maximal correlation $c_{\rm alt}\lt1$ is uniform in $\hbar$ (row (MIX_alt)). Its value is read as $c_{\rm alt}\approx0.73$ at $C=2$ and overlap $\theta=l/L=\tfrac14$ Reading (a family value copied from the closed formula for the two-state telegraph chain; not a proof of an upper bound). The type of the reduction to the box has the same form as the box dynamics for Glauber dynamics of discrete spins (Propositions 3.4 and 3.5 of Martinelli's lectures) Cited (discrete spins, in the form of a total-variation coupling; here rewritten in one dimension, in $L^2$, in the form of maximal correlation).

Step 2 Inside the box — condition (H4) in the form of Proposition H4, and where (H1)(H2) enter

Fix a box $B$ ($L\varepsilon=C\xi$) and boundary values $(a,b)$. Let the slow variable inside the box be $Y$ = the arrangement of kinks (the number of kinks, and the position of each kink defined by projection onto the manifold of kinks). Splitting the variance by $Y$ gives

Proved here (the identity for conditional expectations).

First term = the fast side. (H1) and (H2) enter here. By the nearest-neighbour Markov property, the conditional measure with $Y$ fixed splits into a product of the following parts. (i) Scraps of the phase without kinks between kinks (length $\le C\xi$, with values at both ends). (ii) The tube around each kink or kink pair, and the core inside it (the interval that stays near the top). The Poincaré constant of a product measure is the maximum of the constants of the components (the variance inequality for independent products), so

Proved here (the Poincaré constant of a product measure is the maximum of the constants of the components. That it splits into a product depends on how $Y$ is defined, and its regularity is included in the row of Step 0). The constant of (i) is the scrap version of condition (H1), and the constants of (ii) are the tube and core rows of condition (H2) Assumption (all three are assumption rows).

Second term = the slow side. Longitudinal transport along the family of saddles enters here. $\mathbb E[f\mid Y]$ is a function of $Y$ alone, and its fluctuations relax by the diffusion of kink positions and by the creation and annihilation of kink pairs. The Dirichlet form $\mathcal E_Y$ on $Y$ has mobility per kink of the form $D=\hbar/S_0$ (row (Mob): the projection formula, 10-04 §03). Here we put two rows.

• Row (KG): the Poincaré constant of the box for the reduced kink gas, $C_{\rm kg}\le(C\xi)^2/\bigl(c_{\rm kg}(C)(1-\eta_{\rm KG})D\bigr)$, with $\eta_{\rm KG}\lt1$ uniform in $\hbar$ and the boundary values Assumption (the assumption that the kink gas falls into the same type as the Glauber dynamics of the one-dimensional Ising model).
• Row (X): the cross term $\mathcal E_Y(\mathbb E[f\mid Y])\le(1+\chi)\,\mathcal E(f)$, with $\chi$ uniform in $\hbar$ and the boundary values Assumption (of the type of the cross constant of a two-scale inequality; a statement of this form is not taken from a known theorem).

$c_{\rm kg}$ of (KG) comes from reading in the continuum limit the closed formula for the gap of the Glauber box with fixed ends, $(1-\gamma)+\gamma(1-\cos(\pi/(L+1)))$,

which gives $c_{\rm kg}=C^2+\pi^2$ Proved here (the algebra of (11)) Leankg_constant (assumption: $C\xi\ne0$). The first term $D/\xi^2$ is the creation and annihilation of pairs, and the second is the diffusion of kink positions. When a boundary value is near the top, half a kink can sit at the end of the box (corresponding to a free end), and in the $\sup$ including that case $c_{\rm kg}=c_{\rm ff}(C)$, read as $7.17$ at $C=2$ Reading (a family value obtained from lattice values of the gap of the free-end box of the one-dimensional Ising model).

Combining (9)(10) with (KG)(X) gives

Proved here LeanH4_two_scale (assumptions: the inequality of decomposition (9), first term $\le C_{\rm loc}\mathcal E$, (KG) $\le C_{\rm kg}\mathcal E_Y$, (X), $C_{\rm kg}\ge0$), H4_gap_form (assumptions: $D\gt0$, $C\xi\ne0$, $c_{\rm kg}\gt0$, $\eta_{\rm KG}\lt1$, $1+\chi\ge0$, the two inequalities above). This is Proposition H4.

The second term of $K_4$ is $C_{\rm loc}$ multiplied by $D/(C\xi)^2$, and $D/\xi^2=\hbar/(S_0\xi^2)$ is of size $e^{-441.6}／e^{-1328.6}／e^{-4433.1}$ at the three values of $\hbar$ ($\hbar=3\cdot10^{-3}／10^{-3}／3\cdot10^{-4}$) Reading (Table 2 of 10-04 §07). Unless $C_{\rm loc}$ grows exponentially in $\hbar$, this term vanishes as $\hbar\to0$. In this sense the demand of (H1) for $C_{\rm loc}=O(1)$ is a demand with room to spare.

A note on where (H2) goes. In the two-valley split of §01, (H2) was the mean-difference constant $C_{\rm md}$ itself. In the box split, the direction along the family of saddles (the kink positions) was taken as the slow variable $Y$, so transport in that direction is handled by (KG) and (X) of the second term, and what remains in (H2) is the Poincaré of the tube and the core with $Y$ fixed. In this book the tube and core rows are counted as rows of (H2), and (KG)(X) as rows of (H4) (the same counting as the integrated table, 10-12).

Step 3 Composition — $C_P\le K\xi^2S_0/\hbar$

Putting (12) into $C_{\rm box}$ of (8) and using $L\varepsilon=C\xi$ and $D=\hbar/S_0$ gives

Proved here LeanH3_H4_compose (assumptions: the conclusion (8) of Proposition H3, the conclusion $C_{\rm box}\le K_4(C\xi)^2/D$ of Proposition H4, $D=\hbar/S_0$, $n\gt0$, $\hbar\gt0$, $S_0\gt0$, $c_{\rm alt}\lt1$). That $K$ does not depend on $\hbar$ is the uniformity of $c_{\rm alt}$, $\chi$, $\eta_{\rm KG}$, $c_{\rm kg}$ and $C_{\rm loc}$, that is, the very content of the assumption rows.

Putting in the values of the reduced model ($\chi=\eta_{\rm KG}=0$, $n=3$, $\theta=\tfrac14$) gives $K\approx2.76$ at $C=2$ (the $\sup$ including boundary values at the top), $1.43$ (ends on the well side only), and $K\approx1.78$ at $C=4$ Reading (family values, not upper bounds on $K$) LeanK_ff_C2, K_ff_C4 (assumptions: the measured values of $c_{\rm ff}$ and $c_{\rm alt}$; arithmetic only). The value is the same at the three values of $\hbar$ because $\hbar$ does not enter the reduced model.

Step 4 Squeezing the ratio $R$ — $1\le R\le K'/\hbar$

Dividing (13) by $\xi^2$ gives $R=C_P/\xi^2\le KS_0/\hbar=:K'/\hbar$ Proved here (division). From below, Proposition L (10-04 §04) gives $R\ge1$ Proved here (continuous time, $T\to\infty$; the proof is placed in 10-04). Together,

. Proposition L is a lower bound that uses no conditions, and the left inequality of (14) does not depend on (H1)–(H4).

Step 5 The power form of H — $\ln C_P/\ln\xi\to2$

$\ln C_P/\ln\xi=2+\ln R/\ln\xi$, and (14) gives $0\le\ln R\le\ln K'+\ln(1/\hbar)$. The remaining input is how the ruler grows, $\ln\xi\ge s/\hbar$ (for some $s\gt0$ and small $\hbar$), which comes from $\xi\propto\hbar^{1/2}e^{S_0/\hbar}$. The semiclassical leading term of the continuum model is a copy of a known formula Cited, and for the lattice model it is a reading that $\ln\xi-S_0^{\rm lat}/\hbar-\tfrac12\ln\hbar$ settles to a constant, $-0.774／-0.777／-0.778$, at the three values of $\hbar$ Reading (10-04 §07). On this input

and the power form of H follows Proved here Leanpower_H_of_kink_ratio (assumptions: $0\lt K$, $0\lt s$, $0\lt\delta$, and $1\le R\le K/\hbar$ and $s/\hbar\le\ln\xi$ for $0\lt\hbar\lt\delta$. $K$ in Lean is $K'$ in the text).

Checking the single line

Shrinking Steps 1–5 to one line gives

. Of the inputs written above the arrows, the assumption rows are (H1^seg), (H2^tube), (T0′_P), (KG), (X), (MIX_alt) and the premises of Step 0, and the rest are elementary algebra, known theorems or readings.

Here we also check that the statements of the assumption rows do not contain the quantities of the conclusion. The quantities of the conclusion are $C_P(\mu)$ of the whole measure with $T=\infty$ and the ratio $R$. (H1^seg), (H2^tube) and (T0′_P) are constants of the measures of scraps, tubes and cores, (KG) is a constant of the reduced gas, (X) is a comparison of forms, and (MIX_alt) is a quantity of correlation. None of them contains $C_P(\mu)$ or $R$. A form that puts the conclusion among the premises (a circle of dependence) does not appear in the way these steps are built (for how to check this, see 10-13).

To be precise, the conclusion of Proposition 4.5 is the first-step statement that "the relaxation time $C_P$ of Langevin dynamics grows as a power of $\xi^2$". $C_P$ is a different quantity from the gap of the transfer operator (the tunnelling splitting), and even if every row of Proposition 4.5 were in place, the statements from the second step on would not follow (10-01 §08, 10-03 §06).

Within the range searched, no known theorem gives the upper bound (5) of Proposition 4.5. Table 1 summarises how far nearby theorems reach and where they stop.

As the last three rows of Table 1 show, the known tools sit at the two ends, "a single kink" and "an upper bound uniform in the volume but coarse", and what lies between them (an upper bound on long paths sharp up to the diffusion time $\xi^2/D$ of the kink gas) is the place of Proposition 4.5. The Bauerschmidt–Dagallier bound becoming $e^{c\xi}$ is read as follows: the criterion looks at a single quantity, the susceptibility, so it cannot distinguish slow but local relaxation from slow and global relaxation, and $\xi$ enters the exponent Reading.

We put the steps of §04 into one table in the format of the five grades (10-00 §01). Each row has one grade, and the column of machine checking is outside the grades. The symbols of the rows correspond to the numbers of the nodes of the dependency graph in §07.

The "Proved here" rows of Table 2 are all algebra that takes assumption rows as input. That "Proved here" rows line up does not mean that the conclusion of Proposition 4.5 has been shown. The conclusion becomes unconditional only when every "Assumption" row and the "Reading" row (Xi) have been replaced.

We turn the rows of Table 2 into a list of numbered nodes. The format is common to this part (the integrated graph of 10-12 is the same), and the numbers of the premises to the right of "⟸" are always placed smaller than the number of the conclusion on the left. The leaves (assumptions, known results, readings) have small numbers, and the conclusion has the largest number.

[01] ASM P0        uniqueness of the T=∞ measure, limit step, regularity     ħ: —
[02] ASM (H1seg)   Poincaré of scraps without kinks                         ħ: —
[03] ASM (H2tube)  Poincaré of the tube (length ≤ Cξ)                       ħ: —
[04] ASM (T0'P)    Poincaré of the core (uniform in core length)            ħ: —
[05] ASM (KG)      box gap of the reduced kink gas                          ħ: —
[06] ASM (X)       cross term ℰ_Y ≤ (1+χ)ℰ                                  ħ: —
[07] ASM (MIXalt)  alternating maximal correlation c_alt < 1               ħ: —
[08] CIT Mob       mobility of a single kink D = ħ/S₀                       ħ→0
[09] RDG Xi        ln ξ ≥ s/ħ (ξ ∝ ħ^½ e^(S₀/ħ))                            three points, leading term
[10] PRV TP        lemma of two projections                                 independent of ħ
                   Lean two_proj_var (assumption: A+B ≤ (1+c)V)
[11] PRV ES        conditional independence of even boxes, product variance (7) ⟸ [01]   independent of ħ
[12] PRV DEC       decomposition of variance (9), into a product (10) ⟸ [01]   independent of ħ
[13] PRV KGc       c_kg = C²+π² (algebra of (11))                           independent of ħ
                   Lean kg_constant (assumption: Cξ ≠ 0)
[14] PRV Prop H3   (8) ⟸ [07][10][11]                                       independent of ħ
                   Lean box_two_family, box_shift_average (assumptions: two-family inequality, C_box ≥ 0, sum over overlaps ≤ ℰ)
[15] PRV Prop H4   (12) ⟸ [02][03][04][05][06][08][12][13]                  independent of ħ
                   Lean H4_two_scale, H4_gap_form (assumptions: decomposition, C_loc, (KG), (X), D > 0, η < 1 and others)
[16] PRV Prop 4.5  (13) C_P ≤ Kξ²S₀/ħ ⟸ [08][14][15]                        ħ→0
                   Lean H3_H4_compose (assumptions: (8), (12), D = ħ/S₀, n, ħ, S₀ > 0, c_alt < 1)
[17] PRV Prop L    R ≥ 1 ⟸ [01]                                             every ħ
                   Lean local_rayleigh_ratio_le_one (skeleton of the finite sum only)
[18] PRV (14)      1 ≤ R ≤ K'/ħ ⟸ [16][17]                                  ħ→0
[19] PRV power H   ln C_P / ln ξ → 2 ⟸ [09][18]                             ħ→0
                   Lean power_H_of_kink_ratio (assumptions: 0 < K, s, δ; 1 ≤ R ≤ K/ħ; s/ħ ≤ ln ξ)

How to check. In every row with "⟸", look to see that the numbers on the right are smaller than the number on the left. [11] and [12] use only [01], [14] uses [07][10][11], [15] uses [02]–[06][08][12][13], [16] uses [08][14][15], [17] uses [01], [18] uses [16][17], and [19] uses [09][18]. Every row keeps the rule. If the rule is kept, following the premises always lowers the number and eventually reaches a leaf, so there is no circle.

Among the leaves, "Assumption" covers the seven [01]–[07], "Cited" is [08], and "Reading" is [09]. The conclusion [16] of Proposition 4.5 depends on all of [01]–[08] (on [01] also through [11][12]), and the power form of H [19] depends further on [09]. Only Proposition L [17] depends on nothing but [01] and does not use (H1)–(H4).

To be precise, this graph is the dependency of the construction of the steps of Proposition 4.5, and it does not include the content of each assumption row (for example Proposition H1, which supports [02] at the nodes). Those are in the graphs of the chapters on each condition, and the graph that joins them all into one is the integrated graph of 10-12.

There is no $\hbar$ at which all rows are in place. The node propositions of condition (H1) are at six points with $\hbar\ge0.005$, the conditional propositions that support the core row are at the lower end of the window, of order $\hbar\le10^{-3}$, and the two do not meet. (KG), (X) and (H2^tube) are assumption rows at every $\hbar$. The band figure and the way of fitting them together are placed in 10-12.

We list where each of the four conditions stands, one line each. The details are in each chapter.

This part is a study that chose Assumption H as its foothold, and Proposition 4.5 is one way of splitting to build the upper side of the first step. Other views are treated in Part 11.

What this chapter can say

Sources and reproduction

Words used in this chapter — splitting of the variance, local Poincaré, transport of the mean difference, conditions (H1)–(H4), Proposition 4.5, phase (phase without kinks), scrap, tube, core, box dynamics (reduction to the box), maximal correlation, alternating maximal correlation, lemma of two projections, Efron–Stein inequality, reduced kink gas, cross term, dependency graph. Their meanings are in A-02 Glossary.

Next chapter to read: 10-06 (H1) Local Poincaré within a phase — turns the Poincaré constant of the measure of the phase without kinks into node propositions at six values of $\hbar$, by a two-scale construction of kept sites and bridges, and describes the method and cost of filling in between the nodes.

← 10-04 The path measure of the double well, and kinksContents10-06 (H1) Local Poincaré within a phase →