2026-10-15 · chapter Yang–Mills and the mass gapPart 14 Closing — conditional statementsquestions, an index by chapter and item number · 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

The table of questions (a cross-index)

What this chapter shows — The questions that the parts of this book left open, gathered into one index. The seventeen questions of Part 10 (Q1–Q17), and the nineteen questions (F1–F19) and candidate rules (N1–N6) of Part 11, are looked up by chapter and item number, without moving the items. The questions left by the text of Parts 12 and 13 are given numbers on this page (Question 12a onwards, Question 13a onwards), listed with the section of the chapter they come from and the rows of the ledger of conditions they concern (①–⑩ of 14-02 Table 2, and Outside 1–4 outside the ledger). The two tables of "the next move" listed by the record of the study are also looked up in the language of the ledger rows.

Prerequisite chapters — 14-02 (the ledger of conditions; the numbers and names of the rows are fixed there). The content of the questions is in the chapters in the right-hand column of each row.

Said first — This chapter adds no new results. Questions are questions, not claims. The status of a question (done, partly done, not yet, new) and "what does not move even if settled" are kept only in the tables of the chapters they come from, and are not copied here. Whichever question is settled, the problem of the mass gap does not move. What moves is one row of the ledger, or one label in one chapter.

Order of this chapter

How to read the index — four sources, the numbering convention, correspondence with the ledger rows (Table 1)
The questions of Part 10 — Q1–Q17 (Table 2)
The questions and candidate rules of Part 11 — F1–F19, N1–N6 (Tables 3 and 4)
Questions from Part 12 — Questions 12a–12p (Table 5)
Questions from Part 13 — the assumptions of the reduction of the band, criteria, β-monotonicity and more, Questions 13a–13p (Table 6)
The next moves listed by the record of the study — two tables looked up by ledger row (Table 7)
Looking up from the ledger rows — for each row, the questions that concern it (Table 8)

How to read the index — four sources and the numbering convention

The questions have four sources. Table 1 gives, for each source, how the numbers are assigned and the table that the numbers point to. The numbers are those of the source chapter, used as they are and not renumbered on this page. Parts 12 and 13 do not yet have a chapter gathering their questions, so we pick up sentences in the form of questions from the sections "what cannot be said", "what remains" and "what is lacking" of the text of those parts, and number them on this page. The numbers are lower-case letters by part (12a, 12b, … and 13a, 13b, …), chosen so as not to clash with the symbols of other chapters (numbers of propositions, conditions and assumptions).

Table 1 The four sources of the questions
Source	Numbers	Count	Table to look in	How the size of a question is written
Part 10 (the first road, first half)	Q1–Q17	17	10-17 Table 3	Tools (SAT, paper, python3, interval arithmetic, Lean, literature) and the form of settlement
Part 11 (the first road, second half)	F1–F19, N1–N6 and three combinations	19 + 9	11-08 Table 1, Table 2	Kind of work (paper, reading, computation, Lean) and status
Parts 12 and 13 (the second road and the band)	Questions 12a–12p, Questions 13a–13p	16 + 16	Tables 5 and 6 of this page	As in the sentences of the source sections. No estimate of size is written
"The next move" of the record of the study	Order 1–7 of the record's table, and order of work 1–4	7 + 4	Table 7 of this page	Looked up after replacing them with the language of the ledger rows

The numbers of the ledger rows are as in 14-02 Table 2: ① ultraviolet control, ② the gap on a small T², ③ uniform acceptance of the insertion, ④ the base point on a small T², ⑤ reflection positivity in z, ⑥ the uniform slab gap (frame decorrelation), ⑦ the renormalisation comparison (breakdown: ⑦a lower end, ⑦b upper end, ⑦c middle, plus the two rows strong-coupling end and one loop), ⑧ the band, ⑨ what remains of the first step, ⑩ the four-dimensional goal (side of time). The differences outside the ledger are Outside 1, the limit to $\mathbb R^4$ ; Outside 2, joining the area law and the gap; Outside 3, all compact simple groups; Outside 4, verification (14-02 Table 3).

The column "rows concerned" of the tables gives, as this book's arrangement, which ledger rows' content a question would touch if it were settled. Settling a question does not necessarily make a row disappear. 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 (14-02 §01).

The questions of Part 10 — Q1–Q17

The questions of Part 10 are "small questions" expected to be settled at the size of an undergraduate thesis. Table 2 lists only a one-line summary and where to look. The form of settlement and "what does not move when settled" are in each row of 10-17 Table 3. The assumption around which the questions of Part 10 turned is now called the diffusive-relaxation-time assumption (symbol Assumption ξ², upper side ξ²↑, lower side ξ²↓; formerly Assumption H).

Table 2 The questions of Part 10 (summaries; the full text of each row is in 10-17 Table 3)
Q.	Summary	Tools	Rows concerned	Where to look
Q1	Does a complete family of $k=3$ cells exist in $d=9$ ?	SAT, Lean	— (the combinatorics of one-link integration)	10-17 Table 3, 10-02 §02
Q2	The largest dimension in which a complete family exists, $d_{\max}(k)$	Exhaustive search, paper	— (same as above)	10-17 Table 3
Q3	The constants of $1-\lambda_k$ for the Hermite family	Paper, python3	— (worked example)	10-17 §05
Q4	The $SU(N)$ version, and the Lean statement, of the saturation of the ratio of functions on the forest	Paper, Lean	⑩	10-17 Table 3
Q5	$C_P(\beta)$ in the link variables of two-dimensional lattice Yang–Mills	Paper, python3	⑩ (control)	10-17 Table 3
Q6	The best constant of the curvature after one-link integration, $c^*\in[21,27]$	Global optimisation, interval arithmetic	— (the strong-coupling side)	10-17 Table 3, 5-11
Q7	The coefficient of $N^{-3}$ in the column model, analytically	Paper, python3	⑨	10-17 Table 3
Q8	In the column model, the sign of $1/P$ at finite $u$	python3	⑨	10-17 Table 3
Q9	How much does the shortfall $c_2$ change with the choice of gauge in the block transformation?	python3	⑩	10-17 Table 3
Q10	$C_P$ and $\xi^2$ of the two-dimensional $O(3)$ sigma model	python3, literature	⑩ (an asymptotically free control)	10-17 Table 3
Q11	$C_P/\xi^2$ of the three-dimensional $\mathbb Z_2$ gauge theory	python3	⑩ (control)	10-17 Table 3
Q12	Monotonicity in volume of $C_P(V)$ , and boundary conditions	python3	⑩	10-17 Table 3
Q13	Collect the primary sources on the dynamical exponent of local algorithms for four-dimensional $SU(2)$	Literature	⑩	10-17 Table 3
Q14	Bound from above, by interval arithmetic, the two-point maximal correlation of stage 2 of the core ( $8\le k\le14$ )	Interval arithmetic, paper	⑨	10-17 Table 3, 10-15 Table 1
Q15	Read the power of the prefactor $\hbar^{-p}$ of $C_P$ for the core by the capacity of the degenerate point	python3, paper	⑨	10-17 Table 3
Q16	A comparison theorem showing the alternating maximal correlation between boxes $c_{\rm alt}\lt1$	Paper	⑨	10-17 Table 3
Q17	When a small term that breaks the conservation is added to a dynamics with conservation, the length at which the gap departs from the square of the side	Paper, python3	⑩ (the smallest model of Part 11)	10-17 Table 3

Reading Table 2. The questions that concern row ⑨ (what remains of the first step) are Q7, Q8, Q14, Q15 and Q16; the questions opened by "the road of blocks and mixing" in the supplement to Part 10 (10-15 §road C) are gathered as Question 13k in Table 6 of Part 13. The questions concerning ⑩ are questions of how to set up the statement of Assumption ξ² on the side of the four-dimensional lattice; whichever of them is settled, ⑩ itself (ξ²↑ and the bridge, uniform as $\beta\to\infty$ ) does not move.

The questions and candidate rules of Part 11 — F1–F19, N1–N6

The questions of Part 11 came out of the study that repairs the form of the statement of Assumption ξ² (the length at which conservation breaks, the sector and local versions, the form the one-sided bridge needs, the vacuum measure). Table 3 gives only summaries and where to look; the status (done, partly done, not yet, new) and "what moves and what does not move if settled" are in 11-08 Table 1.

Table 3 The questions of Part 11 (summaries; the full text of each row and its status are in 11-08 Table 1)
Q.	Summary	Size	Where to look (the section the question came from)
F1	For the free field, compare the excitation energies of observables supported in the box	Paper	11-04 §05
F2	Monotonicity of the factor that cuts off times, $\varphi(x)/x^2$	Elementary, Lean	11-04 §03
F3	Whether the static ratio catches the slowness, for the sign of the Polyakov loop on a small periodic box	Small computation	11-01 §02
F4	For $U(1)$ on a two-dimensional layer, how the gap decreases as the strip-shaped region is lengthened	Computation	11-02 §03
F5	Primary sources on the autocorrelation of the topological charge in pure $SU(2)$ gauge theory	Reading, computation	11-07 §07
F6	A many-family version of the lemma of two projections, and its condition on maximal correlation	Paper	11-03 §06
F7	$\inf\operatorname{Hess}V_t$ of the one-link Wilson weight convolved with the heat kernel, and the flow of the weight of the bad set	Paper + small computation	11-06 §07
F8	For Wilson's transfer operator, show assumption (D) restricted to functions cut off at low energy	Paper	11-04 §08
F9	For finite-state reversible Markov chains, close in Lean $\tau_{\rm int}\le C_P$ and the identity of the ground-state transformation	Lean	11-05 §01
F10	Reread the autocorrelations of local modes and topological modes in two-dimensional $CP^{N-1}$ through the eyes of the local version	Reading, computation	11-02 §07
F11	For two-dimensional $SU(2)$ , find $\lambda_W$ and $\Gamma$ of families of loops at finite $\beta$	Computation	11-04 §06
F12	On a four-dimensional box, the values for families of loops the size of the span	Computation	11-04 §06
F13	For the 0+1 double well, sandwich the local ratio by a reduction to one dimension with the position of the kink as coordinate	Paper + small computation	11-04 §07
F14	Add one point $\beta=7$ for the two-dimensional strip-shaped region, and see whether the slope settles	Computation	11-02 §05
F15	For fields whose ratio lies between the threshold and 1, another argument that sees the mass while staying in the ratio form	Paper	11-04 §06
F16	On Wilson's lattice with discretised time, write the version of the bridge that cuts off times (Proposition W and the lower bound in finite volume) under assumption (D)	Paper	11-04 §03
F17	Whether the Kogut–Susskind Hamiltonian becomes frustration-free when it is split with terms grouped by sector	Paper	11-05 §04
F18	On the $CP^{N-1}$ test bed, add seeds and pairs of $N$ , and read the exponent of the slowness of the topological mode	Computation	11-02 §07
F19	Measure the definition (2) of the length at which conservation breaks with the charge of a four-dimensional subregion	Computation	11-02 §01

The summaries of Table 3 are shortened from the sentences of the source table. The terms of Part 11 "window factor" and "window version of the bridge" are restated here as "the factor that cuts off times" and "the version of the bridge that cuts off times", so that they are not confused with the band of Part 13 (note on terms: 13-00 §02). The "strip-shaped region" of F4 and F14 is a width-2 region of a two-dimensional lattice, a different thing from the crossover band. In terms of ledger rows, only F13 touches ⑨; the others are questions that repair the form of the statement of ⑩.

The candidate rules are candidates for the question "if we were to make new mathematics, what would we put as rules?", and each has one question attached (11-08 Table 2). Table 4 lists only the symbol, the name of the rule and where to look. The relation to the three roots of the third step is in 11-08 Table 3.

Table 4 The candidate rules (the one-line candidate and its question are in 11-08 Table 2)
Symbol	Name of the rule	Section where the candidate came from
N1	Axiomatising scale	11-07 §06
N2	The algebra of loops	11-07 §06
N3	Conserved quantities and sectors	11-03 §04
N4	The class of measures that pay with rarity	11-06 §07
N5	An axiom of asymptotic freedom	11-07 §06
N6	Rules for curvature (Bakry–Émery on the space of orbits)	11-07 §06
N1+N4, N3+N4, N2+N3	Three combinations	The three rows at the bottom of 11-08 Table 2

Seen from the study of Parts 12 and 13, among the candidates N5 (an axiom of asymptotic freedom) and N1 (axiomatising scale) sit closest to the problem of the band. In the statements that cross the band, the place where asymptotic freedom enters is narrowed to one (the $\beta'\lt\beta$ of the renormalisation comparison, 13-06 §01), and the question of N5 (what follows from the rule of approaching a free field in the ultraviolet) becomes a question of restating that one place in the language of rules. This is an estimate of position; none of the candidates promises a prospect of crossing the band Reading (a physical estimate).

Questions from Part 12 — Questions 12a–12p

These are the questions that the text of Part 12 (the second road: from the direction of space) left open. The sentence in each row is the sentence of the source section shortened into the form of a question, with its conditions kept in the same row. The group is $SU(2)$ , and the small T² carries the 't Hooft twist $n_{12}=1$ . The questions from the three chapters 12-06 to 12-08 (driven insertion, fixed maps, the measured record) are the five rows at the bottom of the table (12l–12p).

Table 5 Questions from Part 12 (the rows concerned are rows of the ledger of 14-02; this book's arrangement)
Q.	Question (with its conditions in the same row)	Rows concerned	What does not move even if settled	Source
12a	The converse of Theorem A of Paper 3: at fixed $N$ , an upper bound by a dilute gas uniform in $M$ (a bound on the string tension from above). In the draft it is reduced to a single positivity lemma	⑦b	A bound uniform in $N$ , the continuum limit	12-02 §06, A-07 §05
12b	Agreement of the minimal actions $S_*(N,\ell)\to S_{\min}(N,\infty)$ (the minimal action of a column with its frame frozen to the vacuum, and the minimal action of a free fractional instanton). The numbers agree to five digits (a reading)	⑦b	Same as above	12-02 §06
12c	On the twisted $T^2\times\mathbb R^2$ , existence of a solution of topological charge $\tfrac12$ attaining the lower bound $4\pi^2/g^2$ of the action (at present an outline with assumptions)	④	The rate on the lattice, the band	12-02 §02, §07 (c)
12d	The $SU(N)$ version of uniqueness of the vacuum that absorbs the twist, the gap of the adjoint field, and the absence of flat configurations in the (1,1) sector	Outside 3	Groups without a centre	12-01 §09 (c)
12e	The second term when moving to a fixed sector: that the bias of the outer container is typically small (0 as $M\to\infty$ )	⑥	A uniform estimate of the container free energy	12-03 §04, §08 (b)
12f	Estimate the container free energy $\Delta F_j$ uniformly over good frames and uniformly in the lattice spacing (needs a multiscale analysis of the conditional measure with the frame held fixed)	①, ⑥	The band (the stage of widening sideways)	12-03 §08 (e)
12g	With the map of a thick vortex, bound the cost of the insertion independently of $N$ (an improvement of the slab-confinement theorem towards the continuum limit; at present an outline, Assumption)	①, ③	The band	12-04 §04, §07 (c)
12h	The three things lacking between the insertion estimates and the area law: (i) that the event that insertions are refused simultaneously in all cells is exponentially rare in the number of cells, (ii) that the lower bound on the acceptance rate is uniform in the lattice spacing, (iii) an upper bound on the variance of the insertion ratio	③	The band	12-05 §07
12i	A negative control that applies the insertion estimates to four-dimensional $U(1)$ (not confining at weak coupling) and checks where they stop working	⑦c (the condition that distinguishes groups)	The estimate for $SU(2)$	12-05 §07, §08 (f)
12j	That the map composing insertions in separate cells satisfies the identity "odd under insertion" (at present an assumption of Theorem 8, and an assumption in Lean too)	③	The identity for one cell	12-05 §08 (c)
12k	Comparing the slab-confinement theorem with arguments of Tomboulis–Yaffe type and with the area law for spatial Wilson loops at high temperature (Borgs 1985)	— (a question about the label)	The content of the theorem	12-04 §06
12l	Constructing a kernel whose flip probability $a$ is close to 1. The flow lemma $\theta\le1-a$ gives only a naive form, because for a one-step kernel $a$ saturates around $\tfrac12$ (a judgement from measurement and the record)	⑧ (a form usable for testing the band)	Ultraviolet control ①	12-06 §06, §08 (d)
12m	The quasi-static limit: that the work $W$ approaches the conditional free energy of the container as the number of driving steps $m\to\infty$ (a proof including an estimate of how the kernel mixes)	③	The band	12-06 §04, §08 (b)
12n	Bound the error of the map that reshapes the fluctuations (the local relative entropy near the vortex surface, and higher loops) at intermediate coupling and for large fields. At one loop it reproduces $\beta-\beta'=0.258$	⑦ (from one loop to non-perturbative)	The statement that crosses the band itself (⑦c)	12-07 §08 (a)
12o	Whether the extra ultraviolet cost of fixed maps grows in the same way in four-dimensional gauge theory (whether cancellation by Ward identities occurs). The present numbers are three points in a two-dimensional model	①	The band	12-07 §06, §08 (d)
12p	Measurements that widen the T² beyond the sizes measured (up to about $L\sqrt\sigma\approx2$ ), and an estimate of the errors of driven insertion and replica exchange (the treatment of autocorrelation)	— (a question about a reading)	Any ledger row (numbers are not proofs)	(b)(c) of 12-08 §07

Reading Table 5. 12a and 12b are exactly what remains of ledger ⑦b (the upper end); if they are settled, both sides of the rate match on the lattice at fixed $N$ , but the direction of increasing $N$ is the same problem as ⑦c. 12f and 12g are questions that touch the content of ultraviolet control ①, a wall separate from the band (12-00 §02). (ii) of 12h is a restatement of ledger row ③, uniform acceptance of the insertion.

Questions from Part 13 — the assumptions of the reduction of the band, criteria, β-monotonicity and more

These are the questions that the text of Part 13 (the crossover band) left open. The first three rows put into question form the three assumptions of the reduction of the band (13-02 §02), and the difficulty of the band is concentrated in the third row, the lateral-widening inequality (13-02 §08). An argument that closes the band must be able to distinguish $U(1)$ (criterion (m), 13-06 §08). This condition applies to every row of Table 6.

Table 6 Questions from Part 13 (the rows concerned are rows of the ledger of 14-02; this book's arrangement)
Q.	Question (with its conditions in the same row)	Rows concerned	What does not move even if settled	Source
13a	Show the lateral-widening inequality (strong form $\theta(R;T^2_{L'})\le C(L'/L)^2\theta(R;T^2_L)$ , or the weak form in which the total loss is finite) non-perturbatively across $L\sim1/\Lambda$ , between the dilute regime and the confining regime. It does not follow from structural inputs alone (reflection positivity, duality, the transfer matrix)	⑧ (the place of the input to the reduction is ⑦)	Outside 1–4	13-02 §02, §08
13b	Show frame decorrelation $\theta(sR;T^2_L)\le C_{\rm D}\,\theta(R;T^2_L)^{s^2}$ uniformly in $L$ . With $L$ fixed it is replaced by the chessboard density $\kappa_{\rm cb}$ , so the question becomes a comparison of $\kappa_{\rm cb}(L)$ across scales	⑥	The non-abelian input to the band (13a)	13-02 §02, 14-02 Table 2 ⑥
13c	Show the base point on a small T², $\theta(L_0;T^2_{L_0})\le1-\varepsilon_0$ , under ultraviolet control and uniform acceptance of the insertion (the expected size $\varepsilon_0\propto(L_0\Lambda)^{11/3}$ is a one-loop estimate)	④ (premises ①, ③)	The band	13-02 §02, 14-01 Table 1 [03]
13d	Intermediate milestone (a): $\Delta F(L,L)\to0$ (at any speed). It rules out the known non-confining phases, but gives no uniform lower bound on the tension	Partway to ⑧	The mass gap, a uniform tension for the area law	13-01 §06, 13-06 §08
13e	Intermediate milestone (b): $\ln(1/\theta(L,L))\ge cL^2/\ln L$ (almost an area law; by the Tomboulis–Yaffe inequality, decay faster than the perimeter law)	Partway to ⑧	A uniform tension	13-01 §06
13f	β-monotonicity: $\partial_\beta\Delta F\ge0$ with the lattice geometry held fixed. Rewritten in terms of electric flux, it is a problem of the same weight as $\beta$ -monotonicity of the string tension in lattice units, and as far as we searched, not even $\beta$ -monotonicity of Wilson-loop expectation values has been shown. In the chain of the band it is not needed if the renormalisation comparison is shown directly	⑦ (an auxiliary role)	The renormalisation comparison itself	13-06 §02
13g	Checking that, on a slab with twist $n_{12}$ , β-monotonicity for $\mathbb Z_2$ and $U(1)$ does not follow from Ginibre's inequality (without the twist it does)	⑦	β-monotonicity for $SU(2)$	13-06 §01
13h	The criterion at the lower end: push confinement and the comparison up to $\beta\approx0.3$ – $1$ , by a finite-size criterion of Dobrushin–Shlosman type using the influence $\delta(L_0,\beta)$ on the inside when the centre flux on the boundary of the box is changed, and by a computer-assisted count of the growth constant of admissible surfaces	⑦a	Above $\beta\approx1$ (the middle)	13-06 §04
13i	Write out the bookkeeping of Proposition S at the strong-coupling end (the interaction of volume polyhedra and surface contours), and make the constant $u_0\approx10^{-80}$ larger	The strong-coupling end of ⑦	Beyond roughening ( $\beta_R\approx1.9$ )	13-06 §03
13j	Show any one of the four faces of the middle (softness in the direction of the twist $\kappa_w$ , mixing of the box flux $\delta(L_0)$ , the cost of twisting a column $f(\ell)$ , the margin of the pressure) uniformly in the scale, in a form that fails for $U(1)$ . As a preliminary step, test $\kappa_w$ and $\delta(L_0)$ numerically with $U(1)$ as the control	⑦c	Ultraviolet control ①	13-06 §07
13k	At each $\beta$ in the band, the area law on both sides at fixed $\beta$ (with errors of the order of the perimeter, assumption (A)). If this can be said, the comparison inside the band shrinks to finitely many steps (Lemma 1)	⑦	The seam between the upper end of the band and ultraviolet control	13-06 §06
13l	What remains of the road of blocks and mixing for the first step: the second and third stages of strong mixing (the Laplace method with constraints), checking against the original sources the Varadhan-type limit, Witten's Morse theory (corners of the Neumann box) and the standard theorem for one-dimensional chains, and interval arithmetic for the curvature at the minimum $\lambda_*(12)$ and the reach of the optimal path	⑨	⑩ (⑨ is a miniature, not an implication)	10-15 §road C, 13-07 §06
13m	Why, in the first step, the convexity threshold (the first conjugate point at the top, $\approx7.84$ ) and the mixing length (about 9) are numerically close	⑨	⑩	13-07 §07 (c)
13n	Can the smallest band be crossed: test a tool claimed to cross the band first on two-dimensional asymptotically free models (the $O(N)$ model at large $N$ , the Gross–Neveu model), with the ultraviolet cutoff still removed (screening criterion (j). Exponential decay at all temperatures for the two-dimensional $O(3)$ model is also said to be unsolved; not yet checked)	⑧ (a touchstone)	The four-dimensional band	13-08 §07, Q10
13o	The asymptotic form of the comparison principle for large loops only, $\sigma_{\rm tw}(L)\le\sigma_\infty$ . The form for small loops was broken numerically, but this form has not been refuted (there is no mechanism to show it)	⑧	The comparison principle for small loops (broken numerically)	13-08 §04
13p	Put each of the three new candidates (reduction by compactness of the band, log-concavity of the Weyl measure and a height/azimuth decomposition, local step functions of the column) through the tests of the screening criteria	⑦c	None of the candidates closes the band	13-10 §05, §07

Reading Table 6. 13a–13c are the three assumptions of the reduction theorem (13-02 §03); if all three are in place, the correct form of the band follows. But the three differ in character, and non-abelian dynamics enters only in 13a (13b is replaced by a structural fact once $L$ is fixed, and 13c is a statement on the weak-coupling side). 13f–13k are the breakdown of the renormalisation comparison (⑦): 13h is the lower end, 13j the middle, 13i the strong-coupling end. 13n–13p are questions coming from the chapters on the roads tried and stopped and on new candidates (13-08, 13-10). 13l and 13m are questions about the first step on the side of time (the first road); even if they are settled, the band does not move (13-07 §04).

The next moves listed by the record of the study — two tables looked up by ledger row

Among the material of Parts 12 and 13 there are two tables of "the next move", kept as a record of the study. One lists seven rows in decreasing order of the prospect of closing; the other lists four rows in the order of work. Both are tables in the record, not chapters of the book. Table 7 looks up each of their rows in the language of the ledger rows and the chapters of the book. The "amount of work" scale of the record's tables is a number of prospects, not a number with grounds, so it is not copied.

Table 7 The rows of the record's two tables, with the ledger rows and the chapters of the book (the sentences for the content of each row are on the side of the chapters of the book)
Record's table	Order	Content (in the language of the book)	Ledger row	Chapter of the book, question on this page
Order of the prospect of closing	1	What remains of strong mixing for the first step (checking the Varadhan-type limit, a uniform form of the Laplace method with constraints)	⑨	10-15 §road C, Question 13l
	2	The logarithmic Sobolev inequality for blocks in the first step (checking Witten's Morse theory and the decomposition against the original sources, interval arithmetic for $\lambda_*$ )	⑨	Same as above, Question 13l
	3	Stitching 1 and 2 together (the standard theorem for one-dimensional chains)	⑨	Same as above, Question 13l
	4	The positivity lemma needed for the converse bound of Paper 3 (by a low-temperature cluster expansion at one scale)	⑦b	Question 12a, A-07 §05
	5	Agreement of the minimal actions in Paper 3 (exponential decay of the minimisers, and cutting off)	⑦b	Question 12b
	6	Writing out the bookkeeping of the strong-coupling end, and improving the constant $u_0$	The strong-coupling end of ⑦	Question 13i
	7	The middle: show the comparison between scales by one of the four faces	⑦c	Question 13j
Order of work	1	Of the faces of the middle, test $\kappa_w$ and $\delta(L_0)$ numerically with $U(1)$ as the control	⑦c	Question 13j
	2	The upper end: writing out the semiclassical theorem at fixed $N$	⑦b	Questions 12a, 12b, 12-02 §06
	3	The lower end: a computer-assisted estimate of the growth constant of surfaces, and formulating the criterion by mixing of the box flux	⑦a	Question 13h
	4	Add to Paper 2 the scan with step functions and the $U(1)$ control	— (a revision of the paper)	A-07 §04

None of the rows of the two tables touches the differences outside the ledger (Outside 1–4). Questions aimed at the differences outside the ledger are in the table of tasks of Part 9 (9-04) and in the unknown column of the table of seven stages (9-06 Table 1).

Looking up from the ledger rows — for each row, the questions that concern it

Table 8 looks up Tables 2–7 again from the side of the ledger rows. The content of each row, its label and what is needed for it to disappear are in 14-02 Table 2; here we list only the numbers of the questions concerned. A row without questions is a row in which no step small enough to be shrunk into the form of a question is yet in sight.

Table 8 The ledger rows and the questions concerned (this book's arrangement)
Row	Questions concerned
① Ultraviolet control	Questions 12f, 12g, 12o (in part). This book has no table that splits this row into small questions (6-07)
② Gap on a small T²	— (the twist part is on paper in 12-01; the form uniform in $a$ depends on ①)
③ Uniform acceptance of the insertion	Questions 12g, 12h, 12j, 12m
④ The base point on a small T²	Questions 12c, 13c
⑤ Reflection positivity in z	— (closed in the record; what remains is placing the paper in the book)
⑥ The uniform slab gap	Questions 12e, 12f, 13b
⑦ The renormalisation comparison	Strong-coupling end: Question 13i / ⑦a: Question 13h / ⑦b: Questions 12a, 12b / ⑦c: Questions 12i, 13j / tests of candidates for ⑦c: Question 13p / outside the breakdown: Questions 12n, 13f, 13g, 13k
⑧ The band	Questions 13a (the content), 13d, 13e (intermediate milestones), 12l, 13o (forms usable for testing the band), 13n (a touchstone)
⑨ What remains of the first step	Q7, Q8, Q14, Q15, Q16, F13, Questions 13l, 13m
⑩ The four-dimensional goal (side of time)	Q4, Q5, Q9–Q13, Q17 and F1–F12, F14–F19 (all are questions that repair the form of the statement of ⑩, not ⑩ itself)
Outside 1, the limit to $\mathbb R^4$	—
Outside 2, joining the area law and the gap	— (even the statement of what it suffices to assume is not settled, 14-01 §04)
Outside 3, all compact simple groups	Question 12d (only the $SU(N)$ part)
Outside 4, verification	—

What to read from Table 8. The rows with many questions are ⑦, ⑨ and ⑩, and the questions on the band itself, ⑧, gather in 13a. 13d and 13e are intermediate milestones weaker than ⑧, and settling them does not make ⑧ disappear. The differences outside the ledger have almost no steps that can be shrunk into the form of a question. Outside 2 starts, before any question is set, from making a statement of what needs to be shown.

What this chapter can say

What is said	What is not said
Which section of which chapter the questions of Parts 10 and 11 (Q1–Q17, F1–F19) and the candidate rules (N1–N6) came from (Tables 2–4)	The answer to any question. The status and the full text of "what does not move even if settled" are in the source tables
A list of the questions that the text of Parts 12 and 13 left open, with their source sections (Tables 5 and 6)	Estimates of the size of the questions. Of the sentences of chapters 13-03 to 13-05 and 13-09, only those that overlap with rows of Table 6 have been picked up
The correspondence between questions and ledger rows (Table 8, this book's arrangement)	That settling a question makes a row disappear (the rule for rows disappearing is 14-02 §01)

Sources and reproduction

Item	Kind	Source / tools
Q1–Q17 of Table 2	This book	10-17 Table 3
F1–F19 and N1–N6 of Tables 3 and 4	This book	11-08 Table 1, Table 2
The questions of Tables 5 and 6	This book	The sections of the chapters in the right-hand column ("what cannot be said", "what remains", "what is lacking", "map")
The record's two tables in Table 7	This book	From the material of Parts 12 and 13, the table of next moves (seven rows in order of the prospect of closing) and the "order of the next work" in the list of tasks up to a proof (four rows). The content of the rows has been replaced with the language of the chapters of the book
Numbers and names of the ledger rows	This book	14-02 Table 2, Table 3

Terms used in this chapter — ledger row, the differences outside the ledger, what does not move even if settled, intermediate milestone, the lateral-widening inequality, frame decorrelation, the base point on a small T², β-monotonicity, the renormalisation comparison, the four faces, candidate rules, the diffusive-relaxation-time assumption. Their meanings are in the A-02 Glossary and the A-06 Index of assumptions.

Next chapter to read: 14-04 What is not claimed (a cross-index) — the reverse side of the questions: what this book does not show, does not read strongly, and does not call false either.

Revised 2026-10-15: first version.