2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasecondition (H1), Proposition H1, six nodes · Kiichi, Shiori and Rin

(H1) Local Poincaré within a phase

Statement. On the lattice of paths of the double well ($\varepsilon=\tfrac12$), consider the measure $\mu^{(+)}$ restricted to $x_i\ge0$ at every site. Condition (H1) says "$C_P(\mu^{(+)})$ is at most a constant independent of $\hbar$ and of the length". Bakry–Émery on the convex core works only deep in a valley ($x_i\ge a\gt1/\sqrt3$), and exponentially many excursions within the valley fit in an interval of length $\xi$, so this measure is not convex (10-04 §06).

Construction. Keep one site in every 12 (the coarse side), and treat the 11 sites between them as bridges with both ends fixed (the fine side). Split the variance into "the conditional variance with the kept sites fixed" and "the variance of its expectation". The fine side is controlled by splitting each bridge further into three small bridges and two sites, and the coarse side is judged on the chain of kept sites by the gap at one site and the strength of the coupling between neighbours. The key to the criterion on the coarse side is the margin inequality

, and interval arithmetic is used to check it for every combination of end values.

Proposition H1. At each of the six nodes of $\hbar$, the margin $\ge0.067$ is checked computer-assisted for all neighbour and end values in $[0,2.5]$, and on top of the remaining rows (assumptions and known theorems) $C_P(\mu^{(+)})\le C$ with $C\le1.6\times10^{10}$ (from paper and interval-arithmetic inputs only). With numerical readings as input, $C\le658$. Neither depends on the length of the ring.

What remains. For the $\hbar$ between the nodes, there is a scheme that fills in subinterval by subinterval by making $\beta=1/\hbar$ a third direction of tilting, and its cost is estimated at tens of hours on one core. $\hbar\lt0.005$ needs infinitely many subintervals and so needs another method (asymptotics as $\hbar\to0$). End values $\gt2.5$, the sharpness of the constant, the scrap version used inside the box, and checking the bibliography of the cited theorems also remain.

The measure of the phase. From the lattice action on a ring of $N=12n$ sites ($n\ge3$),

we form the measure $\mu^{(+)}\propto e^{-S/\hbar}\mathbf 1\{x_i\ge0\ \forall i\}$ restricted to configurations with $x_i\ge0$ at every site. The orthant is convex, so this measure is the equilibrium of Langevin dynamics reflected at the wall $x_i=0$ (Neumann condition). The Dirichlet form is $\mathcal E(f)=(\hbar/\varepsilon)\sum_i\mathbb E|\partial_if|^2$. The path stays on the side of the valley $x=1$ and has no kinks. This is "the phase without kinks".

The version used inside the box ((H1^seg) of 10-05 §04 Step 2) uses, instead of the ring, scraps of length $\le C\xi$ with values at both ends, and asks for uniformity in the end values as well.

What is hard. Three naive roads all lose a factor exponential in $\hbar$.

• The convex core: restricting to $x_i\ge a\gt1/\sqrt3$ gives $C_P\le2/(3a^2-1)$ by Bakry–Émery, but the action of an excursion out of the core (0.1535 at $a=1/\sqrt3$) is smaller than the kink action $2/3$, and $e^{(S_0-c_a)/\hbar}$ excursions fit in an interval of length $\xi$ Proved here (10-04 §06).
• Global perturbation (of the Holley–Stroock type): the oscillation of the difference from a convex action is proportional to the length of the path, and the constant is of the type $e^{T\delta/\hbar}$ Cited (the Holley–Stroock perturbation lemma; type checked in the restatement as Proposition 6.1 of Ledoux).
• Perturbation site by site: the non-convexity per site is $\varepsilon/(8\hbar)$, but the coupling between neighbours $(x_{i+1}-x_i)^2/(2\varepsilon)$ is strong, so sites cannot be detached and treated one at a time Reading.

The numbers do not contradict (H1). Measuring the integrated autocorrelation time of the magnetisation for Langevin dynamics on the ring with a wall gave $0.86／1.40／1.50／1.39$ at $\hbar=0.3／0.15／0.1／0.07$, sizes that do not depend on $\hbar$ Reading (the autocorrelation of a single observable gives only a lower bound on the Poincaré constant, and is not evidence for an upper bound).

Kept sites and bridges. Keep the sites $y_j:=x_{12j}$, and treat the 11 sites $x_{12j+1},\dots,x_{12j+11}$ between $y_j$ and $y_{j+1}$ as a bridge with both ends fixed at $y_j,y_{j+1}$. Splitting the variance by the kept sites $y$ gives

(row (T5): the decomposition of the variance is elementary; the regularity and density of the functions used are assumptions). Because the interaction is nearest-neighbour, the bridges are independent once $y$ is fixed, and the marginal measure of $y$ is a nearest-neighbour chain of the kept sites (row (T2): the Markov property).

The fine side: nesting inside the bridge. A bridge of 11 sites is not convex (the smallest eigenvalue of the Hessian of the bridge is $(1/\varepsilon)\lambda_{\min}(L_{11})-\varepsilon/2=-0.11$). So we keep sites 4 and 8 of the bridge as one more level and split it into three small bridges (3 sites) and two sites $(x_4,x_8)$.

• A small bridge of 3 sites is log-concave: Hessian $\ge(1/\varepsilon)\lambda_{\min}(L_3)-\varepsilon/2=2(2-\sqrt2)-\tfrac14\approx0.92$. Bakry–Émery gives the gap $\rho_f=0.922$ Proved here (lower bound on the curvature) Cited (Bakry–Émery on a convex region).
• The cross term between the small bridges and the two sites is $\kappa_f=1.98$ by the Brascamp–Lieb inequality Cited.
• The gap of the marginal measure of the two sites $(x_4,x_8)$ is $\lambda_c\ge1.3\times10^{-4}$ (end values $[0,4]^2$, $\hbar\in(0,0.2]$) Computer-assisted (row (T3)).

Combining the three by the nested two-scale algebra, the gap of the bridge is at least the smaller root of $t^2-(\rho_f+\lambda_c+\kappa_f^2/\rho_f)t+\rho_f\lambda_c=0$, and $\lambda_S\ge2.3\times10^{-5}$ Proved here (arithmetic of the root).

The coarse side: the gap at one site and the coupling. The one-dimensional conditional measure of $y_j$ with the neighbour values $c,d$ fixed is $\propto e^{-U(t)/\hbar}$, with

($\mathbb E_{a,b}$ is the expectation over the bridge with ends $a,b$). Write $\hbar\rho(c,d)$ for its Poincaré gap multiplied by $\hbar$, and let the strength of the coupling between the two ends of the bridge be

(the second equality is the identity obtained by differentiating $\partial_a\log Z(a,b)$ in $b$ Proved here; non-negativity follows from $e$ being increasing in $a$ Cited (the FKG correlation inequality)). Using the Poincaré version of Helffer's criterion (Proposition 3.1 of Ledoux) in the Gershgorin form on the ring, the gap of the chain on the coarse side is at least the margin

Cited (Proposition 3.1 of Ledoux) Proved here (the Gershgorin form) Leanring_gershgorin (the inequality for quadratic forms on the ring; assumption: $\kappa\ge0$). (1) means $\lambda\gt0$.

The cross term and the algebra of the rate. The strength of the cross term linking the fine side and the coarse side is $\kappa^2=2(v/\hbar)\lambda_S/\varepsilon^2$, with $v/\hbar\le2\bigl(644/1455+(60/97)^2/\lambda_c\bigr)\le5888$ (row (T4)) Proved here (arithmetic) LeanT4_paper_numeric (assumption: $\lambda_c\ge1.3\times10^{-4}$) Cited (Brascamp–Lieb). Both $\lambda$ and $\lambda_S$ are in units of the gap for the flat Dirichlet form $\int|\nabla f|^2$ multiplied by $\hbar$. Then if $0\le t\lt\lambda$ satisfies

, we have $C_P(\mu^{(+)})\le\varepsilon/t$ Proved here LeanH1_assembly (algebra with the assumptions named: decomposition (3) [hT5], the fine side [hFine; $\rho$ in Lean is $\lambda_S$ in the text], the conclusion of the criterion on the coarse side [hLedoux], the margin $\lambda\le\hbar\rho-2\hbar\kappa$ [hT1], $0\lt\lambda$, the cross term [hT4], the rate condition (7)).

The whole construction, as a tree of decompositions of the variance, is as follows.

Var f = E[Var(f|y)] + Var(E[f|y])                                 (T5: decomposition of the variance)
 ├ fine side: bridges of 11 sites with y fixed
 │    └ nesting {4,8}: small bridges BE ρ_f=0.922, cross term BL κ_f=1.98, two sites λ_c (T3) → λ_S
 ├ coarse side: ring of kept sites (nearest-neighbour, T2) ── Poincaré version of Helffer's criterion (Gershgorin form)
 │    └ (T1): gap at one site ħρ (Muckenhoupt, FKG band, tail) − 2·coupling ħκ (Taylor of the tilt, quadrature intervals) ≥ λ
 └ cross term: κ² = 2(v/ħ)λ_S/ε² (T4 ← T3)
 → rate condition tκ² ≤ λ_S(λ_S−t)(λ−t) (H1_assembly) → C_P ≤ ε/t

The key row is the margin (6), and the values of Table 1 are obtained at the six nodes for every combination of neighbour and end values in $[0,2.5]$.

The interval arithmetic is implemented by adding outward widths to float64 operations (rounding of non-negative sums is widened outwards by a relative $10^{-10}$, interval arithmetic operations by a relative $10^{-12}$, weights below $e^{-700}$ are replaced from above, and the region $\varepsilon V/\hbar\gt690$ is covered by a tail upper bound). It is not rounding with a direction specified for each operation, but an added width that presupposes correct IEEE rounding. This premise is written in the same row, row A1 of Table 2. The arithmetic that gives the margin $\ge0.067$ from the six pairs of values is LeanT1_margin_six_nodes (no assumptions, numbers), and the step from the interval output to the margin at one node is T1_node_margin (assumptions: the two inequalities of the interval output and $0.067\le r-2k$).

Tools that extend the intervals to the continuum between nodes. Table 1 gives values "for every combination of neighbour and end values", and to pass from computations on finitely many boxes to the continuum we use three lemmas.

• The $\hbar\rho$ side: Muckenhoupt's estimate, which squeezes the Poincaré constant of a one-dimensional measure by a Hardy-type quantity $B$ as $B\le C_P\le4B$ Cited; a lemma that builds a band from the values at the nodes to the continuum of neighbours by the monotonicity (FKG) of $e(c,t)$ in the neighbour value Cited (FKG); and a lemma for the tail on the side of large $t$ (the continuous version of Holley's comparison and a tilted Gaussian bridge) Assumption (the theorem number of the continuous version has not been checked in the original source).
• The $\hbar\kappa$ side: Lemma K2, which tilts the measure at the centre of a box of end values and controls the whole box by the second-order Taylor expansion of the cumulant generating function and a Hölder remainder Proved here (the Taylor remainder and Hölder's inequality). This formula is exact whatever constant is chosen for the centring.
• The smaller of two upper bounds is also an upper bound, and the reflection symmetry $\hbar\kappa(a,b)=\hbar\kappa(b,a)$ covers half Proved here Leanmin_of_two_upper, symm_upper (assumptions: two upper bounds / symmetry and a one-sided upper bound).

The moves that made the margin positive at $\hbar=0.01$ and $0.005$. When $\hbar$ is small the upper bound on $\hbar\kappa$ swells, and in a coarse computation the margin was negative at $\hbar=0.01$, with $0.634$. What limits it differs from place to place, and three moves bring it down. (i) At the corner of the ends $(0,0)$, the centring constant of Lemma K2 is retaken for each subrectangle as the median of that range (the formula is exact for any centre, so only the upper bound goes down). (ii) The second-order term acts as the square of the box width $\delta$, so $\delta$ is halved. (iii) At the large end $(2.5,2.5)$, the bridge jumps far into the well in one step, so the quadrature error is the limit, and the quadrature step is halved. At $\hbar=0.005$, refining the quadrature on the $\hbar\rho$ side raises the lower bound from $0.6945$ to $0.92182$. The last two rows of Table 1 are the results of these moves Computer-assisted.

The constant. $C\le1.6\times10^{10}$ in (8) is the $\varepsilon/t$ that follows from $t=3.2\times10^{-11}$ satisfying the rate condition (7) with the paper and interval-arithmetic inputs ($\lambda\ge0.067$, $\lambda_S\ge2.3\times10^{-5}$, $v/\hbar\le5888$) Proved here (arithmetic) LeanH1_rate_paper_six_nodes (no assumptions, numbers). Taking the gap of the two sites to be the numerical $\lambda_c=0.57$ gives $\lambda_S\ge0.093$, $v/\hbar\le1.0015$, then $t=7.6\times10^{-4}$ and $C\le658$ Reading (the input is a floating-point reading) LeanH1_rate_numeric_six_nodes (no assumptions, numbers). Most of the difference between the two values comes from the weakness of the paper value of $\lambda_c$.

We put the inputs of Proposition H1 into one table in the format of the five grades. Items whose paper proof is placed in this book are "Proved here", those whose proof is not are "Assumption", interval arithmetic is "Computer-assisted" with the premises of its implementation written in the same row, and cited theorems are split into "Cited" or "Assumption" according to the level to which their bibliography has been checked.

The "Computer-assisted" rows A1 and A9 of Table 2 are inequalities at the computed nodes and within the computed ranges. They say nothing about $\hbar$ or end values outside those ranges, and that is what the "Assumption" rows B1–B5 take on.

Proposition H1 is a statement at six points. Condition (H1) is a statement on the continuum of $\hbar$, so the gaps between the points must be filled. This section describes the method for that, the result of trying the method on one subinterval, and an estimate of the cost for the whole range.

(a) $\beta$ as a third direction of tilting. $\beta=1/\hbar$ enters the measure $e^{-\beta S}$ only as an exponent of the weight, so we regard an interval $[\hbar_1,\hbar_2]$ of $\hbar$ as an interval $[\beta_{\rm lo},\beta_{\rm hi}]$ of $\beta$, and add a third direction (the observable $-S$) to the two directions of tilting in the end values in Lemma K2. $\beta$ enters the margin (6) in six places (the weight, the width of the tilt, the prefactor, the quadrature step, the tail, the monotonicity band), and for each it is determined at which end of the interval the worst case lies Proved here. A guide for the ratio of one step, $r=\beta_{\rm hi}/\beta_{\rm lo}$, is $r\approx1.17$, from the fluctuation of the action $\operatorname{Var}(\beta S)$ (about half the degrees of freedom of the bridge, near $11/2$ Reading), and $\hbar\in[0.005,0.2]$ can be covered by 24 subintervals Proved here ($0.2/1.17^{24}\le0.005$) Leannodes_reach_117 (no assumptions, numbers).

(b) The chord form breaks. With the end values $(a,b)$ fixed, $f(\beta)=\hbar\kappa(a,b;1/\beta)$ is smooth in $\beta$, and a lower bound on the second derivative $f''\ge-C$ gives the chord upper bound

Proved here Leanchord_beta_upper (assumptions: $f\in C^2$, $f''\ge-C$, $0\le C$, $\beta_{\rm lo}\le\beta_{\rm hi}$, upper bounds at the ends). But $f''$ is a connected cumulant, and controlling it by a product of absolute values (Hölder) gives $5\times10^3$–$1.8\times10^6$ times the true value Reading, so at $r=1.17$ the remainder is $0.25$–$0.64$, far beyond what is allowed Leanholder_r117_exceeds (numbers).

(c) A step-by-step scheme (a scheme that carries $\hbar$ as an interval). Compute the unnormalised integral $J_X(\beta)=\int Xe^{-\beta H}dx$ ($X\ge0$ is the weight of the product of end observables) at three points of $\beta$, and interpolate by a quadratic. The interpolation error is $|\omega(\tau)|/6\cdot\sup|J_X'''|$, with $\max|\omega|=2\Delta^3/(3\sqrt3)$ Proved here ($\omega$ is the cubic that vanishes at the three nodes. Choosing $K$ so that $g(t)=J_X(t)-p_X(t)-K\omega(t)$ vanishes at the three interpolation points and at $\tau$, four points in all, Rolle's theorem used three times gives a zero of $g'''$, and $K=J_X'''(\eta)/6$). The third derivative is controlled by $|y|^3\le c_3(\lambda_+)e^{\lambda_+y}+c_3(\lambda_-)e^{-\lambda_-y}$ and the log-convexity of $J_X$, through the maximum at the ends of the integrals at four shifted values of $\beta$ Proved here. Splitting the interval of $\beta$ further into small pieces, on each piece the expectation is a linear-fractional function, and its value lies between the values at the two ends of the piece, so it can be enclosed by the interval hull Proved here Leanmobius_between (assumption: the denominator is positive at both ends of the piece), lebesgue_le (assumption: $|u|\le1$), cube_le_exp_split (assumption: $\lambda_\pm\gt0$). This scheme never uses monotonicity. The remainder enters as a third-order quantity of raw integrals that needs no cancellation, and the Hölder loss of (b) does not appear.

(d) A trial on one subinterval. Table 3 is the result of running (c) in interval arithmetic on $\hbar\in[0.008547,0.01]$ ($r=1.17$) and the rectangle of end values $[0,0.18]^2$ Computer-assisted (the same implementation premises as Table 1).

The allowance. What is needed on the subinterval at $\hbar=0.01$ is "point upper bound + swelling $\le\hbar\rho/2=0.4335$" in every box. If the same amount is allowed to every box, the uniform allowance set by the worst box (upper bound $0.39996$) is $0.4335-0.39996\approx0.0335$, and the swelling $0.0464$ of Table 3 exceeds it. On the other hand, the allowance for the boxes in the rectangle of Table 3 is $0.4335-0.19\approx0.24$, and the swelling falls within it.

What limits it. The limit is not the remainder but the growth of the node upper bound itself with $\beta$ (the quadrature width widens with $\beta$). The upper bound $0.39996$ of the worst box is mostly quadrature width, and the floating-point value for the same box is $0.0076$ Reading. Recomputing the band containing this box with finer quadrature brings the upper bound of the box down to $0.0751$ Computer-assisted, but the maximum over the whole range only comes down to $0.39971$. This is because about 32,000 boxes with upper bounds above $0.19$ remain at the coarse setting in a frame along the four sides of the square of end values.

Cost. Table 4 is the estimate for the whole range $\hbar\in[0.005,0.2]$ (computation time on one core; all are estimates, none has been run yet). The cost of one subinterval is about 3.4 times that of a point computation (measured in the trial of Table 3), and the smaller $\hbar$ is, the larger the cost of one point.

Which of the two models is right cannot be decided from a trial on one rectangle. The road that recomputes the frame first passes at $r=1.17$ regardless of the model if the upper bound in the frame falls below $0.318$, so it also narrows the range of the cost estimate. The $\hbar\rho$ side (a one-dimensional measure, 23 degrees of freedom) is read by the same guide as $r\approx1.12$ and 33 subintervals Reading.

To be precise, Table 3 is the result for one rectangle of one subinterval, and does not close that subinterval. To close a subinterval, every rectangle in it must be computed by the scheme of (c).

Table 5 lists the rows that remain in going from Proposition H1 to condition (H1), and to (H1^seg) used inside the box. The cost column is filled only where there is an estimate.

Proposition D is a small proposition of a negative type. The variance $p(1-p)$ of the measure $(1-p,p)$ on two points $\{0,1\}$ is not monotone when $p$ is raised in the order of likelihood ratios ($3/16\lt1/4\gt3/16$) Proved here Leantilt_variance_not_monotone (no assumptions, numbers). The same holds for the family obtained by tilting the symmetric double well linearly, and FKG alone does not give "the gap is monotone in the neighbour".

We turn the rows of Table 2 into a list of nodes in the same format as 10-05 §07. The numbers of the premises are always placed smaller than the number of the conclusion.

[01] CMP A1   (T1a) margin ≥ 0.067 (neighbours and ends [0,2.5]²)      six nodes
[02] PRV A2   Lemma K2 (Taylor of the tilt, any centre)                independent of ħ
[03] CIT A3   Muckenhoupt squeeze B ≤ C_P ≤ 4B                         independent of ħ
[04] CIT A4   MTP₂ correlation inequality (monotone e, ħκ ≥ 0)         independent of ħ
[05] ASM A5   continuous version of Holley's comparison (tail lemma)   six nodes
[06] CIT A6   Poincaré version of Helffer's criterion (Ledoux Prop 3.1) independent of ħ
[07] CMP A9   (T3) λ_c ≥ 1.3×10⁻⁴ (ends [0,4]²)                         (0,0.2]
[08] ASM A10  Poincaré–Miranda theorem                                  (0,0.2]
[09] CIT A12  Brascamp–Lieb (cross term κ_f, (T4))                      independent of ħ
[10] ASM A15  (T5) regularity, density                                  —
[11] PRV A8   (T2) kept sites form a nearest-neighbour chain            independent of ħ
[12] PRV A14  (T5) decomposition of the variance (3) ⟸ [10]             independent of ħ
[13] PRV margin  continuum statement of λ ⟸ [01][02][03][04][05]        six nodes
                Lean T1_margin_six_nodes (numbers), T1_node_margin (assumption: interval output)
[14] PRV A7   coarse-side gap ≥ λ (Gershgorin) ⟸ [06][11][13]           six nodes
                Lean ring_gershgorin (assumption: κ ≥ 0)
[15] PRV A11  nesting λ_S ≥ 2.3×10⁻⁵ ⟸ [07][08][09]                    (0,0.2]
[16] PRV A13  (T4) v/ħ ≤ 5888 ⟸ [07][09]                               (0,0.2]
                Lean T4_paper_numeric (assumption: λ_c ≥ 1.3×10⁻⁴)
[17] PRV A16  algebra of the rate C_P ≤ ε/t ⟸ [12][14][15][16]          six nodes
                Lean H1_assembly (assumptions: hT5, hFine, hLedoux, hT1, 0 < λ, hT4, rate condition)
[18] PRV A17  Proposition H1: C ≤ 1.6×10¹⁰ ⟸ [17]                      six nodes
                Lean H1_rate_paper_six_nodes (numbers)

In every row with "⟸" the numbers on the right are smaller than the one on the left, so there is no circle. Among the leaves, "Assumption" covers [05][08][10] and "Computer-assisted" covers [01][07]. $C\le658$ from numerical inputs (row A18) puts reading values into the algebra of [17], and does not enter the conclusion [18] of this graph. The rows B1–B5 of Table 2 are not premises of Proposition H1 but rows for going from Proposition H1 to conditions (H1) and (H1^seg), and in the integrated graph (10-12) they come between [18] and the node of condition (H1).

The construction of this chapter is one way of handling one condition within the first step, with Assumption H as the foothold. Other views are treated in Part 11.

What this chapter can say

Sources and reproduction

Words used in this chapter — phase (phase without kinks), kept site, bridge, nesting, Helffer's criterion, margin, node, subinterval, allowance, swelling, frame, scrap version (H1^seg), computer-assisted, interval arithmetic. Their meanings are in A-02 Glossary.

Next chapter to read: 10-07 The map of (H2) — seven routes and the seam of the ring — puts on one sheet the map that splits transport along the family of saddles (H2) into seven routes, and the seam of the ring (the outer chain closes on the whole window, the inner chain has finite coefficients).

← 10-05 The statement of the first step — four conditions and Proposition 4.5Contents10-07 The map of (H2) — seven routes and the seam of the ring →