2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasereconciliation, distance to known theorems, bibliography · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

Reconciliation — the numbers of the record and the known numbers

What this chapter shows — For each of the four conditions of the first step, how far the nearest known theorem reaches and where it stops. A summary of the results of checking the statements of the known theorems cited in the first step against original sources and secondary sources, and the types of discrepancy that had occurred before the check. We lay out, row by row, the known numbers (closed formulae) used as positive controls against the numbers of the record, and divide them into agreement, conditional agreement, disagreement and not comparable. The nineteen-row cross-check on the side of the four-dimensional lattice is kept as it is.

Prerequisite chapters — 10-05 §05 (where known theorems reach), 10-12 (the integrated table), 10-13 §07 (translating conventions).

Said first — The level is [Y3]. Numbers agreeing shows neither that the assumptions are true nor that the mass gap exists. Many of the numbers that agreed are checks that recompute known values by another road. This part places Assumption H at the axis of the study, and this chapter holds the numbers and tools used in its construction up against outside numbers and tools.

The order of this chapter

The conventions of reconciliation — four verdicts, and three things to align
Distance to known theorems — for each condition, the nearest known result and what differs (Table 1)
Summary of the bibliographic checks — the support of the main citations, and the types of discrepancy (Table 2)
The numbers of the record and the known numbers — cross-check against the closed formulae used as positive controls (Table 3)
Rows that cannot be compared — numbers with no counterpart to check against
The nineteen rows on the side of the four-dimensional lattice — one table of cross-checks (Table 4)
A way of reading dependency graphs learned from outside documents — two types

The conventions of reconciliation — four verdicts, and three things to align

There are four verdicts.

Agreement: the same quantity compared under the same conditions agrees within the error or the digits shown.
Conditional agreement: it agrees, but how it agrees depends on one choice (the lattice spacing, the box, the convention, the form of extrapolation).
Disagreement: compared as the same quantity, it differs beyond the error. Even when the reason is known, we write disagreement.
Not comparable: the quantities or conditions differ, and no counterpart to check against is found.

Before comparing, we align three things: the convention (the same $C_P$ changes value with the normalisation of the Dirichlet form, Table 4 of 10-13), the conditions (the lattice spacing $\varepsilon$ , the boundary, whether inside or outside a model), and the kind of number (closed formula, computer-assisted upper bound, floating-point reading, lower bound of a family). Especially in the first step, the counterpart is often "a model whose answer is known by a closed formula"; agreement there is a check that the tool works correctly (a positive control), not a check of the claim on the double well.

Distance to known theorems — condition by condition

Table 1 of 10-05 listed where known theorems reach for Proposition 4.5 as a whole. Table 1 divides it by condition, and writes in one line the nearest known result for each condition and what differs.

Table 1 The nearest known result for each condition (the grade is the grade of the row citing the known theorem; statements checked in the text)
Condition	Nearest known result	Where it reaches	What differs	Grade
(H1) Local Poincaré within the phase	Poincaré version of Helffer's criterion (Ledoux Proposition 3.1), Bakry–Émery on convex domains (Kolesnikov–Milman Theorem 1.2)	The two-scale construction of the chain on the coarse side and the bridge on the fine side, the convex core at the bottom of the valley	The inputs to the criterion (the margin, the gap of two sites) are values at each node of $\hbar$ , and it gives no input uniform in $\hbar$	Cited
(H2) Transport along the family of saddles	Poincaré version of Eyring–Kramers (Menz–Schlichting Lemma 2.4, Theorems 2.9 and 2.12)	Asymptotics and prefactor of the gap between valleys of a Morse function	Fixes the dimension: the constants are not uniform in dimension, and a degenerate family of saddles such as the translations of a kink is outside the assumptions	Cited
(H2) Same as above	Eyring–Kramers for the stochastic Allen–Cahn equation on a fixed interval (Di Gesù–Le Peutrec, Brooks–Di Gesù)	Asymptotics of the gap with the length of the interval fixed short of the bifurcation threshold	Excludes the family: the regime in which the interval becomes infinitely long, and the family of translations, are stated by the authors themselves to be outside the scope	Cited
(H3) Reduction to the box	The box dynamics of Glauber dynamics (Propositions 3.4 and 3.5 of Martinelli's lectures)	The type that bounds the whole gap by the box dynamics and the gap inside the box	Discrete spins, in the form of total-variation coupling. The rewriting into one dimension, $L^2$ and the form of maximal correlation is on this book's side (10-11 §02)	Cited (type)
(H4) The gap inside the box	The closed formula for the Glauber dynamics of the one-dimensional Ising model, diffusion of a single interface (Bertini–Brassesco–Buttà Theorem 2.1)	The formula for the gap of a box with fixed ends, the diffusion coefficient $D$ of the position of a single kink	The former is for discrete spins, and that the reduced kink gas falls in that type is assumption (KG). The latter is a statement at the level of the process, not a statement on the box gap with many kinks	Assumption (the original source of the closed formula has not been checked), Cited
The whole (uniform in volume)	A criterion uniform in volume (Bauerschmidt–Dagallier Theorem 2.1)	Applies directly to the lattice of paths, an upper bound uniform in length	Uniform in volume but doubly exponential: the value is $C_P\le e^{c\xi}$ ( $c\approx1.4\to1.8$ , read at $\varepsilon=0.25$ , $\hbar=0.4\to0.1$ ), and since $\xi$ grows exponentially in $\hbar$ , it is doubly exponential in $\hbar$ . It does not reach the power form of H	Cited, Reading (values)
The whole (an analogous case of a theorem of the same type)	The gap of the dynamics of hierarchical $\|\varphi\|^4_4$ (Bauerschmidt–Bodineau)	A theorem of the same type as the power form of H: the gap vanishes with the same power as the free field (with a logarithmic correction)	Hierarchical: a model of nested boxes, neither the lattice model itself nor the path measure of the double well. Cited as evidence that a question of this type stands as mathematics (10-01 §06)	Cited

From Table 1, the known tools lie in four places, "asymptotics between valleys with the dimension fixed", "a single kink", "an upper bound uniform in volume but coarse" and "a theorem of the same type in another model", and within the range searched we found no theorem that gives a sharp upper bound on a long path up to the diffusion time of the kink gas, $\xi^2/D$ . No row can replace any one of the four conditions wholesale by a known theorem; they take the form of citing a step within a condition (a criterion, a type, a coefficient).

Summary of the bibliographic checks — support, and the types of discrepancy

We picked up every row in the assumption tables of the four conditions of the first step marked "cites a known theorem", and checked it against the original source or a secondary source restating the statement. 55 items were picked up, of which 26 are main items entering the columns of the condition tables directly. Table 2 gives where the 26 main items went.

Table 2 Where the 26 main citations went (level of checking. "Changed into assumptions" means renamed as an assumption row rather than a citation row)
Destination	Items	Content
Statement checked in the text of the original source	4	Poincaré version of Helffer's criterion (Ledoux), Bakry–Émery on convex domains and Brascamp–Lieb (by a single theorem of Kolesnikov–Milman), Sturm's oscillation theorem
Statement checked in a restatement in a secondary source	8	Muckenhoupt's bracket (in the form of Barthe–Roberto), correlation inequality for MTP₂ (Karlin–Rinott), the type of Lyapunov and local Poincaré, the Poincaré constant of the exponential distribution, IMS localisation, the bathtub principle, the maximal correlation of $n$ steps of a reversible chain, maximal correlation of Gaussians
Proof given in this book (elementary)	7	Decomposition of the variance and the steps around it, the direction of min-max, the ground-state transformation (integration by parts), the lemma of two projections, the Efron–Stein type
Original source needs to be obtained	4	The Poincaré–Miranda theorem, regularity of the weighted Neumann operator on a box, stochastic monotonicity (Herbst–Pitt), the principal eigenvalue of a positive integral kernel on $\mathbb R$
Changed into assumptions	3	Cross term (X), the type of the kink gas (KG), Eyring–Kramers for minimisers on a wall (ΔS_core)
Total	26	19 with support (the top three rows)

The 4 items that need to be obtained are counted in the integrated table as "Assumption" rows ([010], [024], part of P0 and others in 10-12 Table 1). The rule of this part is that citations that cannot be checked in the text of the original source do not go into "Cited" (10-00 §01).

The type of discrepancy — one word of the setting. Between the record before checking and the original sources there were 15 discrepancies. 12 were reattributions that do not change the mathematics (citing the version for convex domains through another theorem, correcting attributions, correcting pages), and 3 needed rewriting of assumption rows. The 3 rewrites were (a) a row that had put regularity of eigenfunctions on a box up to the corners as "standard" (for a box with corners and an asymmetric weight it does not follow from the standard results, so it was split off as an assumption row), (b) a row in which the theorem number of the continuous version of monotonicity in the end values was unconfirmed (left in the row as unconfirmed), and (c) a row that used a logarithmic Sobolev lemma in uncentred form (the restatement of the original source is in centred form).

The discrepancies all arose from one word of the setting — whether there is a boundary, whether it is finite, whether it is centred. And they arose not in rows where the record wrote the source as "memory" but in rows where it wrote "standard". Rows written as "standard" have a weak motive to be checked. This is why the five grades of this part have no grade "standard". Every cited theorem is always assigned to one of "statement checked in an original or secondary source (Cited)", "proof given in this book (Proved here)" and "not checked (Assumption)".

The numbers of the record and the known numbers — closed formulae used as positive controls

The tools of the first step (criteria, the machinery of interval arithmetic, reduction to the box) were first applied to models whose answers are known by closed formulae, and applied to the double well only after confirming that they work correctly. Table 3 is the list of those cross-checks; the ratio is "record side ÷ known side".

Table 3 Positive controls and calibration of the first step (grades on the record side by label; row numbers continue after Table 4 of §06)
#	Quantity	Record side	Known side	Ratio	Verdict	Conditions, notes
20	The best constant from applying the criterion of Lyapunov and local Poincaré (of BBCG type) to the one-dimensional Gaussian	Reading 5.828427 by scanning weights ( $u=0.41422$ )	Closed formula $3+2\sqrt2$ ( $u=\sqrt2-1$ )	1.000000	Agreement	6 digits. 8 digits in the dual form too. Lean`gauss1d_bbcg`, `gauss1d_equality` (no assumptions)
21	The true Poincaré supremum of the same model	Reading 4.4979 (400 finite elements), 4.4994 (800)	The value of the criterion $3+2\sqrt2$	0.77	Conditional agreement	The loss of the criterion is 1.30 times, and the direction (criterion ≥ true value) is right
22	$C_P$ of the pure Gaussian $V=x^2/2$ and the right-hand side of Proposition H3	Reading right-hand side 1.10–1.33 ( $L=4$ –128)	$C_P=1$ (equality in Bakry–Émery)	1.10–1.33	Agreement	A check of the direction of the inequality. Approaches 1 as the proportion of overlap → 0 Lean`gauss_box_le_one`
23	Both sides of Proposition H3 for heat-bath Glauber of the one-dimensional Ising model	Reading holds in all 9 cases, loss 1.17–1.35 times	The gap by diagonalisation over all states	—	Agreement	Rings of 12 and 16 sites, couplings 0.3, 0.6, 1.0. A check of the direction of the inequality
24	The gap of the Glauber box with fixed ends	Reading diagonalisation over all states of small chains	Closed formula $1-\gamma\cos(\pi/(L+1))$	1.00000	Agreement	The 6 digits shown. The same for boundary values $(+,+)$ and $(+,-)$ . The original source is bibliographic only
25	Alternating maximal correlation ( $C=2$ , $\theta=\tfrac14$ )	Proved here $\varphi(e^{-1/2},e^{-1})$ of Proposition G	The closed formula of the two-state telegraph chain, 0.7308	1	Agreement	Two expressions of the same quantity (10-11 §02)
26	The action of the kink on the lattice	Reading $S_0^{\rm lat}=0.665266$ ( $\varepsilon=\tfrac12$ )	The continuum $S_0=2/3$	0.998	Conditional agreement	The difference of 0.2% is due to the lattice spacing. There is no estimate of the difference between continuum and lattice
27	The logarithm of the correlation length $\ln\xi$	Reading two routes: the difference of eigenvalues, and the transfer-operator version of Herring's formula	The semiclassical leading term $\xi\propto\hbar^{1/2}e^{S_0/\hbar}$	—	Conditional agreement	The two routes agree to 6 digits at intermediate $\hbar$ . $\ln\xi-S_0^{\rm lat}/\hbar-\tfrac12\ln\hbar$ settles towards a constant, $-0.774／-0.777／-0.778$ , which agrees with the form of the leading term. There is no counterpart for comparing the constant term
28	The diffusion coefficient of a single kink	Reading $D/(\hbar/S_0)=1.00$ – $1.14$	$D=\hbar/S_0$ (the coefficient of the theorem at the level of the process)	1.00–1.14	Conditional agreement	Point values at $\varepsilon=0.25$ , $\hbar=0.02,0.07,0.1$ . Corrections for finite $\hbar$ and the spacing
29	The machinery for fitting the criterion uniform in volume	Reading 1.0049 for the harmonic model	Closed formula 1	1.005	Agreement	A positive control for the transfer-matrix computation applying the criterion to the lattice of paths ( $\varepsilon=0.25$ )
30	The machinery of interval arithmetic: the pure Gaussian chain	Computer-assisted every point is inside the enclosure with $\hbar$ kept as an interval	Variance per step $(k-1)/k$ , mean $(k-1)s/k$	1	Agreement	With the interval width set to 0, the same as the point machinery to the order of $10^{-12}$ (10-08 §07)
31	The machinery of interval arithmetic: another implementation and point values	Computer-assisted upper bound	Reading 75 floating-point recursions, 26 point values of the transfer matrix	—	Agreement	0 outliers. All point values below the upper bound (smallest difference $2.85\times10^{-4}$ )
32	The gap of the hierarchical Gaussian free field	—	$L^{-2N}$ at depth $N$ of the nested boxes (known)		Not comparable	A known number put as a counterpart for comparison. The number on the side of the free field of the Langevin dynamics of links has not yet been computed

The rows of agreement in Table 3 are all checks that "the tool worked correctly on a model whose answer is known"; they do not show that the rows on the double well are correct. The conditions of the rows of conditional agreement are the lattice spacing (rows 26 and 28), finite $\hbar$ (row 28), and the missing constant term of the counterpart (row 27). Row 21 measures, on a model whose answer is known, the loss by which the criterion comes out 1.30 times larger than the true value; it serves as a rough guide when comparing the upper bound of the criterion with the needed value on the double well (10-09 §05).

Rows that cannot be compared — numbers with no counterpart to check against

We list the numbers of the first step for which no counterpart to check against is found. All of them are numbers at the places where the construction comes closest to the upper half of Assumption H, and only there is there no counterpart.

Conversion between the two lattice spacings. The readings of the diffusion coefficient of kinks and of the lower bound of the ratio $R$ are at spacing $\varepsilon=0.25$ , while the table of correlation lengths and the steps of Proposition 4.5 are at $\varepsilon=\tfrac12$ , and the conversion between the two spacings is not written (the table of ranges in 10-04).
The coefficient of $1/\hbar$ of the logarithmic Sobolev constant of the core. A quantity needed in the Feynman–Kac strand; the Monte Carlo reading and the pathwise upper bound are three orders of magnitude apart (10-09 §05).
The reading of $\sup|\nabla\psi|^2$ . A reading that it is within 0.54 times the threshold; it is not an upper bound. No known value of the same quantity was found in the range searched.
The lifting assumption of Proposition P5. An assumption that lifts the measured values of the two-site lump linearly beyond the measured range; the counterpart for comparison is the lifted value itself (10-09 §05).
Comparison with the hierarchical free field. Row 32 of Table 3. The known number $L^{-2N}$ exists, and the number on the record side does not yet.

The nineteen rows on the side of the four-dimensional lattice

We keep one table that holds the numbers computed by the record on the side of the four-dimensional lattice (the one-loop slope $\kappa$ , the shortfall $\delta(b)$ , the ratio $m/\sqrt\sigma$ , upper bounds of mass, $C_P$ and autocorrelation times, strong-coupling thresholds, the exponent of stochastic time) up against existing computations and the numbers of primary lattice sources. The ratio is "record side ÷ existing side", and it is left blank for rows that cannot be compared. The details of each row and the conversion of conventions are in the chapter on the side of the four-dimensional lattice (10-02) and the source tables of Part 5.

Table 4 Cross-checks on the side of the four-dimensional lattice (rows 1–19)
#	Quantity	Record side	Existing side	Ratio	Verdict	Conditions, notes
1	κ (window model)	0.371521	0.371511	1.00003	Agreement	Within the one-loop model
2	Two-loop exponent	51/121	b₁/(2b₀²)	1	Agreement	Arithmetic of rational numbers
3	Lattice slope / one loop	0.304(2)	0.3715	0.819	Disagreement	β≤2.60, before asymptotic scaling
4	Lattice slope / two loops	0.304(2)	0.3972	0.766	Disagreement	Two loops widen the gap
5	c₂ of δ(b)	0.0197, 0.4200	—		Not comparable	Different variables from Symanzik and MCRG
6	m/√σ (p=2) / LT	3.844(43)	3.844(61)	1.000	Conditional agreement	The same data table; choosing p=2
7	m/√σ (p=1) / LT	4.053(73)	3.844(61)	1.054	Conditional agreement	χ² cannot choose p
8	m/√σ (p=2) / AT	3.844(43)	3.781(23)	1.017	Conditional agreement	1.3 times the combined error
9	Upper bound β=2.40 (APE)	1.025(140)	0.953(19)	1.076	Agreement	The same spatial side 12 (L√σ≈3.2)
10	Upper bound β=2.55 (APE)	0.531(50)	0.586(10)	0.906	Conditional agreement	−1.1σ. 12³×24 (L√σ≈1.9) against 20⁴
11	Upper bound β=2.60 (APE)	0.558(51)	0.514(8)	1.086	Conditional agreement	12³×24 (L√σ≈1.6) against 24⁴
12	Upper bound β=2.60 (flow)	0.501(71)	0.514(8)	0.975	Conditional agreement	12⁴ (L√σ≈1.6) against 24⁴
13	Ratio m(2.40)/m(2.60)	1.837(302)	1.854	0.991	Conditional agreement	β=2.60 is a small box with L√σ≈1.6
14	Straight-line threshold / SZZ	1/12	1/48 (converted by β=4β_SZZ)	1.000	Agreement	Rederivation of a known result
15	Threshold after one-link integration	√(2/27)=0.2722	—		Not comparable	Threshold of the marginal measure. The constant moves by 3.27 times within the same criterion. Not a claim of a widened range
16	C_P of the free field	C_P^(A)=ξ², 4C_P(time 0)²	Gaussian measure, ground-state transformation	1	Agreement	Recomputation
17	τ_int/ξ² of Langevin	0.5–1.4	—		Not comparable	No primary source for the same quantity at zero temperature towards the continuum limit was found in the range searched
18	The pole in stochastic time	13/6, −11/3	Baulieu–Zwanziger	1	Agreement	Only one independent primary source
19	Exponent s	−9/22	g₀^{9/11} (Lüscher–Schaefer equation (1.1))	1	Conditional agreement	Conditional on the one-loop leading logarithm and H. Not an independent check

On the side of the four-dimensional lattice there are 6 agreements, 8 conditional agreements, 2 disagreements and 3 not comparable (rows 3 and 4 are two readings of the same disagreement). On the side of the first step (rows 20–32 of Table 3) there are 8 agreements, 4 conditional agreements and 1 not comparable. Read together, the rows of agreement are recomputations of known values or checks of tools on models whose answers are known, and the rows with no counterpart gather, on both sides, at the numbers that come closest to Assumption H ( $\tau_{\rm int}/\xi^2$ on the side of the four-dimensional lattice; the coefficient of the core and the lifting assumption on the side of the first step).

A way of reading dependency graphs learned from outside documents

We read two outside documents that claim to solve the existence and mass gap of Yang–Mills, and drew from them, not their correctness, but two types useful as ways of reading the construction of a claim. We do not give the sources; we write only the types. This book does not judge whether those documents are right or wrong.

The type that puts the conclusion as a premise of intermediate steps. One document piled up steps in the form of theorems and proofs; intermediate steps (clustering, uniqueness of the vacuum, exponential decay) took as a premise the positivity of the mass gap, "to be shown later", and the positivity of the mass gap itself was supported by physical relations and dimensional analysis. Read step by step, each step derives its conclusion from its premises, but when the steps are arranged as numbered nodes, an edge goes back from the final conclusion to the intermediate steps, and the rule of numbering is broken. It is a type that is invisible to step-by-step checking, and visible once arranged in a dependency graph.
The type with an object for its name and numbers for its content. The other document closed its main theorem in Lean. Tracing the conjuncts of the main theorem back to their definitions, "lies in the spectrum" was defined as the same proposition as the conclusion, and the value of the gap was a rational number put literally; operators, fields and measures appeared only as names. It is a type that is invisible even when machine checking passes and the list of axioms is clean, and visible once the definitions are traced back (10-13 §05).

These two types should be applied in the same way to the construction of this book. In the integrated graph of the first step, we checked the rule of numbering on every edge, and checked by reading that the statements of the assumption rows do not contain the quantities of the conclusion (10-12 §03). For the Lean theorems, that there is no name on the side of the conclusion whose definition needs tracing back (only inequalities of real numbers) can be checked by the six questions of 10-13.

This part places Assumption H at the axis of the study, and has held the numbers and tools of its first step up against outside numbers and tools. Other viewpoints are taken up in Part 11.

What this chapter can say

Can say	Cannot say
Cited Where the nearest known result for each condition stops (Table 1). Within the range searched, no known theorem gives a sharp upper bound on a long path up to the diffusion time of the kink gas	That no such theorem exists (outside the range searched)
Of the 26 main citations, 19 are supported by an original source, a secondary source or a proof in this book, 4 need to be obtained, and 3 were changed into assumption rows (Table 2). The type of discrepancy is one word of the setting	The statements of the 4 items that need to be obtained
The cross-check of positive controls (rows 20–32 of Table 3): that the tools worked correctly on models whose answers are known	That this shows the rows on the double well are correct
The nineteen-row cross-check on the side of the four-dimensional lattice (Table 4)	Explaining the disagreement between the slope of the lattice data tables and the one-loop value beyond the reason that it is before asymptotic scaling
The two types drawn from outside documents, and the check of applying them to the construction of this book	Whether the outside documents are right or wrong

Sources and reproduction

Item	Kind	Source, tool
Statements of known theorems (Table 1)	Cited	G. Menz, A. Schlichting, Ann. Probab. 42 (2014), arXiv:1202.1510. G. Di Gesù, D. Le Peutrec, arXiv:1506.04434; M. Brooks, G. Di Gesù, arXiv:2606.16083. F. Martinelli, LNM 1717 (1999) 93–191. L. Bertini, S. Brassesco, P. Buttà, Arch. Ration. Mech. Anal. 190 (2008) 307–345. R. Bauerschmidt, B. Dagallier, Comm. Pure Appl. Math. 77, arXiv:2202.02295. R. Bauerschmidt, T. Bodineau, arXiv:1809.02075. M. Ledoux, LNM 1755 (2001) 167–194. A. V. Kolesnikov, E. Milman, J. Geom. Anal. 27 (2017), arXiv:1310.2526. The same as the source table of each chapter
Closed formula for Glauber dynamics	Assumption	R. J. Glauber, J. Math. Phys. 4 (1963) 294 (bibliographic only). The formula was checked independently by diagonalisation over all states of small chains
Summary of the bibliographic checks (Table 2)	This book	A summary of the table checking the 55 items cited in the assumption tables of the first step against original sources, secondary sources and bibliographic queries. The bibliography of each item is in the source table of each chapter
Numbers on the record side of Table 3	Reading, Computer-assisted, Proved here	Floating-point computation (transfer matrices, finite elements, diagonalisation over all states), interval arithmetic, closed formulae in the text. In the source table of each chapter
The nineteen rows of Table 4	This book	One table of cross-checks on the side of the four-dimensional lattice (the bibliography of the original materials is in the source tables of Part 5 and 10-02)
The two types	This book	A reading of two outside documents that claim a construction or a proof (sources withheld)

Terms used in this chapter — reconciliation, agreement, conditional agreement, disagreement, not comparable, positive control, calibration, distance to known theorems, one word of the setting, the type that puts the conclusion as a premise of intermediate steps, the type with an object for its name and numbers for its content. Their meanings are in the A-02 glossary.

Next chapter: 10-15 What remains and its cost — the distance to a theorem — lays out what remains and its cost for each assumption row of the integrated table, and separates the distance within the first step from the distance from the second step on.

Revision 2026-10-09: first edition.