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 (H2), seven routes, the seam of the ring · Kiichi, Shiori and Rin

The map of (H2) — seven routes and the seam of the ring

The statement of condition (H2) is as in 10-05 §03. We copy it here.

What the box split leaves in (H2). In the box split of 10-05, the direction along the family of saddles (the kink positions) was taken as the slow variable, so longitudinal transport in that direction is handled by rows (KG) and (X) of (H4) (the note in 10-05 §04 Step 2). What remains in (H2) are the tube and core rows after the arrangement of kinks is fixed, namely (H2^tube) and (T0′_P). The map of this chapter is a map of the tools that support these two rows.

Why it is hard. On the core $Q_k=(-x_c,0)^k$ ($x_c=0.89$; $k$ sites of boxes, each running from the side of the valley $-1$ to just short of the top $0$), the action is almost flat at the top and the curvature at one site is negative (row W of Table 4 in §05; for details see 10-09 §06). Convexity tools (Bakry–Émery) do not apply to the whole core, and the core length $k$ grows up to the correlation length, so that the constant is uniform in $k$ must be shown by another means.

Two places — the tube of the bridge and the seam of the ring

The rows of (H2) appear in two places where the shape of the path differs.

• The tube of the bridge. The tube around a kink, for configurations with one kink on an interval with both ends fixed in the two valleys (a bridge). The core is the interval inside the tube that stays near the top. (H2^tube) and (T0′_P) inside the box have this shape.
• The seam of the ring. Configurations on a short periodic path (a ring, length $T\ll\xi$) with one kink–antikink pair, at the place where the separation $\ell$ of the pair shrinks and the two meet. The configuration just before the pair disappears is the plateau at the top itself (§03). The argument for the tube of the bridge uses the fact that the kinks are apart, so the seam needs a different treatment.

In both places, the tool that controls the core constant is of the same type. We attach a coarse mark "valley side or top side" to the sites in the core, and combine the logarithmic Sobolev constant $A_W$ of the marginal of the marks on a range of 17 sites around one site (the site window $W$) with the contraction ratio $q$ of the chain extending outside the window. The contraction condition $q\lt1$ of this "chain outside the window" is the outer chain of routes 1 and 6, and the size of $A_W$ is the centre of routes 2, 3 and 5. The site window $W$ is different from the range of $\hbar$ treated computer-assisted (the window of $\hbar$, $[3\cdot10^{-4},3\cdot10^{-3}]$). The window in "the chain outside the window" and "the stages inside the window" is the site window, while the window in "the whole window" and "the lower end of the window" is the window of $\hbar$. Where they could be confused we add "site".

Seven routes were set up to close the core rows of (H2). The numbers are names of the routes and indicate neither order nor weight. "Grade" in Table 1 uses the symbols of the five grades of 10-00 §01, and when two or more grades are mixed in one route they are listed from the main one.

Three notes on how to read Table 1.

(a) Only two rows are computer-assisted. They are route 1 and the outer chain of route 6. Both are closed in the sense that "within the computed window of $\hbar$, the inequality holds given the table of outputs of interval arithmetic", and say nothing outside the window or as $\hbar\to0$ (10-08).

(b) Even if route 1 closes, (H2) does not close. The ratio $q_{\rm free}$ of route 1 enters only one factor $S_W=1+2\bigl(16+1/(1-q_{\rm free})\bigr)$ of the upper bound on the logarithmic Sobolev constant $A_W$ of the site window ([12] of the dependency graph in §06). In the current construction of the upper bound, the upper bound on $A_W$ lies outside the needed value (by 4–5 orders of magnitude in the product form of the factors for the 17-site window, and by 29–74 times even in the form for the 3-site window with the needed side raised, 10-09 §03), and the one remaining proposition (route 5) is being attacked by other strands.

(c) Route 7 was decided to be out of scope. The power form of H is a statement as $\hbar\to0$, so in the end (H1)–(H4) must be stated in the asymptotics $\hbar\to0$, and computer assistance on a fixed window alone does not reach that. Outside the window, paper asymptotics are needed. In this part we first check the grades of the rows inside the window, and how to fit them together is described in 10-12 and 10-15.

A ring is a periodic path of length $T\ll\xi$ with one kink–antikink pair, and we write $\ell$ for the length of the arc on the negative side. The reduced plateau model of 10-05 §02 (c) kept only this $\ell$. The seam is where $\ell$ becomes small and the pair disappears.

The seam is the plateau at the top itself

The continuum minimal action $S(\ell)$ with the length of the negative arc fixed at $\ell$ tends, as $\ell\to0^+$, not to $0$ but to $S_0$ (the path touches the top and returns). For $\ell\le\pi\sqrt2$ the minimising configuration has no kinks: it is a plateau lying flat on the top $0$, with action $S_0+\ell/8$. For $\ell\gt\pi\sqrt2$ a dip appears and the action tends to $2S_0$ ($2S_0-S\approx8e^{-\ell}$) Assumption (a continuum variational argument and a cross-check with floating-point minima on the lattice; the proof is not placed in the book). On the lattice ($\varepsilon=\tfrac12$) too, the minimum with $m$ sites on the negative side lies on $S_{\rm lat}(1)+(m-1)/16$ for $m\le9$ Reading (floating-point minima, difference $\le5\times10^{-11}$). The configuration at the seam is the core itself (the interval that stays near the top), and the same difficulty as for the core of the tube of the bridge appears here.

The $\ell$ direction is easy. One-dimensional transport in $\ell$ alone can be controlled, seam included. On each of the band, the collar and the plateau of the seam, the Muckenhoupt quantity, which squeezes the one-dimensional Poincaré constant within a factor 4, is at most $64S_0\hbar$, $L_0^2/(8D)$ and $T^2/D$, and the algebra passes machine checking Assumption (paper estimates on the one-dimensional model; the reduction to the model is itself an assumption). The hard part is the side of the coordinates: at the seam, the number of sign changes does not become a slow variable. So we fold by sectors that do not depend on the sign, namely runs of sites with $|z|\lt\theta$ ($\theta=\tfrac12$) (lumps). With the rest fixed, the action of a lump of $k$ sites is convex for $k\le7$, from $V''\ge-\tfrac12$ Proved here (the smallest eigenvalue $8\sin^2\frac{\pi}{2(k+1)}-\tfrac14$ of the tridiagonal matrix $\mathrm{tridiag}(-2,4+w,-2)$, $w\ge-\tfrac14$, is positive for $k\le7$).

The outer chain and the inner chain

On a site window $W$ of the ring (17 sites) we put marks: one lump (length $k\le7$, possibly absent), the other sites of the window on the valley side, and outside the window the side of fixed sign. The construction of the logarithmic Sobolev constant of the window involves contraction ratios of two kinds of chain.

• The outer chain: the chain extending outside the site window. It moves in the region of fixed sign $[0,\infty)$, and mapped to $(-\infty,0]$ by the reflection $x\mapsto-x$ it becomes a chain of the same type as the outer chain of the core of the bridge. The action is invariant under the reflection Proved here ($V$ is even) Leanchain_action_reflect (no assumptions; the image of the measure and the reversal of the Stein kernel are outside Lean), but it is not the same chain as the chain of the bridge. The box of the bridge has a lower wall $-x_c$, and the outer chain of the ring has none.
• The inner chain: the stages inside the site window. From stage to stage the box alternates between "valley side" and "lump", and the lump box $(-\tfrac12,\tfrac12)$ straddles $0$.

The difference between the two chains can be read from the corner at the wall. The classical corner ratio at a wall at position $b$ is $r(b)=\Delta/(\Delta-\varepsilon^2V'(b))$. It is exactly 1 at a wall with $V'(b)=0$ (the wall $0$ of the outer chain), and exceeds 1 at a wall where the force pushes out of the box Proved here (the algebra of the sign of the ratio) Leancorner_eq_one (assumption: $\Delta\ne0$), corner_gt_one (assumption: $0\lt c\lt\Delta$). The formula for the ratio itself comes from a computation of the envelope, and its regularity is Assumption. At the upper wall $+\tfrac12$ of the lump, $\varepsilon^2V'(\tfrac12)=-3/64$, so the force points towards the wall and the corner exceeds 1 (the classical value for an isolated wall is $1.2857$).

In one line, Table 2 says: the outside is closed (inside the window, conditional on pin $\le0$), and the inside has a finite coefficient. "Closed" here means that within the computed window of $\hbar$ the contraction of the outer chain became a computer-assisted inequality, not that (H2) is closed on the ring. The result for the outer chain was first closed for the chain conditioned on the box (upper bound $0.998888525$) and then extended to the unconditioned chain (the tail term put into the remainder of the interval arithmetic, deep pins covered by a closed-form upper bound). The extension removed the need for the four assumption rows that returned the conditioned chain to the original measure.

Why the inner chain exceeds 1, and the cost of putting it into intervals. What limits the inner chain is the stage with the pin at the edge of the far wall of the lump box, which is exactly the corner pushing outwards described above (close to the classical value $1.2857$). The interval machinery of the outer chain is built on the premises that the right wall is at $0$ with $V'(0)=0$, that there is one box, and that the pins and the grid coincide, so it cannot be used for the inner chain just by swapping the inputs. The estimated cost of putting the inner chain into interval arithmetic is 1–2 paper steps + 1–2 scripts + about 50 runs, about 42 hours (one core), and what it gives is only an upper bound on the finite coefficient $\bar q\approx1.3$. This coefficient enters $A_W$ as a factor $\bar q^{32}\approx10^{3.6}$, so it is not a road to meeting the closing condition of the sector framework (10-09 §02).

The rows that remain at the seam of the ring. (lim) (that the limit of stacking stages without end falls inside the enclosure of the fixed point at each $\hbar$), (END_rand) (the case where the value at the end of the chain is given not by a point but by a distribution), C2′ (the ratio conditioned on the length of the plateau), the boundary stage, and putting the inner chain into intervals. All remain Assumption or Reading.

In the current construction (Proposition LSσ_W), the upper bound on the logarithmic Sobolev constant of the site window has the form of a product of factors $A=c_W\cdot C_{\rm 1D}^W\cdot S_W\cdot T_W$. $c_W$ is a constant of the marginal measure of the marks, built from the minimal weight $\pi_*$ and the gap of the mark dynamics $\mathrm{gap}_W$, and $S_W$ and $T_W$ are built from the contraction ratios of the chains. Route 2 is the paper reduction and estimate of the number of runs for moving the values of the factors from floating point at three points to interval arithmetic.

The estimate totals about 42 runs and about 21 hours (one core). We decided not to run these 42. The reason is the floor of the product form itself. From $S_W\ge1+2/(1-q_{\rm free})$, $T_W\ge|W|$ and $c_W\ge\ln\bigl(1/\nu(\sigma_i=\text{valley side})\bigr)\approx E_U/\hbar$, even with the true values of the factors, $A\gtrsim1.4\times10^4〜1.4\times10^6$ Reading (estimate), which lies outside the needed value of the sector framework (10-09 §03, tens to hundreds) by tens to tens of thousands of times. Even if the factors are confirmed in interval arithmetic, this form as it stands does not reach the closing condition. The paper reduction of route 2 (screening, the extreme sequence) is kept as a tool that enters unchanged into upper bounds not of product form (the strands of route 5).

We put the rows of Tables 1–3 into one table in the format of the five grades. Each row has one grade, and the column of machine checking is outside the grades. The symbols of the rows correspond to the nodes of the dependency graph in §06.

The two "Computer-assisted" rows of Table 4 (QF and QO) are both computer-assisted on the premise row IA, and what Lean checked is "the construction with the table of outputs of interval arithmetic placed as an assumption" (10-08 §07). The "Proved here" row CORE is algebra on the assumption row LSW, and does not mean that a lower bound on the gap of the core has been shown.

We turn the rows of Table 4 into numbered nodes. The numbers of the premises to the right of "⟸" are always placed smaller than the number of the conclusion on the left (the format common to this part, 10-05 §07).

[01] ASM IA     premises of interval arithmetic (correct IEEE rounding, summation error, own exp)  ħ: —
[02] ASM LIM    (lim), (END_rand), regularity                         ħ: —
[03] ASM LSW    (LS_W♭) A_W < A_max(ħ)                                ħ: —
[04] ASM SM     small assumptions of route 4                          ħ: —
[05] ASM PROD   Proposition LSσ_W (product form A_W ≤ c_W·C_1D·S_W·T_W)  ħ: —
[06] RDG N2     constants of route 2                                  ħ: three points
[07] RDG QIN    inner chain of the ring q_in ≈ 1.24–1.28              ħ: four points
[08] RDG EDGE   boundary stage q ≈ 1.004–1.045                        ħ: four points
[09] CMP QF     q_free ≤ 0.998854904 ⟸ [01]                           ħ: window
               Lean win_qfree_window_le_of_interval_table (assumption: table of interval outputs)
[10] CMP QO     q_out ≤ 0.998888530 (pin ≤ 0) ⟸ [01][02]              ħ: window
               Lean unc_window_lt_one_of_table (assumption: table of interval outputs)
[11] PRV SIG    sector framework (Σ.1)–(Σ.5) ⟸ [02][04]               independent of ħ
               Lean cell_sum_sup, comp_close (assumptions in the table of 10-09)
[12] PRV SWB    S_W of the bridge ≤ 1779.6 ⟸ [09]                     ħ: window
[13] PRV SWR    S_W of the ring ≤ 122745 ⟸ [07][08][10]               ħ: window
               Lean sw_inner_le (assumptions: q_in ≤ 13/10, q_out ≤ 0.998888530)
[14] RDG AV     A_W in product form ≈ 2.5·10⁶–4.7·10⁷ ⟸ [05][06][12]  ħ: three points
[15] PRV CORE   positive lower bound on the core gap uniform in k ⟸ [03][11]  ħ: —
               Lean comp_close (assumptions: the three inequalities of Barta, (Σ.3), Bochner)
[16] PRV T0P    the form of row (T0'_P) of 10-05 ⟸ [02][15]           ħ: —

How to check. [09] has only [01] on the right, [10] only [01][02], [11] only [02][04], [12] only [09], [13] only [07][08][10], [14] only [05][06][12], [15] only [03][11], and [16] only [02][15]. In every row the numbers on the right are smaller than the one on the left, so there is no circle.

What to read from this graph. (1) The road to the core row [16] passes through [15], and [15] depends on the assumption row [03] (the one remaining proposition of route 5). (2) The computer-assisted [09][10] enter only the product-form value [14] through [12][13], and [14] does not satisfy [03] (it lies outside the needed value). That is, in the current graph the two computer-assisted rows are not on the road to [16]. When an upper bound not of product form satisfies [03], [09][10] become part of the grounds of [03] if the ratio $q$ enters that bound. (3) None of the reading rows [06][07][08] is used in the proof of an upper bound.

We also check that the statements of the assumption rows do not contain the quantities of the conclusion. The quantities of the conclusion are the gap of the measure of the core ([15]) and the constant of row (T0′_P) ([16]). [03] is the logarithmic Sobolev constant of the marginal of the marks on the site window, [04] constants of the inputs of the sectors, [05] the construction of the product-form upper bound, and [02] limits and regularity. None of them contains the gap of the core itself.

This part places Assumption H at the axis of the study and treats (H2) as one of the pillars of its first step. Other views are treated in Part 11.

What this chapter can say

Sources and reproduction

Words used in this chapter — route, core, site window, window (of $\hbar$), chain outside the window, Stein-kernel ratio, ring, seam, lump, outer chain, inner chain, boundary stage, corner at the wall, computer-assisted, floor of the product form. Their meanings are in A-02 Glossary.

Next chapter to read: 10-08 What computer-assisted means — the conditions on the chain inside the window — explains the mechanism and premises by which interval arithmetic turns floating-point computations into proofs of inequalities, using the contraction condition of the chain outside the window as an example.

← 10-06 (H1) Local Poincaré within a phaseContents10-08 What computer-assisted means — the conditions on the chain inside the window →