2026-10-09 · chapter Yang–Mills and the mass gapPart 10 Towards a proof of the mass gap — building the first step of the staircasenot shown, not read strongly, not called false · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

What is not claimed — not shown, not read strongly, not called false

What this chapter shows — The list of things Part 10 does not claim (fourteen items), and a table matching each item to the sentence in the body, and the chapter, that supports it. The difference between not claiming something and claiming its opposite. For the first-step proposition, the Lean theorems, the computer-assisted results and the readings, examples of how each is misread once its conditions are removed.

Prerequisite chapters — None. This chapter can be read from anywhere. The terms are defined in 10-01 §05 (Assumption H, H↑, H↓), 10-00 §01 (the five labels) and 10-05 (Proposition 4.5 and conditions (H1)–(H4)).

Said first — The level is [Y2]. This book does not solve the problem of Yang–Mills existence and the mass gap. It contains no construction of the continuum limit and no proof of the mass gap. This chapter only splits that fact into items and writes them out; it adds no new result. This part places Assumption H at the axis of the study, and it does not claim that H is the only road or the right road.

The order of this chapter

Three meanings — not shown / not read strongly / not called false (Table 1) / Detour: negating a quantifier
The fourteen items and the sentences in the body that support them — a table of correspondence (Table 2)
On the main problem — the claim of the problem statement, footnote 2
On the first step — Proposition 4.5 is algebra on top of four conditions (Table 3)
Not reaching is not evidence of falsity — the death of detour plans, the reading for the periodic box, H↓ and the free field (Table 4)
Conditions fall away in summaries — sentences of the body and one-line summaries (Table 5)
Analogies and other documents — what is not judged, and the convention for putting things out

Three meanings — not shown, not read strongly, not called false

When a mathematical text says "not claimed", there are three meanings. The first is not shown. On the main problem, this part shows nothing. The second is not read strongly. Whatever has been shown always carries conditions or assumptions, and reading it with the conditions removed gives a sentence that has not been shown. The third is not called false. That a method did not reach something is a property of the method, not a claim that the proposition it did not reach is false.

The third becomes clear when written in logical form. Writing " $P$ is claimed" as $\vdash P$ , we have

\neg(\vdash P)\quad\text{and}\quad \vdash\neg P\quad\text{are different sentences} \tag{1}

On the left: "we do not have a proof of $P$ ". On the right: "we have a proof of the negation of $P$ ". The "not claimed" of this chapter always has the meaning on the left. As a metaphor: a road not drawn on a map is not evidence that there is no road there.

Table 1 How the items of this chapter are divided (numbers are the items of Table 2)
Kind	What is not claimed	Item
Not shown	Existence and the mass gap on $\mathbb R^4$ , the construction of the continuum limit, the power form of H for the first step	1, 2, 3, 10
Not read strongly	Computer-assisted results, readings, Lean theorems, lifting, the relation between H and the mass gap	4, 5, 6, 7, 11
Not called false	Propositions a tool did not reach, H↑ as read to fail in the periodic box	8, 12, 13
Not judged, not counted	Rederivations of known results, the correctness of other documents, putting unclosed propositions out	9, 14

Detour: negating a quantifier

Detour

A subsection you may skip. The negation of "for every $f$ , $Q(f)\le C$ " is "for some $f$ , $Q(f)\gt C$ ". In symbols, $\neg\forall f\,A(f)\iff\exists f\,\neg A(f)$ . To show H↑ false, a single sequence of counterexamples $f$ suffices; to show H↓ false, one must show "for every $f$ the ratio is $o(\xi^2)$ ". The latter is a sentence of the same "for every" type as H↑, and it does not come out of listing test functions. That some families did not reach it is a "for every" restricted to a family ( $\forall f\in\mathcal F$ ), not a "for every" with the family removed.

The fourteen items and the sentences in the body that support them

Table 2 writes, one row at a time, what this part does not claim, and lists the places in the body that support each row. "Support" means that the sentence in that chapter already states the conditions, assumptions and scope, and the row merely copies it onto one line.

Table 2 What is not claimed, and the sentences in the body that support it
Item	What is not claimed	Kind	Supporting sentence in the body (gist)
1	The solution of the problem of Yang–Mills existence and the mass gap. Neither a construction of the continuum limit nor a proof of the mass gap in the limit is here	Not shown	10-01 §08: in four dimensions, what we have are only the theorems (S2)(S7) and (S1) for small $\beta$ . Even with H and the bridge, (S1)(S3)(S6)(S7)(S8) do not follow
2	Even for the first step (the double well in 0+1 dimensions), Assumption H (the power form of H) has not been shown. Proposition 4.5 is algebra on top of four conditions, and each of the four conditions leaves assumption rows	Not shown	10-05 §08: where the four conditions stand. 10-12 §07: the assumption rows needed for the conclusion number 15 or 20
3	There is not yet a single $\hbar$ at which all rows of the four conditions hold at once	Not shown	10-12 §06: the nodes of (H1) and the range of (H2) do not intersect. 10-15 §05
4	Computer-assisted results are inequalities inside the windows and nodes that were computed, and say nothing about outside the window or about $\hbar\to0$	Not read strongly	10-08 §10: what computer assistance does not say (outside the window, $\hbar\to0$ , the difference between continuum and lattice)
5	Readings (floating point, Monte Carlo, fits, lower bounds over families) are used neither as upper bounds nor as proofs of lower bounds	Not read strongly	10-00 Table 1: what the label "Reading" does not guarantee. 10-12 Table 1: all 19 Reading rows keep the grade "Reading"
6	A Lean theorem checks the algebra from assumptions to conclusion; it does not check that the assumptions are true, nor the reading of real numbers as physical quantities	Not read strongly	10-13 §05: four limits (truth of the assumptions, the encoding of the objects, putting the conclusion into a definition, circles of dependence)
7	From the first step to the second step on, we only state the prospect that the form of the sentences and the form of the tools lift; there is no theorem of implication	Not read strongly	10-03 §06: what lifts and what does not (constants and the range of $\hbar$ do not lift)
8	That a tool did not reach something is a property of that tool, not a claim that the proposition it did not reach is false	Not called false	the table of scope in 10-10: the "death" of a detour plan is a property of the tool. 10-02 §08: where the roads on the four-dimensional lattice stopped
9	Rederivations of known theorems are not counted as results. The correctness of other documents that claim a construction is not judged	Not judged	10-14 §03: the support of the cited theorems. 10-14 §07: from outside documents we take only patterns, give the source, and do not judge
10	The record on the side of the four-dimensional lattice (Assumption H, the three chains, the sum of shortfalls) also remains assumption	Not shown	10-01 §05: the three chains are all assumptions. 10-02 §07: what remained is algebra and readings
11	Under the assumption of the one-sided bridge, H↑ implies a lower bound on the gap of the lattice in physical units, but the converse is false in general and undecided for the class of Wilson measures. H is not the mass gap itself	Not read strongly	(4) of 10-01 §06 (the one-sided bridge) and (5) (Proposition C: on the circle, $\Delta\gt0$ and $C_P=\infty$ )
12	The "death" of a detour plan means that the ceiling of that construction does not exceed the needed value, not that the one remaining proposition is false	Not called false	10-10 §03: the line that killed it is a line of a tool's inequality
13	The reading that H↑ fails on the four-dimensional lattice in the periodic box is a conditional reading, resting on numbers from the literature and on the conjecture that "HMC and Langevin are in the same universality class"; it is not a theorem that H↑ is false	Not called false	the note in 10-01 §05 (topological sectors and the order of volume), 10-02 §05
14	A proposition with assumption rows left is not put out as a result. Only what has had all its conditions closed is put out	Not counted	10-12 §08: the integrated table is a map, not a fraction of progress

Items 1–10 are the skeleton of the part, and 11–14 were added from the text of the individual chapters. Every row compresses the conditions of the body onto one line, and is written so that reading the body gives the conditions in full.

On the main problem — nothing is said about the claim of the problem statement

What the problem statement (0-03) asks is that, for every compact simple gauge group, a quantum Yang–Mills theory on $\mathbb R^4$ exists, satisfies the axioms (4-09, 4-10), and has a mass gap.

\text{existence (satisfying the axioms)}\quad\wedge\quad \operatorname{spec}(H)\subset\{0\}\cup[\Delta,\infty),\ \ \Delta\gt0 \tag{2}

This part shows nothing about either part of (2) (item 1). Footnote 2 of the problem statement excludes from solutions limits obtained only by weak existence (compactness) Cited. That subsequential limits come for free from compactness is a matter at that level, and neither uniqueness nor the mass gap follows from it.

Assumption H, which this part treats, is a sentence that restates $\Delta$ of (2) (the gap in Euclidean time) in terms of the relaxation time $C_P$ of Langevin dynamics. The two are different quantities, and only when the one-sided bridge holds can one cross, in one direction, from H↑ to a lower bound on $\Delta$ (item 11, 10-01 §06). We placed H at the axis of the study because it is language that can be computed on the lattice, has the tools of convexity and Poincaré inequalities, and is within reach of machine checking; we do not claim that this restatement is the only road to solving the problem. Other views are placed in Part 11.

On the first step — Proposition 4.5 is algebra on top of four conditions

The central sentence of the first step is Proposition 4.5: "on conditions (H1)–(H4), $C_P\le K\xi^2S_0/\hbar$ , and hence the power form of H" (10-05 §04). Reading this sentence as "the power form of H has been shown for the double well" is too strong (item 2). Beneath the four conditions there are 15 or 20 assumption rows of the integrated table (depending on which of the two roads) and 3 reading rows (10-15 Table 1).

The algebra of the construction is checked in Lean, but when only the theorem name remains in a summary, the conditions fall away. Table 3 lists the theorem names of this part that are easily read strongly once their assumptions are dropped (item 6).

Table 3 Theorem names, assumptions, and what is not to be read strongly
Theorem name	Assumptions (gist)	Not to be read strongly
`H3_H4_compose`	The box product inequality, the box gap inequality, $D=\hbar/S_0$ , $n,\hbar,S_0\gt0$ , $c_{\rm alt}\lt1$	Not "Proposition 4.5 has been shown". Conditions (H1)(H2) are inside the assumption of the box gap
`power_H_of_kink_ratio`	$0\lt K,s,\delta$ , $1\le R\le K/\hbar$ , $s/\hbar\le\ln\xi$	Not "the power form of H has been shown". $R\le K/\hbar$ is the conclusion of Proposition 4.5 itself
`H1_assembly`	Seven named ones (decomposition, fine side, Helffer's criterion, margin, $0\lt\lambda$ , (T4), the rate condition)	A proposition at six nodes, not (H1) uniform in $\hbar$
`cloc_term_vanishes`	(ΔS_core), the reading of $\xi$ , $\Delta S_*\lt2S_0$	The local term vanishes when (ΔS_core) is assumed. The value of the action barrier is a reading at $\hbar=0$
`path5_conditional`	The remainder inequality, the proxy ratio, rarity, the arithmetic margin (named)	Proposition P5 at the lower end of the window; there is no edge of implication to the one remaining proposition
`win_qfree_window_le_of_interval_table`	Table of interval-arithmetic output	Inequalities inside the window (item 4). Not on the road to the conclusion in the integrated table
`hup_is_gap_positivity_of_lower`	$0\lt m_{\rm lat}$ , $\kappa/m_{\rm lat}^2\le C_P$ , $C_P\le K\xi^2$	The one-sided bridge enters as an assumption. That the bridge holds for the Wilson measure has not been shown (item 11)
`drainExp_eq_zero_iff`	$0\le\eta\le2$ , $\theta\ge0$	Algebra of exponents on the side of the four-dimensional lattice. Applying it to physical quantities rests on assumptions (item 10)

Example. power_H_of_kink_ratio is the algebra "if the ratio $R$ lies between $1$ and $K/\hbar$ , and $\ln\xi$ is at least $s/\hbar$ , then $\ln C_P/\ln\xi\to2$ ". Removing the assumption $R\le K/\hbar$ and writing "the power form of H in Lean" gives a sentence that has not been shown. The correct summary is "the power form of H follows by algebra from the conclusion of Proposition 4.5 (Lean)".

Not reaching is not evidence of falsity

The death of detour plans (items 8, 12). Of the detour plans aimed at the one remaining proposition of route 5, 13 were judged "dead" (10-10 §03). Death means that the ceiling of the upper bound obtained by that construction does not exceed the needed value. It does not mean that the one remaining proposition [026] is false, and it says nothing about the possibility of closing it by another construction.

The reading for the periodic box (item 13). On the four-dimensional lattice, it is reported that in a box with periodic boundary the autocorrelation time of the topological charge grows faster than $\xi^2$ , and together with $\tau_{\rm int}\le C_P$ this reads as H↑ failing in the periodic box (the note in 10-01 §05). This reading stands on numbers from the literature Reading and on the conjecture that HMC and Langevin are in the same universality class Assumption. So this part does not claim "H↑ is false on the four-dimensional lattice" either. What it claims is only that how to correct the statement of H remains as a question.

H↓ is not non-triviality. What H↓ avoids is only that the physical mass collapses to infinity; it does not ask whether there is interaction. This can be confirmed by computing H in a theory without interaction. For the free field with lattice mass $m$ , the Poincaré constant of the path measure is the largest eigenvalue of the covariance,

C_P=\frac1{m^2},\qquad m^2=4\sinh^2\frac1{2\xi},\qquad \frac{C_P}{\xi^2}=\frac{1}{4\xi^2\sinh^2(1/2\xi)}=1-\frac1{12\xi^2}+O(\xi^{-4}) \tag{3}

Table 4 $C_P/\xi^2$ for the free field on the lattice Proved here (values of the closed formula (3))
$\xi$	$C_P/\xi^2$	$1-1/(12\xi^2)$
1	0.920674	0.916667
2	0.979425	0.979167
8	0.998699	0.998698
64	0.999980	0.999980

The ratio tends to 1 as $\xi\to\infty$ , so both the upper and the lower side of H hold. H and H↓ are sentences that hold in a theory without interaction, and non-triviality (S8) does not follow from H (10-01 §06).

Conditions fall away in summaries — sentences of the body and one-line summaries

The error of reading a claim strongly occurs, in most cases, at the stage of compressing a sentence of the body onto one line. Table 5 sets the conditional sentences of the body beside one-line versions with the conditions dropped completely. Every sentence in the right column is one this part has not shown.

Table 5 Sentences of the body (conditional) and one-liners with the conditions dropped (every one on the right is wrong, or not shown)
Topic	Sentence of the body	One-liner with the conditions dropped
Chain outside the window	$q_{\rm free}\le0.998854904$ over the whole $\hbar$ window $[3\cdot10^{-4},3\cdot10^{-3}]$ Computer-assisted (assuming IEEE correct rounding, outward widths)	The chain condition of (H2) is closed
Chain outside the ring	$q_{\rm out}\lt1$ over the whole window Computer-assisted (conditional on the pin $\le0$ ; the boundary stage is a reading of 1.004–1.045)	The seam of the ring is closed
Mixing between boxes	$c_{\rm alt}=0.731/0.410/0.138$ ( $C=2/4/8$ , values of the reduced model) Reading	(MIX_alt) holds
(H1)	Margin $\ge0.067$ at six nodes (ends $[0,2.5]^2$ ) Computer-assisted, constant $C\le1.6\times10^{10}$ (a proposition at the nodes)	(H1) has been shown
Proposition 4.5	Algebra on top of conditions (H1)–(H4) (Lean has the algebra of composing boxes)	The power form of H has been shown for the double well
Strand 3	A reading of $\sup\|\nabla\psi\|^2$ from below, $0.433\pm0.007$ (0.54 times the threshold, $\hbar=10^{-3}$ ) Reading; the paper upper bound is about 1000 times the threshold	The assumption of strand 3 falls within the threshold
Periodic box	On numbers from the literature and a conjecture, H↑ reads as false in the periodic box	H↑ is false in four dimensions

The errors in Table 5 are of two types. One drops the scope (window, nodes, model values); the other drops the grade (Reading, Computer-assisted, Assumption). Both are prevented by keeping the words of the condition in the one-line summary.

Analogies and other documents — what is not judged, and the convention for putting things out

Where words are borrowed from the side of physics, they inherit the level of where they were borrowed from. Holography is a conjecture, and a table of correspondences that uses it is a dictionary, not a proved equivalence. The "4" in a complete family existing for $d\le4$ is a combinatorial number of the lattice, a coincidence unrelated to the dimension of physical spacetime (10-02 §02). The correspondence between the first step and the four-dimensional lattice ( $\hbar\to0$ and $\beta\to\infty$ ) is also an analogy, not a theorem of correspondence (10-03 §06).

On other documents that claim a construction, this part does not judge correctness (item 9). What we took from outside documents is only patterns for reading dependency graphs (closing by putting the conclusion into a definition, circles of dependence), and even those were written as patterns with the source withheld (10-13 §05, 10-14 §07).

Finally, the convention for putting things out (item 14). As long as even one assumption row remains, the propositions of this part are not put out as results. What is put out is only what has had all its conditions closed and has passed machine checking and bibliographic checking. Until then, the text stands as a map of which rows are closed and which are not.

What this chapter can say

Said	Not said
That this part shows nothing about (2) of the problem statement. The fourteen items that are not claimed, and the places in the body that support them (Table 2)	Whether (2) is true. The existence of the continuum limit. A gap uniform in the volume
The assumptions of the theorem names in Table 3 (copies of the statements in each chapter)	The truth of the assumption rows
Proved here That $C_P/\xi^2=1-1/(12\xi^2)+\cdots$ and H hold for the free field on the lattice (closed formula)	A theorem comparing $C_P$ and $\xi^2$ for an interacting four-dimensional measure (in neither direction)
The conditions of the reading that H↑ fails in the periodic box (numbers from the literature and a conjecture)	That H↑ is false, or true, on the four-dimensional lattice

Sources and reproduction

Item	Kind	Source / tools
Problem statement, footnote 2	Cited	A. Jaffe, E. Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute (2000) (the source table of 0-03)
Each row of Table 2	This book	The chapters and sections in the right column of the table
Theorem names and assumptions of Table 3	Lean	Statements in 10-13 §03 and the grade tables of each chapter
(3), Table 4	Proved here	Closed formula. The expansion $1/(4\sinh^2(u/2))=1/u^2-1/12+O(u^2)$
Sentences of the body in Table 5	This book	10-07 §03, 10-08 §05, 10-11 §02, 10-06 §04, 10-09 §05, 10-01 §05

Terms used in this chapter — not shown, not read strongly, not called false, negating a quantifier, the one-sided bridge, the death of detour plans, conditions that fall away in summaries. Their meanings are in the A-02 Glossary.

Next chapter to read: 10-17 Insights and small questions — the views that repeatedly proved useful in studying the first step, and questions that can be settled at the size of an undergraduate thesis.

Revised 2026-10-09: first version (inherits the three meanings, the detour on quantifiers and the table of the free field from "What is not claimed" in the former 10-10).