2026-10-15 · chapter Yang–Mills and the mass gapPart 14 Closing — conditional statementsnot shown, not read strongly, not called false · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

Part 0 Entrance — how to read this book
0-01 How to read this book
0-02 One picture
0-03 Reading the problem statement word by word
0-04 Four dimensions in mathematics and four dimensions in physics
Part 1 Mathematical preparation
1-01 Vector spaces and linear maps
1-02 What a group is
1-03 Lie groups and Lie algebras
1-04 SU(2) and SU(3)
1-05 Manifolds and tangent spaces
1-06 Differential forms and the exterior derivative
1-07 Vector bundles and connections
1-08 Probability and measure
1-09 Probability in infinite dimensions
1-10 Hilbert spaces and self-adjoint operators
1-11 Fourier analysis and distributions
1-12 Detour: what happens only in four dimensions
Part 2 Physical preparation
2-01 Lagrangians and the action
2-02 The idea of a field
2-03 Electromagnetism is a gauge theory
2-04 Special relativity and spacetime
2-05 The bones of quantum mechanics
2-06 The harmonic oscillator and creation and annihilation
2-07 The idea of the path integral
2-08 Statistical mechanics and phase transitions
2-09 Detour: the gap and the correlation length in the Ising model
2-10 The relaxation time and the dynamical exponent z
Part 3 Yang–Mills theory (classical)
3-01 The gauge principle
3-02 Non-abelian gauge fields
3-03 The action and the equations
3-04 Gauge theory as geometry
3-05 Instantons and topology
3-06 Detour: Donaldson and exotic R⁴
3-07 Yang–Mills inside the Standard Model
Part 4 Quantisation
4-01 Canonical quantisation and the Hamiltonian
4-02 The path integral and Euclideanisation
4-03 Perturbation theory and Feynman diagrams
4-04 Divergences and renormalisation
4-05 The algebra of divergences
4-06 Asymptotic freedom
4-07 Dimensional transmutation and Λ
4-08 Gauge fixing and Faddeev–Popov
4-09 The axioms of quantum field theory
4-10 The Osterwalder–Schrader axioms and reconstruction
4-11 The definition of the mass gap
4-12 Detour: the perturbation series diverges
Part 5 Lattice gauge theory
5-01 Wilson's lattice
5-02 The strong-coupling expansion
5-03 Reflection positivity and the transfer matrix
5-04 The infinite-volume limit and the cluster expansion
5-05 Weak coupling and the continuum limit
5-06 How U(1) differs from the non-abelian case
5-07 The Monte Carlo method
5-08 Measuring glueballs and the string tension
5-09 What to hold fixed when taking the limit
5-10 Lattice gauge theory with finite groups
5-11 Detour: Bakry–Émery and one-link integration
Part 6 Constructive field theory
6-01 What constructive field theory is
6-02 Solvability in two dimensions and the Yang–Mills measure
6-03 Constructing scalar fields
6-04 The cluster expansion
6-05 The sequence of renormalisation group steps
6-06 Ultraviolet stability of three-dimensional Yang–Mills
6-07 Four dimensions
6-08 Triviality of φ⁴ in four dimensions
6-09 Stochastic quantisation and regularity structures
6-10 Where four dimensions stops
6-11 Obstacles other than divergence
6-12 Detour: "if 1/0 is given a symbol"
Part 7 From the side of physics
7-01 How physics sees it
7-02 Mechanisms of confinement
7-03 The string picture
7-04 Large N
7-05 String theory and extra dimensions
7-06 How extra dimensions become invisible
7-07 Where gauge fields come from
7-08 Holography
7-09 The mass gap becomes geometry
7-10 Back to four dimensions
7-11 Supersymmetry and Seiberg–Witten
7-12 The sentence that plays the role of an equivalence principle
7-13 Where the way physics digs parts from mathematics
Part 8 A dictionary of two languages — the physical and the mathematical view
8-01 How to read the dictionary
8-02 Dictionary A
8-03 Dictionary B
8-04 Dictionary C
8-05 The physical counterparts of our assumptions
The map
Part 9 Where things stand, and open tasks
9-01 How far the world has come
9-02 Where the two ways of digging part
9-03 Candidates for new concepts
9-04 The list of open tasks
9-05 Common misconceptions
9-06 The stages a proof must contain — the table of seven stages
The first road — from the direction of time
Part 10 The first road, first half — from the direction of time: the mass gap and the diffusive-relaxation-time assumption
10-00 Entrance — the staircase, and the random walk, the ruler and the clock
10-01 The line of construction and Assumption ξ²
10-02 Roads tried on the four-dimensional lattice, and walls
10-03 The four steps of the staircase — why we start from 0+1 dimensions
10-04 The path measure of the double well, and kinks
10-05 The statement of the first step — four conditions and Proposition 4.5
10-06 (H1) Local Poincaré within a phase
10-07 The map of (H2) — seven routes and the seam of the ring
10-08 What computer-assisted means — the conditions on the chain inside the window
10-09 The sector framework and the three strands of route 5
10-10 The list of detour plans — types of breakage
10-11 (H3)(H4) and the constant of the core
10-12 The integrated table and the dependency graph
10-13 Lean statements and the limits of machine checking
10-14 Reconciliation — the numbers of the record and the known numbers
10-15 What remains and its cost — the distance to a theorem
10-16 What is not claimed
10-17 Insights and small questions
Part 11 The first road, second half — repairing the form of the assumption, making time continuous
11-00 Further considerations — entrance
11-01 One form that obstructs Assumption ξ² — a nearly conserved global quantity
11-02 Choosing the ruler and the coordinates again — the length at which conservation breaks
11-03 Two ways to repair Assumption ξ² — the sector Assumption ξ² and the local Assumption ξ²
11-04 Which Assumption ξ² the one-sided bridge needs
11-05 Making time continuous — the Hamiltonian lattice and the vacuum measure
11-06 Paying with rarity instead of convexity — renormalisation on groups and log-Sobolev
11-07 Because it is four dimensions — the dimensionless coupling and topology
11-08 Questions, and what is not claimed
11-A1 Table of notation for Part 11
The second road — from the direction of space
Part 12 The second road: from the direction of space — a small twisted T² and confinement
12-00 Entrance — space, not time
12-01 The twisted T² and 't Hooft flux
12-02 Vortices and fractional instantons
12-03 Containers — conditioning on the frame makes things independent
12-04 Slab confinement
12-05 Vortex-insertion identities
12-06 Driven insertion and the flow lemma
12-07 Why fixed maps fail
12-08 The measured record
12-09 Where this road stops — widening the T² is the band
The crossover band — two roads reach the same wall
Part 13 The crossover band — two roads reach the same wall
13-00 Entrance — confluence
13-01 The correct form of the band
13-02 Reduction to a single inequality
13-03 The only place where non-abelian enters
13-04 Structure alone cannot cross
13-05 The chessboard and simultaneous flips
13-06 Renormalisation comparison — a comparison that distinguishes groups
13-07 The band seen from the side of time
13-08 A map of the roads tried and stopped
13-09 The measured record
13-10 Candidate new viewpoints
13-11 Ultraviolet control — what it means to close at infinitely many steps
Closing
Part 14 Closing — conditional statements
14-00 Entrance — the promise of this part
14-01 The conditional statement — the mass gap and confinement, arrow by arrow
14-02 The ledger of conditions
14-03 The table of questions (a cross-index)
14-04 What is not claimed (a cross-index)
Appendices
A-01 Table of notation
A-02 Glossary
A-03 Guide to the literature
A-04 Lean and machine checking
A-05 How this book was made
A-06 Index of assumptions
A-07 Papers that came out of this book

What is not claimed (a cross-index)

What this chapter shows — What this book does not claim, gathered across the parts into one index. The fourteen items of Part 10 (10-16 Table 2) and the sixteen items of Part 11 (11-08 Table 4) are looked up by chapter and item number, without moving the items. From the sections "what cannot be said" of the text of Parts 12, 13 and 14 we pick up items, number them on this page (items 12a onwards, 13a onwards, 14a onwards), and list them with the places in the text that support them. Finally, a table of examples in which sentences of Parts 12 to 14 lose their conditions in a one-line summary.

Prerequisite chapters — None. You can start reading anywhere. The three meanings (not shown, not read strongly, not called false) are used as in 10-16 §01. The numbers of the ledger rows are those of 14-02 Table 2.

Said first — This book does not solve the problem of Yang–Mills existence and the mass gap. There is no construction of the continuum limit, no proof of the mass gap, and no proof of confinement in the continuum limit. This chapter only writes that fact out, split into items, and adds no new results.

Order of this chapter

How to read the index — the three meanings, and the number of items per part (Table 1)
The fourteen items of Part 10 — looking up 10-16 Table 2 (Table 2)
The sixteen items of Part 11 — looking up 11-08 Table 4 (Table 3)
The items of Part 12 — on the tools of the second road (Table 4)
The items of Part 13 — on the band (Table 5)
The items of Part 14 — on the conditional statement and the ledger (Table 6)
Conditions drop out in summaries — sentences of Parts 12 to 14 and one-line summaries (Table 7)

How to read the index — the three meanings, and the number of items per part

"Not claimed" has three meanings (10-16 §01). Not shown: on that proposition this book shows nothing. Not read strongly: what has been shown carries conditions or assumptions, and read in a form with the conditions removed it becomes a sentence that has not been shown. Not called false: that a method did not reach something is a property of that method, not a claim that the proposition it did not reach is false. Besides these, there are items of "not judged, not counted": rederivations of known results are not counted as results, and the correctness of other documents is not judged.

The third becomes clearer when written in logical form. Writing " $P$ is claimed" as $\vdash P$ , $\neg(\vdash P)$ and $\vdash\neg P$ are different sentences, and the "not claimed" of this chapter always has the meaning on the left. The items of Part 13 have one more direction: cases in which a numerical reading points to the opposite of a proposition (the measurement of the comparison principle, item 13d). There too, the book does not claim the opposite proposition as a theorem, but places it as a "reading".

Table 1 The items per part (how they are numbered, and how many)
Part	Numbers	Count	Table or section to look in
Part 10 (the first road, first half)	Items 1–14	14	10-16 Table 2
Part 11 (the first road, second half)	Items 1–16	16	11-08 Table 4 (the five items on the candidate rules are at the end of 11-08 §02)
Part 12 (the second road)	Items 12a–12m	13	Table 4 of this page (the "what cannot be said" of each chapter)
Part 13 (the crossover band)	Items 13a–13l	12	Table 5 of this page
Part 14 (closing)	Items 14a–14f	6	Table 6 of this page

The items of Part 10 apply as they are to Part 11 (11-08 §04), and in the same way the items of Parts 10 and 11 apply as they are to Parts 12 to 14. In particular, "not reaching something is not evidence that it is false" (items 8 and 12 of 10-16) and "conditions drop out in summaries" (10-16 §06) are common to all parts.

The fourteen items of Part 10 — looking up 10-16 Table 2

Table 2 lists only the gist of each item, its kind, and the main places in the text that support it. The full text of the items and all the supporting text are in 10-16 Table 2. The assumption named by the items of Part 10 is now called the diffusive-relaxation-time assumption (symbol Assumption ξ², upper side ξ²↑, lower side ξ²↓; formerly Assumption H), and the gists in Table 2 use this name.

Table 2 The items of Part 10 (gist; full text in 10-16 Table 2)
Item	Gist	Kind	Supporting text (main places)
1	The solution of the problem of existence and the mass gap; a construction of the continuum limit	Not shown	10-01 §08
2	Even for the first step, the power form of Assumption ξ² has not been shown. Proposition 4.5 is algebra on top of four conditions	Not shown	10-05 §08
3	There is not yet a single $\hbar$ at which all rows of the four conditions hold at once	Not shown	10-12 §06
4	Computer-assisted results are inequalities inside the range that was computed	Not read strongly	10-08 §10
5	Readings are used neither as upper bounds nor as proofs of lower bounds	Not read strongly	10-00 Table 1
6	A Lean theorem checks the algebra from assumptions to conclusion	Not read strongly	10-13 §05
7	From the first step to the second step on, only the prospect that the form of the sentences and the form of the tools lift	Not read strongly	10-03 §06
8	That a tool did not reach something is a property of that tool	Not called false	the table of scope in 10-10
9	Rederivations of known results are not counted as results. The correctness of other documents is not judged	Not judged	10-14 §03
10	The record on the side of the four-dimensional lattice also remains assumption	Not shown	10-01 §05
11	Under the assumption of the one-sided bridge, ξ²↑ implies a lower bound on the gap, but the converse is false in general. Assumption ξ² is not the mass gap itself	Not read strongly	10-01 §06
12	The "death" of a detour plan does not mean that the remaining proposition is false	Not called false	10-10 §03
13	The reading that ξ²↑ fails in the periodic box is a conditional reading, resting on numbers from the literature and on a conjecture	Not called false	10-01 §05
14	A proposition with assumption rows left is not put out as a result	Not counted	10-12 §08

The sixteen items of Part 11 — looking up 11-08 Table 4

Table 3 gives the gist of the items of Part 11, their kind, and the main places in the text that support them. The full text is in 11-08 Table 4. What is not claimed about the candidate rules (N1–N6) (neither consistency nor a prospect towards the third step is shown, the Boolean-algebra comparison is a metaphor, and five others) is at the end of 11-08 §02; here we only point to it.

Table 3 The items of Part 11 (gist; full text in 11-08 Table 4)
Item	Gist	Kind	Supporting text (main places)
1	The solution of the problem of existence and the mass gap. The continuum limit, recovery of rotations, the OS axioms and non-triviality are not touched	Not shown	11-00 §05
2	That Assumption ξ² is false (the reading for the periodic box extends the numbers of the originals and a conjecture)	Not called false	11-01 §06
3	The sector and local versions are candidate definitions, not shown at any $\beta$	Not shown	11-03 §07
4	The one-sided bridge is an inequality on a lattice with continuous time. On Wilson's lattice with discretised time, assumption (D) remains	Not read strongly	11-04 §08
5	"The gap = the Poincaré constant of the vacuum measure" is an identity, and we do not say it has made the problem easier	Not read strongly	11-05 §07
6	The definition of the length at which conservation breaks, and the restatement of what to compare with, are candidates	Not read strongly	11-02 §08
7	The results for hierarchical models and two-dimensional models imply nothing for four dimensions	Not read strongly	11-06 §09
8	The non-perturbative mechanisms of physics (confinement, instantons, centre vortices, duality) are not used as tools of proof in Part 11	Not shown	11-07 §08
9	The two points of "because it is four dimensions" are estimates as properties of the critical dimension	Not read strongly	11-07 §01
10	The "what is not claimed" of Part 10 applies to Part 11 as it is	—	10-16 §02
11	The local version goes only as far as the definition and the ratio form. The local ξ²↑ has not been shown for the target gauge theory	Not shown	11-03 §03
12	The thresholds of the local bridge depend on the choice of reference and on the family of test observables	Not read strongly	11-04 §06
13	For the 0+1 double well, the local ratio is not 1. Taking it to approach 1 at short distances in four dimensions is an assumption that depends on asymptotic freedom	Not read strongly	11-04 §07
14	The length at which conservation breaks was measured only on the width-2 region of two-dimensional $U(1)$	Not read strongly	11-02 §03
15	For $SU(2)$ , it is undecided whether the freezing of the topological charge breaks Assumption ξ²	Not called false	11-07 §07
16	The results of the two-dimensional $CP^{N-1}$ test bed imply nothing for four dimensions	Not read strongly	11-02 §07

Item 8 of Part 11 needs care in reading when set beside Parts 12 and 13. Part 11 did not use centre vortices or instantons as tools of proof, but Part 12 puts vortex surfaces and fractional instantons at the centre of its tools. What it uses, however, are only finite identities on the lattice, estimates at fixed lattice spacing and conditional sentences; it does not bring the physical mechanisms into proofs as they are (items 12b, 12d, 12e).

The items of Part 12 — on the tools of the second road

These items are picked up from the sections "what cannot be said" of the chapters of Part 12 (from the direction of space). The group is $SU(2)$ , and the small T² carries the twist $n_{12}=1$ . The kind column gives the three meanings (and "not judged, not counted").

Table 4 What Part 12 does not claim, and the sentences of the text that support it
Item	What is not claimed	Kind	Supporting sentence of the text (gist)
12a	"Weak coupling persists up to the size of the small T²" is not a theorem. It is a physical estimate made on the assumption of ultraviolet control, and an estimate uniform in the lattice spacing is nowhere in this book. That the small T² and $\mathbb R^4$ are connected without a phase transition (adiabatic continuity) is a conjecture and is not used	Not shown	(a)(b) of 12-00 §08
12b	The slab-confinement theorem is a statement with the lattice spacing and the size $N$ of the T² fixed. The lower bound $-\ln\tanh(\beta N^2)\approx2e^{-2\beta N^2}$ goes to 0 in the continuum limit, and also vanishes as $N\to\infty$ . It says nothing about a tension uniform in the lattice spacing, nor about four-dimensional confinement	Not read strongly	(a)(b) of 12-04 §07, §04
12c	The container identity (Lemma 5) is a rewriting of the Mack–Petkova identity, and is known. The rewriting in the summed ensemble, the way of moving to a fixed sector, nesting, the insertion identities and the slab-confinement theorem are also elementary, and have not been checked against the earlier literature. No novelty is claimed for the results of this part	Not counted	12-03 §08 (a), 12-04 §06, 12-05 §08 (d)
12d	None of the insertion identities and inequalities derives confinement. The large-deviation step, the ultraviolet step and the upper bound on the second moment remain	Not shown	12-05 §08 (a), §07
12e	The count for a dilute gas of fractional instantons (string tension $\propto\Lambda^2(\Lambda L)^{5/3}$ , margin $5/3=11/3-2$ ) is a physical estimate in the dilute regime $\Lambda L\ll1$ , at the one-loop level. The action values obtained by cooling are upper bounds on the minimal action, not lower bounds	Not read strongly	(a)(b) of 12-02 §07
12f	Theorem A of Paper 3 is a statement of a draft, and its label in this book is Assumption. It is not uniform in $N$ , and says nothing about the continuum limit or about the mass gap. The converse bound (uniform in $M$ ) is open	Not read strongly	12-02 §06, §07 (e)
12g	The count that, in four dimensions, independent containers give only up to $R/\ln R$ is a count on top of the assumption of the original source (that the free energy of thick containers takes a form uniform in the boundary values), not a four-dimensional theorem. Neither that assumption nor its negation has been shown	Not read strongly	12-03 §08 (d)
12h	The three facts about the twisted T² (uniqueness of the vacuum, the gap of the adjoint field $4\sin^2(\pi/2N)$ , the absence of flat configurations in the (1,1) sector) say nothing about the lattice spacing $a\to0$ . The gap of the adjoint field is a lower bound on classical fluctuations, not a gap of the quantum effective action	Not read strongly	12-01 §09 (a)
12i	The Lean column covers only finite sums and real algebra. The Haar measure of $SU(2)$ , the co-closed sign field, the topology and projections of sectors, and that $\theta$ is the bias of a sector are outside Lean	Not read strongly	12-03 §08 (f), 12-04 §07 (e), 12-05 §08 (b)
12j	No sentence of this part says anything about widening the T² beyond $1/\Lambda$ . That stage is the crossover band, and the second road stops there	Not shown	12-02 §07 (d), 12-09
12k	The identities of driven insertion and the flow lemma do not derive confinement. "For a one-step kernel the flip probability saturates around $\tfrac12$ " is a judgement from measurement and the record, not a theorem. The Crooks relation and the Jarzynski equality are known	Not read strongly	(a)(d)(e) of 12-06 §08
12l	The comparison of isometric maps and reshaping maps is a statement at the one-loop (Gaussian) level. "Upper bounds come only from variation and explicit construction" is a judgement about this book's list of tools, not a theorem. That the leftover relative entropy is proportional to the area is an estimate for general maps	Not read strongly	(a)(c)(e) of 12-07 §08
12m	The measured numbers of Part 12 are not proofs. The trend from widening the T² does not show the band numerically (it goes only up to about $L\sqrt\sigma\approx2$ ), and reading the measurement of simultaneous flips as a numerical picture of the band has been withdrawn	Not read strongly	(a)(b)(f) of 12-08 §07

The items of Part 13 — on the band

These items are picked up from the chapters of Part 13 (the crossover band). The band is the interval around the scale $L\sim1/\Lambda$ at which weak coupling turns into strong coupling; it is a different thing from the "window" of Parts 10 and 11 (13-00 §02).

Table 5 What Part 13 does not claim, and the sentences of the text that support it
Item	What is not claimed	Kind	Supporting sentence of the text (gist)
13a	The band is unsolved. This book does not show, in any form, a statement that crosses the band ( $\ln(1/\theta(L,L))\ge cL^2$ uniformly in $L$ , or an intermediate milestone towards it). Neither the renormalisation comparison nor β-monotonicity has been shown at any $\beta$ of the band	Not shown	13-01 §06, 13-02 §08, 13-06 §09 (a)
13b	The reduction did not make the problem easier. The reduction theorem is a statement on top of three assumptions, and the strong form of the lateral-widening inequality is stronger than confinement (it contains monotonicity of the slab tension). The lemma that shrinks the comparison inside the band to the area law at fixed $\beta$ is also a reduction, and its assumption contains an unsolved problem	Not read strongly	13-02 §05, 13-06 §09 (c), 14-02 §03
13c	Do not read "the renormalisation comparison holds at both ends" with the conditions removed. The strong-coupling end is an outline in the range $u\le u_0\approx10^{-80}$ with part of the bookkeeping only sketched, and the weak-coupling end is a computation at the one-loop level. A non-perturbative weak-coupling comparison has not been shown at any $\beta$	Not read strongly	13-06 §09 (b), §03
13d	The comparison principle (in the form that Wilson loops on $\mathbb R^4$ are bounded by loops on the twisted slab) is a statement at the one-loop level and is not used as a non-perturbative tool. With $SU(2)$ , the Wilson action and $\beta=2.1$ , the plaquette on the twisted slab $16\times16\times2\times2$ is lower than on $16^4$ (7σ with errors taken from both sides, 5.5σ in an independent rerun), so it is broken for small loops. The book places this as a reading that refutes it Reading (not a theorem)	Not read strongly	13-08 §04
13e	The numbers of the lattice-doubling test (small lattices up to $8^4$ , few values of $\beta$ , 1–3 seeds) are a reading of direction, not evidence that $SU(2)$ has no phase transition to a non-confining phase	Not read strongly	13-06 §09 (d), 13-00 §03 (ii)
13f	We do not say that β-monotonicity is false (no counterexample has been found; a reading). The four faces of the middle are candidates, and none has an estimate uniform in the scale	Not called false	13-06 §09 (e)(f)
13g	The wall of the first step, "15/4 against 4", is the band in miniature, not the band. We do not say that the results of the first step imply anything in four dimensions. That four dimensions have no "finite block beyond the crossover length" is not a claim that the road of blocks is impossible in four dimensions	Not read strongly / Not called false	(b)(f) of 13-07 §07, 13-00 §05
13h	We do not say that the road of blocks and mixing (road C) closes the first step. Apart from Proposition K′, its rows are Assumption and Reading	Not shown	13-07 §07 (d), 10-15 §road C
13i	"The only place where non-abelian enters" is a phrase with the restriction "within this reduction". The margin $5/3=11/3-2$ is a physical estimate in the dilute regime at the one-loop level	Not read strongly	13-02 §08, 13-00 Table 1
13j	The negative lemma, that the lateral-widening inequality does not follow from structural inputs alone (reflection positivity, the identities of centre and duality, positivity of the transfer matrix), is a statement of the form "from this set of inputs alone", not a claim that the band cannot be crossed	Not called false	13-00 Table 1, 13-04 §04
13k	None of the candidate new viewpoints closes the band. The reduction by compactness of the band is a reduction, and we do not say it has made the band easier. Nor do we say that the candidates are not combinations of known facts	Not shown	(a)(b)(g) of 13-10 §07
13l	"Stopped" for a road tried and stopped means that its tool could not cross the band, not that the proposition aimed at by the road is false. Roads stopped for reasons of estimate can be reopened if a new input appears	Not called false	13-08 §01

The items of Part 14 — on the conditional statement and the ledger

Table 6 What Part 14 does not claim, and the sentences of the text that support it
Item	What is not claimed	Kind	Supporting sentence of the text (gist)
14a	Confinement (the area law) is not the mass gap. There is no known road that gets the mass gap from the area law, and the arrow joining the two on the same theory is open (even the statement of what it suffices to assume is not settled)	Not shown	9-06 §03, 14-01 §04, 12-00 §08 (c)
14b	The conditional statement is a statement only insofar as the band is placed among the assumptions. Only the arrows whose paper proof has been placed in the book are called "theorems", and the assumptions in the form of the band (the lateral-widening inequality, the uniform slab tension, the uniform slab gap) carry the content of the conclusion	Not read strongly	14-01 §05
14c	The propositions of the research programme on $\mathbb R^2\times$ a small twisted T² (arrow G1 on the gap side, arrow C2 on the confinement side) are conditional and at the stage of an outline. The assumption of G1 (block correlations decay at rate $m\ge c/L$ ) contains the quantity of the conclusion	Not read strongly	14-01 §02, §03
14d	Even if every row of the ledger disappears, that does not mean the problem is solved. Four things remain: the limit to $\mathbb R^4$ , joining the area law and the gap, all compact simple groups, and verification	Not shown	14-02 §05, 14-00 §01
14e	A row closed in the record (mark ■) does not mean it has disappeared from the book's ledger. A row disappears only when its paper proof has been placed in the book and, if it has a Lean entry, the re-check is done	Not read strongly	14-02 §01
14f	The number of rows of the ledger is not a gauge of progress. Rows can also split and increase	Not counted	14-02 §06

Conditions drop out in summaries — sentences of Parts 12 to 14 and one-line summaries

The error of reading a claim strongly arises, in most cases, at the stage of shrinking a sentence of the text into one line (10-16 §06). Table 7 sets the conditional sentences of the text of Parts 12 to 14 beside one-liners from which the conditions have been dropped entirely. Every sentence in the right-hand column is a sentence this book has not shown.

Table 7 Sentences of the text (with conditions) and one-liners with the conditions dropped (everything on the right is wrong, or not shown)
Topic	Sentence of the text	One-liner with the conditions dropped
Slab confinement	With the lattice spacing and the side $N$ of the T² fixed, $\sigma\ge-\ln\tanh(\beta N^2)$ for every $\beta$ Proved here (12-04 §02)	Confinement in lattice gauge theory has been shown
Theorem A of Paper 3	At fixed $N\ge2$ , $\ell\ge2$ , $\beta\ge1$ , $M\in2\ell\mathbb N$ , a lower bound on the area law with rate $S_*(N,\ell)$ (a statement of the draft; the book's label is Assumption, 12-02 §06)	It has been shown that fractional instantons cause confinement
The one-loop comparison	With the comparison that reshapes the fluctuations, $\beta-\beta'=0.258$ (per doubling) Proved here (at the one-loop level, 13-06 §03)	The renormalisation comparison holds
The doubling test	For $SU(2)$ , the $\beta-\beta'$ read off is $+0.13$ – $+0.25$ ( $4^4\to8^4$ and others, $\beta=2.5$ – $3.0$ , 1–3 seeds) Reading	$SU(2)$ has no phase transition inside the band
The reduction theorem	Under the three assumptions of the base point, frame decorrelation and the lateral-widening inequality, the correct form of the band follows Proved here (conditional, 13-02 §03)	The band has been reduced to a single inequality and become easier
The comparison inside the band	Under the area law on both sides at each $\beta$ of the band (assumption (A)), the comparison inside the band shrinks to finitely many steps Proved here (13-06 §06)	The renormalisation comparison inside the band has been shown
The iteration in Lean	`Doubling.iterate` (assumptions: $c\ge0$ , $t_k\ge0$ , $t_{k+1}\le c^3t_k^4$ ; conclusion: $c\,t_k\le(c\,t_0)^{4^k}$ ) is algebra of a sequence of real numbers	The reduction of the band has been checked in Lean
Reflection positivity in z	Closed in the record (mark ■). Until the paper proof is placed in the book, ledger row ⑤ remains (14-02 §01)	Row ⑤ of the ledger has disappeared

The error types in Table 7 are the same two as in Table 5 of Part 10. One drops the scope (fixed lattice spacing, fixed $N$ , the one-loop level, small lattices); the other drops the grade (Reading, Assumption, the reach of Lean). In Parts 12 to 14 one more type is added: reading "reduced" as "made easier". A reduction gathers the difficulty into one place; it does not reduce the difficulty.

What this chapter can say

What is said	What is not said
That this book shows nothing about the existence and the mass gap of the problem statement (0-03). Which sentences of which chapters support the items of Parts 10 and 11 (Tables 2 and 3)	The truth or falsity of the claims of the problem statement. Existence of the continuum limit. A gap uniform in the volume
The items of Parts 12, 13 and 14, and the places in the text that support them (Tables 4–6)	The truth or falsity of the proposition of each item. Of the sentences of chapters 13-03 to 13-05 and 13-09, only those that overlap with rows of Table 5 have been picked up
Pairs of conditional sentences of Parts 12 to 14 and one-liners with the conditions dropped (Table 7)	That the sentences in the right-hand column of Table 7 are false (only that they have not been shown)

Sources and reproduction

Item	Kind	Source / tools
The three meanings	This book	10-16 §01
Each row of Table 2	This book	10-16 Table 2 and the chapters in its right-hand column
Each row of Table 3	This book	11-08 Table 4 and the chapters in its right-hand column
Each row of Tables 4–6	This book	The sections "what cannot be said", "what remains" and "the differences that remain outside the ledger" of the chapters in the right-hand column
Table 7	This book	The statements and labels of the chapters in the left-hand column. The numerical conditions are in 13-00 §03 (ii) and 13-06 §05
The problem statement	Cited	A. Jaffe, E. Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute (2000) (the source table of 0-03)

Terms used in this chapter — not shown, not read strongly, not called false, not judged / not counted, the conditions that drop out in a summary, the crossover band, reduction, the differences that remain outside the ledger, the diffusive-relaxation-time assumption. Their meanings are in the A-02 Glossary and the A-06 Index of assumptions.

Next page to read: A-07 Papers that came out of this book — the main theorem of each paper, with its label and conditions. To go back to the entrance of Part 14, 14-00.

Revised 2026-10-15: first version.