2026-10-15 · chapter Yang–Mills and the mass gapPart 9 Where things stand, and open taskstable of tools, homes and uses, limits · 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
9-07 A map of the tools — where the known tools used by this book live
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)
The ultraviolet wall — the foundation beneath both roads
Part 15 Ultraviolet control — what it means to close at infinitely many steps
15-00 Entrance — what ultraviolet control is
15-01 Why the ultraviolet is hard — dimensionless coupling, group-valued fields, gauge
15-02 A map of the sources — Bałaban's programme, the BBS chain, two rows absent from the sources
15-03 The decomposition statement — a class that closes with finitely many parameters, and three remainders
15-04 The one-link heat-kernel class and the hierarchical model
15-05 The flux-network theorem — the transverse contraction and the parameter flow in one projection
15-06 Counting the parameters — couplings by loop shape and the attracting class
15-07 Background-dependent fluctuations η — absorption step by step
15-08 The four rows that do not close, and proposals for them
15-09 What cannot be said, the index of questions, the dependency diagram
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
A-08 私達の研究における証明までの道標

A map of the tools — where the known tools used by this book live

This chapter is a guide (a map of the tools). It lists the homes of the known tools used by this book's study, and where each stops.

What this chapter shows — The twenty-two tools that the work from Part 10 on uses (the Poincaré inequality, Bakry–Émery, log-Sobolev, reflection positivity, the strong-coupling expansion, the renormalisation step, Migdal–Kadanoff, one-loop perturbation theory, interval arithmetic, Lean and others) in one table. For each tool, with labels: (1) its home — the chapter of Parts 1–8 where its definition and known results are first set out, and the source; (2) how this book uses it — in which chapter from Part 10 on, and for what; (3) where it stopped when carried over to the group, and cautions. Also which chapter to read first when the same topic appears in two or more parts, three reading orders, the dependencies between the tools, and a list of the limits of the tools.

Prerequisite chapters — 9-06 The stages a proof must contain — the table of seven stages (the stages and the labels) and 0-01 §03, the key to the labels.

Said first — This is a guide chapter and makes no new claims. The claims in each cell of the table are copied from the chapters listed there, with those chapters' labels (the conditions are in the text of each chapter). The column "where it stopped" is not a claim that the tool cannot be used; it records where the way this book used it did not reach. This site shows neither the existence of the mass gap nor the construction of the continuum limit.

The order of this chapter

The role of this chapter — in five lines
The table of tools — home, use, where it stopped (Table 1), and how topics found in two places are divided (Table 2)
Three reading orders — from the tools, from the results, from the map
Dependencies between the tools — tools needed first, and tools used later (Table 3)
What cannot be said — a list of the limits of the tools

The role of this chapter

(1) The chapters from Part 10 on take tools from Parts 1–8 into "lattice gauge theory on a group" and use them there. This chapter lets you look up, in one table, where the definition and the known results of each tool are.

(2) When the same tool appears in two or more parts (for example, Bakry–Émery in 5-11, 11-06 and Part 10; Migdal–Kadanoff in 8-04, 13-08 and 15-04), it divides them into "home" and "use", and fixes one chapter to read first.

(3) It lists, from the side of the tools, where they stopped when carried over to the group. In which part's chapter each stopped can be looked up in the third column.

(4) The labels are the five grades, as in 9-06: Proved here = this book gives a proof on paper; Cited = known, checked against the source; Computer-assisted = computer-assisted (interval arithmetic); Reading = a reading of numbers, or a physical picture; Assumption = an assumption. The correspondence with the five labels of Parts 1–8 (Lean, paper, computed, known, physics) is in 0-01 §03.

(5) There are three ways to read (§02). Whether you come in from the tools, from the results or from the map, coming back to this table takes you to the home chapter.

The table of tools — home, use, where it stopped

Table 1 has one tool per row. The first column, "home", is the chapter where the definition and known results of the tool are first set out; it is the place a second-year student reads first. The second column gives the chapters from Part 10 on that use the tool, and what for. The third column gives where the tool stopped when carried over to a lattice on the group $SU(2)$ , and cautions for reading. Numbers and conditions are written on the same row; the detailed conditions are in the chapters listed.

Table 1 The tools used by this book's work (home, use, where it stopped; labels are per claim within a cell)
Tool	Home (Parts 1–8, source)	Use in this book (Part 10 on)	Where it stopped on the group, cautions
1 The Poincaré inequality and the gap The constant $C_P$ that bounds the variance by the squared gradient. The gap is $1/C_P$	1-10 §08 (on the line, $V''\ge\kappa\gt0$ gives $C_P\le1/\kappa$ ) Cited. The reading as a relaxation time is in 2-10	10-06 (local Poincaré within a phase = condition (H1)). 11-05 (on the lattice with continuous time, the gap $\Delta=\hbar^2/(2C_P(\nu))$ , with $\nu$ the vacuum measure) Proved here	The gap in Langevin time and the gap of the transfer matrix are different quantities (8-03). The identity of 11-05 is a rewording; there is no tool that bounds $C_P$ of the vacuum measure uniformly in the coupling
2 The Bakry–Émery criterion Curvature + second derivative of the potential $\ge K\gt0$ gives a gap $\ge K$	5-11 §02 (the criterion and the outline of its proof) Cited. 5-04 §08 uses only the conclusion	5-11 §05–§07 (integrate a perfect family first; curvature lower bound $2-27\beta^2$ ) Proved here. 10-02 (families on the four-dimensional lattice). 11-06 §02	On one link the criterion speaks only for $\kappa\lt2$ , while the true gap grows with $\kappa$ Reading. On the whole lattice $\beta\lt1/12$ Cited; integrating first, $\beta\lt\sqrt{2/27}\approx0.272$ (the strong-coupling side, lattice spacing fixed). The curvature of the group does not move with $\beta$ , so the threshold cannot exceed $O(1)$ . At weak-coupling scales convexity on the group itself fails (Lemma 1 of 11-06 §03) Proved here
3 The log-Sobolev inequality Bounds the entropy by the gradient. Stronger than Poincaré	5-11 §02 (the criterion also gives log-Sobolev) Cited. The language of measures is in 1-08	11-06 §06 (tools that pay with rarity instead of convexity; the entropy currency). 10-10 type 4	The arrangement "Poincaré is the currency of exit speed, and only the entropy currency can pay for rarity" is a picture Reading. "The class of measures that pay with rarity" is a candidate rule and promises no prospect of closing the third step Assumption
4 Brascamp–Lieb and the hierarchical induction An induction whose entry is that the conditional measure at each scale is uniformly convex	No home in Parts 1–8. The definition and the conditions of the source are in 11-06 §02 (Bauerschmidt–Bodineau) Cited	11-06 §05 (carrying the induction over to the group: what applies, what changes, what does not)	The convexity assumption cannot hold, in principle, on the group at weak-coupling scales Proved here. The non-convex region is the reverse of flat $\phi^4$ (on the group, the hemisphere far from the identity), and the wall is only pushed into a region of probability $e^{-c\beta}$ Proved here. Results for the hierarchical model imply nothing for four-dimensional Yang–Mills
5 Reflection positivity and the transfer matrix	5-03 (Osterwalder–Seiler), 4-10 (reconstruction) Cited	13-04 (positivity of the twist-summed ensemble z under reflections through lattice points) Proved here. 13-05 (input to the chessboard estimate). 9-06 stage 1	Transverse reflections break with the twist (13-04 Lemma 4) Proved here. The structural inputs (reflection positivity, duality, the transfer matrix, the centre identities) hold unchanged for $\mathbb Z_2$ and $U(1)$ , and this set alone does not give the widening inequality (the negative lemma; its last step inherits the Reading of 13-03)
6 The Kogut–Susskind Hamiltonian lattice Only space is a lattice; time is continuous	5-03, 4-01 Cited	11-05 §04 (a road that treats the gap directly). In the colour singlet the Poincaré constant of the vacuum measure halves at leading order Proved here	Local-to-global gap theorems do not apply as they stand on this lattice (reasons in 11-05 §04). For the Kogut–Susskind form used in 11-05, read 5-03 first
7 The strong-coupling and cluster expansions Add weak couplings as corrections, and guarantee convergence by a criterion	5-02, 5-04, 6-04 (the Kotecký–Preiss, Dobrushin and Fernández–Procacci conditions) Cited	13-06 (the strong-coupling end). 15-04 §05 (the exit of the chain in the hierarchical model = the strong-coupling decision region) Computer-assisted	They reach only the strong-coupling shore, with the lattice spacing fixed. They do not reach the band (intermediate coupling). How the strong-coupling ranges compare is in 5-11 §07
8 The character expansion and class functions Expand a weight on the group in characters of irreducible representations. Conjugation-invariant functions = class functions	5-02 (the character expansion), 6-02 (the heat kernel in two dimensions), 1-04 Cited	15-04 §01 (the one-link heat-kernel class is a semigroup and closes with one parameter) Proved here. 15-06 (couplings by loop shape and the attracting class)	It closes because in this model one weight returns to one weight in one step. Multi-link couplings, gauge re-fixing and large-field remainders are absent by construction, so nothing is implied for one step of the four-dimensional hypercubic lattice ("Said first" of 15-04)
9 Dobrushin's influence coefficient A criterion of the type: if the total influence of the neighbours on one site is below 1, correlations decay	6-04 (Dobrushin's condition, $4/27$ in the gas example) Cited	13-07 §03 (one site at the top in the first step). 13-05 §04. 13-10 face 2	At the top in the first step, the conditional one-site measure is $\tfrac{15}4$ -convex and the coupling to a neighbour is 2, so the influence from two neighbours is $\tfrac{16}{15}\gt1$ , back to the deficit $\tfrac{15}4$ versus $4$ Proved here. With the frame fixed, the continuous field follows the boundary and the influence coefficient is close to 1 Reading (physics; the original papers not checked)
10 The renormalisation-group step (blocking) Integrate out the fine variables to make the action at the coarse scale	6-05 (the sequence of steps, blocking, small and large fields) Cited. What has been reached is in 6-07	15-03 (the decomposition statement, the ledger of transverse contraction). 15-05. 15-07 (background-dependent fluctuations $\eta$ )	The numbers of 15-05 are for a step seen in the marginal of one coarse plaquette, not a true blocking step. The true two steps do not match two iterated steps (the contraction $\lambda$ is 3.4–3.7 times larger) Reading. Fixing all coarse plaquettes at once makes the contraction a matrix, exceeding $1/16$ in coarse directions (Theorem G; abelian configurations, tree level) Proved here. The multi-step induction for $\eta$ has not been written Assumption
11 Migdal–Kadanoff and the hierarchical model A recursion that approximates blocking by a map on one weight	8-04 Figure 1 (the approximate recursion and its character) Cited	15-04 §02–§05 (pulled to an invariant curve at rate $1/16$ per step, its position set by the balance of source and contraction, $1/3600$ : leading saddle-point term Proved here; invariance of the tube for $\beta\in[2.46,64]$ Computer-assisted). 13-08 road 30	The MK flow goes to strong coupling whether the group is abelian or not, and does not distinguish $U(1)$ from $SU(2)$ (Kanazawa) Cited. Tomboulis's interpolation relies on a common $t^*$ for the untwisted and twisted sectors; this reduces to inequality (3.5) of Ito–Seiler, not shown outside strong coupling Cited. The hierarchical model confines even for $U(1)$ and is no test of non-abelianness
12 One-loop perturbation theory and the coefficient 11/3 Gaussian fluctuations around a background field. $\tfrac{11}3=4-\tfrac13$	4-03, 4-06 (asymptotic freedom and the split of the coefficient) Cited	12-07 (isometric maps give only $\beta'\ge\beta+0.023$ ; maps that reshape the fluctuations reproduce $\beta-\beta'=0.258$ ) Proved here. 13-03, 13-06, 15-06	All are statements at the one-loop (Gaussian) level. Controlling the error of the reshaping maps at intermediate coupling and large fields is open. The $\tfrac{11}3$ at the head of those tables repeats 4-06; read 4-06 first for the split
13 Dimensional transmutation and $\Lambda$ A mass scale $\Lambda$ arises from a dimensionless coupling	4-07 Cited	11-07 (because it is four dimensions). 13-00, 13-07 (the position of the band, $L\sim1/\Lambda$ )	Placing the band at $L\sim1/\Lambda$ is a physical reading of scales Reading. Most of 11-07 is an arrangement of known material and can be read with Parts 4 and 6 alone
14 The Gibbs variational principle and relative entropy Transporting a measure by a map, the remainder in the bound on the free-energy difference is a relative entropy. Crooks and Jarzynski are of this type	2-08 (free energy, relative entropy) Cited	12-06 (dynamically driven insertion). 12-07 Proposition 1 (the principle of direction) Proved here	In a sandwich by fixed maps, the remaining relative entropy grows with the area of the vortex sheet, and the form $e^{-\sigma A}$ does not come out Reading (physics; C3 of 13-08). What can be used is the qualitative target $\Delta F(L,L)\to0$
15 Twisted boundary conditions, the centre and 't Hooft flux	3-05 (topology), 5-01 (the lattice), 1-04 (the centre) Cited. The definition of the twist itself is in 12-01 Cited	12-01 (three facts, including uniqueness of the vacuum that absorbs the twist) Proved here. 12-04, 13-01, 13-04	For groups with trivial centre ( $G_2$ and others) the arrows that use the centre disappear (9-06 stage 7) Cited. Parts 12 and 13 are written for $SU(2)$
16 Centre vortices and fractional instantons	7-02 (pictures of confinement), 3-05 (instantons) Cited	12-02 (definition and action, an upper bound on the minimal lattice action). 13-03 (a picture of where non-abelianness acts)	The dilute-gas count is a physical reading Reading. For the vortices of 13-03, read the definitions of 12-02 first
17 The container identity and the Tomboulis–Yaffe inequality Fixing the frame makes containers independent. From 't Hooft's criterion to the area law	The basis is 5-02, 5-04 (sums over surfaces). The identity is in 12-03 §03 (Mack–Petkova), the inequality in 13-01 Cited	12-03–12-05 (rewriting in the sector-summed ensemble, nesting) Proved here. 13-02 (reduction to a single inequality)	Placing containers of fixed length independently gives, in four dimensions, only $V(R)\gtrsim R/\ln R$ (Kovács–Tomboulis) Cited. The area law needs one container of the size of the loop, which is the same condition as crossing the band. That the sector bias $\theta$ moves away from 1 is not enough (13-01) Proved here
18 The flux network — Thomson's principle, Kirchhoff's laws, orthogonal projection	1-01 (linear maps and projections) Cited. Standard facts of electrical circuits are in the source table of 15-05	15-05 Theorem F (the minimiser is fixed by a projection; contraction $\lambda\le\max f^2$ , $1/(16d-8)\le\lambda\le1/16$ ) Proved here	Statements at tree level (leading term as $\beta\to\infty$ ) restricted to abelian configurations. They do not include non-abelian configurations, the one-loop source, large fields or gauge re-fixing
19 The BCH formula and quaternions The commutator series of $\log(e^Xe^Y)$ . $SU(2)$ is the quaternions of length 1	1-03 (BCH up to third order), 1-02, 1-04 Cited	5-11 (the quaternion inner product and one-link integration). 15-06 (commutators as images under BCH). 15-07 (the group-geometry part = the third-order BCH coefficient $1/9$ ; leading Gaussian term, two links) Proved here	For $SU(2)$ the group-geometry part rides on the background itself and does not contract, and its budget diverges whatever threshold is chosen (arithmetic treating the one-loop flow as exact) Proved here. For $U(1)$ the third-order BCH term is identically 0
20 Computer assistance — interval arithmetic, certificates, the discipline of double precision Hold every number as an interval containing the true value, rounding outward	No home in Parts 1–8. The general theory is in 10-08 §01–§03 (rounding conventions, holding intervals, positive and negative controls) Cited	10-07, 10-08 (the chain condition $q\lt1$ at every point of the $\hbar$ window) Computer-assisted. 15-04 §05 (the tube of the hierarchical model) Computer-assisted. 10-15	Inequalities have been enclosed by interval arithmetic in the first-step window and in the hierarchical model, a map on one weight (plus the upper bounds on the action of explicit configurations in 12-02 Computer-assisted, upper bounds only). Even in the hierarchical model, $\beta\gt64$ (the limit of zero lattice spacing) is not closed. The road of checking finitely many $L$ by interval arithmetic is out of computational reach in four dimensions (13-08 road 3) Reading. Computer assistance says nothing outside the window (10-08 §10)
21 Formalisation in Lean Statements as types, proofs as terms, checked by a small kernel	A-04 (kernel checking, the criterion of the three standard axioms) Cited	5-11 (the curvature lower bound and the combinatorics of families). 10-13 (the bundles of the first step). The Lean column of Table 1 of 9-06	What is guaranteed is only the arrow from the stated hypotheses to the stated conclusion. Physical measures are not formalised. The interval-arithmetic layer lies outside Lean (10-13 §04). The general theory is A-04; the list of bundles and examples is 10-13
22 Reading Monte Carlo Generate lattice configurations at random and measure averages	5-07, 5-08 Cited	12-08, 13-09 (the measured record) Reading. The numbers of 15-05 and 15-07 Reading	All measured numbers carry the "Reading" label and are not used as grounds for theorems. Lattice size, $\beta$ , seeds and errors are on the same row in each chapter. There are examples where the numbers move when the way of measuring changes (the true two steps and two iterated steps in 15-05)

What to read from Table 1. (a) Two tools have no home in Parts 1–8 (the hierarchical induction and computer assistance). For both, chapters from Part 10 on (11-06 §02, 10-08 §01–§03) double as the home. (b) The ways of stopping in the third column fall into types. The curvature and convexity tools (rows 2–4, 9) stop at the closed shape of the group and at the single site at the top; the renormalisation tools (rows 10, 11, 19) stop where one step is stacked into many; the structural tools (rows 5, 17) stop where they cannot be told apart from $U(1)$ . (c) Computer assistance and Lean (rows 20, 21) both say nothing outside what was written.

How topics found in two or more places are divided

When the same tool appears in several parts, read the "home" of Table 2 first and the "use" after the home. The home chapter alone has the definitions and the known results.

Table 2 Topics found in two or more places (home = read first, use = read later)
Topic	Home	Use	Row of Table 1
The Bakry–Émery criterion	5-11 (the criterion and the one-link example)	11-06 §02–§03 (why it fails on the group at weak coupling), 10-02, 10-06	2
Migdal–Kadanoff	8-04 Figure 1 (the character of the approximation)	15-04 (invariant curve and interval arithmetic), 13-08 road 30 (Tomboulis's claim and its critiques)	11
The Kogut–Susskind lattice	5-03	11-05 (the gap and the vacuum measure)	6
Reflection positivity and the transfer matrix	5-03	13-04, 13-05 (checked for $\mathbb Z_2$ and $U(1)$ )	5
The one-loop coefficient $\tfrac{11}3$	4-06	12-07, 13-03, 13-06	12
Dimensional transmutation	4-07	11-07, 13-00, 13-07	13
Vortices (centre vortices, fractional instantons)	The picture in 7-02; definition and action in 12-02	13-03 (a picture)	16
Renormalisation steps and Bałaban's series	6-05, 6-07 (what has been reached)	15-02 (inside the series and the BBS chain), 15-03	10
Interval arithmetic	10-08 §01–§03	10-07, 10-15, 15-04	20
Lean	A-04 (the general theory)	10-13 (the bundles of the first step and two examples of the types)	21

Three reading orders — from the tools, from the results, from the map

In all three, going back to Table 1 along the way to look up the home chapter lets you read without missing prerequisites. They can be used alongside the reading orders of 0-01 §07.

Three reading orders (read in the order of the arrows; [ ] are rows of Table 1)
Way in	Reading order	What you know when you finish
From the tools readers from the mathematical side	1-10 → 5-03 → 5-11 → 6-04 → 6-05 → 8-04 → 10-08 §01–§03 → A-04 → 11-06 → 15-04 → 15-05 [1, 2, 5, 7, 10, 11, 20, 21, 3, 4, 18]	What each tool needs, and where on the lattice on the group it stopped. The three types of stopping ((b) after Table 1)
From the results from this book's conditional statement	9-06 → 14-01 → 14-02 → 12-04 → 13-02 → 15-03, going back from the "use" column of Table 1 to the home chapters as you read	Which tools each arrow of the conditional statement relies on. How the rows of the ledger of conditions correspond to the tools
From the map seeing the whole of what is open first	0-02 → 9-02 → 9-04 → 9-06 → this chapter → 13-08 → 15-08 → 14-04	Which of the seven stages are open, which tools stopped where, and what this book does not claim

Dependencies between the tools — tools needed first, and tools used later

Table 3 connects the tools of Table 1 by "tools needed first". The numbers are rows of Table 1. A tool needed first may have a later row number, but there are no cycles. Following this column from the left fixes the order in which to read the home chapters.

Table 3 Dependencies of the tools (this book's arrangement; "used together" is where they are combined from Part 10 on)
Tool (row of Table 1)	Tools needed first	Where used together
1 Poincaré inequality	— (linear algebra and spectra of 1-10)	The form of the conclusions of 2 and 3. The identity of 11-05
2 Bakry–Émery	1, 19 (the curvature of the group comes from $SU(2)$ being the three-sphere)	Gives 3. Entry to 4. Combined with 21 in 5-11
3 log-Sobolev	2	The tools of 11-06 that pay with rarity
4 Hierarchical induction	2, 10	Compared with 11 in 11-06 §05
5 Reflection positivity	— (the axioms of 4-10)	The chessboard of 13-05. With 15 and 17, the negative lemma of 13-04
6 Kogut–Susskind lattice	5	Combined with 1 in 11-05
7 Strong-coupling expansion	8	The exit of the chain in 15-04. A criterion of the same type as 9
8 Character expansion and class functions	19	Combined with 10 and 11 in 15-04 and 15-06
9 Dobrushin's influence coefficient	7	Set beside 2 in 13-07 (back to the same deficit)
10 Renormalisation step	12	The frame for 11 and 18. The decomposition of 15-03
11 Migdal–Kadanoff	10, 8	Combined with 20 in 15-04. Road 30 of 13-08
12 One loop	— (perturbation theory of 4-03)	Combined with 14 in 12-07. 13-06, 15-06
13 Dimensional transmutation	12	The reading of the band's position (13-07)
14 Gibbs variational principle	— (2-08)	Combined with 12 in 12-06 and 12-07
15 Twist, centre, flux	— (3-05, 5-01)	The stage for 16 and 17. 13-04
16 Vortices	15	12-02, 13-03
17 Containers and Tomboulis–Yaffe	15, 7	12-03–12-05, 13-02
18 Flux network	10, 11 (comparison with $1/16$ )	Theorems F and G of 15-05
19 BCH and quaternions	— (1-03, 1-04)	The group-geometry part of 15-07
20 Interval arithmetic	— (10-08 §01–§03)	In 10-07, 10-08 and 15-04, encloses inequalities for maps such as 11
21 Lean	— (A-04)	Checks the algebraic steps of 2, 5, 17, 19 and others. The output of 20 is placed as a hypothesis
22 Monte Carlo	— (5-07)	The numbers of 12-08, 13-09, 15-05, 15-07 (Reading)

What to read from Table 3. The rows with no tool needed first (1, 5, 12, 14, 15, 19, 20, 21, 22) are the homes a second-year student can read first. Of these, 20 and 21 have their homes in Part 10 and the appendices, so read them first following the division of Table 2.

What cannot be said — a list of the limits of the tools

The limits of each tool, taken one line at a time from the third column of Table 1. What is not claimed, part by part, is in 14-04 What is not claimed (a cross-index); the limits of the ultraviolet tools of Part 15 are in 15-09 What cannot be said.

The curvature tools (Bakry–Émery, Brascamp–Lieb, Dobrushin) work only on the strong-coupling side with the lattice spacing fixed, and in the first-step window. At weak-coupling scales convexity on the group itself fails (11-06 §03), and at the top in the first step they return to the same deficit $\tfrac{15}4$ versus $4$ (13-07 §03).
The strong-coupling and cluster expansions are tools of the strong-coupling shore and do not cross the band (intermediate coupling).
The statements about one renormalisation step (15-04, 15-05, 15-06) are each restricted to a map on one weight, the marginal of one coarse plaquette, or tree level on abelian configurations; none of them stacks a true step of the four-dimensional hypercubic lattice into many (15-09).
The Migdal–Kadanoff flow and the hierarchical model do not distinguish $U(1)$ from $SU(2)$ . Closing there is no test of non-abelianness.
The structural inputs alone (reflection positivity, duality, the transfer matrix, the centre identities) do not give the widening inequality (the negative lemma of 13-04, in the form "this set of inputs alone").
Comparisons by one-loop tools and by the Gibbs variational principle are statements at the one-loop (Gaussian) level, not non-perturbative bounds.
Interval arithmetic says nothing outside the intervals written. Lean guarantees only the arrow from the stated hypotheses to the stated conclusion, and physical measures are not formalised (10-13, A-04).
Monte Carlo numbers are readings and are not used as grounds for theorems.
No combination of the tools in this table has shown the existence of the mass gap or the construction of the continuum limit (14-04).

What this chapter can say

Can say	Cannot say
Cited The known results in the "home" column of Table 1 (the extent of the sources is in each chapter's source table)	That the twenty-two of Table 1 exhaust the tools of this book. The division into tools is this book's arrangement
Proved here Computer-assisted Reading Assumption The claims in the second and third columns of Table 1 (with the labels and conditions of the chapters listed)	That "where it stopped" in the third column is a limit of the tool in principle (it records the range of this book's use)
The division of Table 2 and the dependencies of Table 3 (this book's arrangement)	That the dependency table exhausts all logical implications
Overall	This site shows neither the existence of the mass gap nor the construction of the continuum limit

Sources and reproduction

Item	Kind	Source, tools
The claims and numbers in each row of Table 1	As labelled in each chapter	The text and source tables of the chapters listed on the row (this chapter copies them and does no new computation)
The Bakry–Émery criterion	Cited	D. Bakry, M. Émery, Diffusions hypercontractives, Séminaire de Probabilités XIX (1985) 177–206 (bibliographic data in the source table of 5-11)
The whole-lattice threshold $1/12$	Cited	H. Shen, R. Zhu, X. Zhu, Commun. Math. Phys. 400 (2023) 805–851 (the source table of 5-11)
The assumption of the hierarchical induction	Cited	Bauerschmidt–Bodineau (the source table of 11-06)
Tomboulis's interpolation and its critiques	Cited	E. T. Tomboulis, arXiv:0707.2179. K. R. Ito, E. Seiler, arXiv:0711.4930 ((3.5)). T. Kanazawa, arXiv:0805.2742 (the source table of 13-08)
The container identity and $R/\ln R$	Cited	Mack–Petkova (1980), Kovács–Tomboulis hep-lat/0108017 (the source table of 12-03)
Thomson's principle, Kirchhoff's laws	Cited	Standard facts of electrical circuits (the source table of 15-05)
Dobrushin's condition	Cited	The source table of 6-04. The original papers of the criteria in 13-05 are not checked
Table 2, Table 3, the three reading orders	This book	Arranged from the prerequisite paragraphs and the cross-references in the text of the chapters (no numerical computation)

Words used in this chapter — home (the chapter where a tool's definition and known results are first set out), use, where it stopped, the Bakry–Émery criterion, the log-Sobolev inequality, the hierarchical model, reflection positivity, class function, influence coefficient, blocking, Migdal–Kadanoff, the flux network, BCH, interval arithmetic, certificate. Their meanings are in A-02 Glossary.

Next chapter to read: 10-00 Entrance — the first road (from the direction of time). If you read from the tools, 5-11 Detour: Bakry–Émery and one-link integration.

Revised 2026-10-15: first version.