2026-10-01 · chapter Yang–Mills and the mass gapPart 0 Entrancehow to read

The whole book　Contents and reading orders

How to read this book — read the strength of a claim from its label, choose an order from the map of prerequisites

What this chapter shows — How the book is put together (twelve parts and the appendices, 124 chapters) and the form shared by its chapters.
What the five labels (Lean, Paper, Computed, Known, Physics) guarantee and do not guarantee.
The map of prerequisites, why detours may be skipped, and five reading orders.

Prerequisite chapters — None. The overall picture is in the next chapter, 0-02, and a reading of the problem statement is in 0-03.

Said first — This book does not solve the problem of Yang–Mills existence and the mass gap. Neither a construction of the continuum limit nor a proof of the mass gap is here. The part with Assumption H as its axis (10-00 onwards) is a study, and does not claim a solution (10-16). The numbers in this chapter were computed from the "Prerequisites" column of the contents of 2026-10-09 (124 chapters), and change if the contents are revised.

Order of this chapter

What the book is — the one sentence of the problem and the roles of the parts
Readers and prerequisite knowledge — from second-year knowledge
The key to the labels — what they guarantee and do not guarantee (Table 1)
The form of a chapter — the three opening lines, numbering, notation, moving figures
The map of prerequisites — how many chapters a given chapter needs (Figure 1, Table 2)
How to skip detours — skipping them leaves no prerequisite missing (Detour: how many orderings do not break the prerequisites)
Suggested reading orders — entrance only, from the mathematical side, from the physical side, roads into Part 10 and Part 11 (Figure 2, Table 3)
What this book does not say — the side that has been shown, and the side the problem needs

What the book is — the one sentence of the problem and the roles of the parts

This book is for reading the Clay Mathematics Institute's prize problem "Yang–Mills existence and mass gap", from second-year university knowledge up to where research stands. The prize problems were announced on 24 May 2000 at the Collège de France in Paris Known, and the core of the official problem statement (the description by Arthur Jaffe and Edward Witten) is the following sentence Known.

Prove that for any compact simple gauge group $G$ , a non-trivial quantum Yang–Mills theory exists on $\mathbb R^4$ and has a mass gap $\Delta\gt0$ .

The mass gap means that the spectrum $\sigma(H)$ of the theory's Hamiltonian $H$ (the energy operator) is empty just above the vacuum energy 0.

\sigma(H)\subset\{0\}\cup[\Delta,\infty),\qquad \Delta\gt0 \tag{1}

Example. What is needed to reach (1) is how the parts are divided. Parts 1 and 2 are tools, Part 3 is the classical theory, Part 4 is quantisation and the exact definition of (1) (4-11), and Parts 5–7 are the lattice, constructive field theory and the physical view. Besides these, there is the dictionary part that sets the physical vocabulary beside the mathematical vocabulary to compare them (8-01 onwards), the part on where things stand with a timeline and open tasks (9-01 onwards), the part of study with Assumption H as its axis (10-00 onwards), the part of further considerations (11-00 onwards), and the appendices, such as the table of notation, the glossary and the guide to the literature (A-01 onwards). A word-by-word reading is in 0-03, and why four dimensions is special is in the existing articles What happens only in four dimensions and One sign is different, and in 0-04.

Readers and prerequisite knowledge — from second-year knowledge

The intended reader is a second-year student of mathematics or physics. What may be taken as known is multivariable calculus, the basics of linear algebra, complex numbers, ordinary differential equations, introductory mechanics and electromagnetism, and elementary probability. Other tools are prepared in Parts 1 and 2, and a tool not in the prerequisite chapters is defined, minimally, where it is used. If you can read the following three formulas, you can start from Part 1.

\int_{-\infty}^{\infty}e^{-x^2/2}\,dx=\sqrt{2\pi},\qquad e^{i\theta}=\cos\theta+i\sin\theta,\qquad \det\begin{pmatrix}-\lambda&1\\1&-\lambda\end{pmatrix}=\lambda^2-1 \tag{2}

Example. The Gaussian integral becomes the Gaussian measure that represents the free field (1-08), and $e^{i\theta}$ , as an element of the group $U(1)$ , becomes a gauge transformation of electromagnetism (2-03). The formula on the right finds the eigenvalues of the matrix $\bigl(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\bigr)$ , and the eigenvalues are $\pm1$ . If this matrix is read as a Hamiltonian, the difference 2 between the two eigenvalues is the energy gap of a two-level system, the simplest example of a gap (2-05). We recommend that readers from mathematics read Part 2 carefully, and that readers from physics read the second half of Part 1 (1-07–1-11) carefully.

The key to the labels — what they guarantee and do not guarantee

This book shows the strength of each claim, sentence by sentence, with a label. It does not mix "proved", "conjectured" and "supported numerically" in one sentence, and puts a label on any sentence that might mix them.

Table 1 The five labels.

Label	Meaning	What it does not guarantee
Lean	A statement machine-checked in Lean 4, given with its theorem name. The number of `sorry`, the marker for an omitted proof, is 0. `native_decide` (a shortcut that hands a computation to compiled code outside the checker) is not used. The axioms are only Lean's three standard ones (propositional extensionality `propext`, the axiom of choice `Classical.choice`, soundness of quotients `Quot.sound`). For details see A-04 and the Lean guide	That the premises are true
Paper	A proof exists on paper, but machine checking has not been done	That there are no errors
Computed	The range checked by computation on this machine	That it holds outside the range, or that it is a theorem
Known	Known theorems, their restatements, checks against outside literature	That it is a result of this site
Physics	Accepted in physics but not a theorem of mathematics (numerics, semiclassics, conjectures)	That a proof exists

In Part 10 these five labels are refined, to suit the material, into the five grades (proved here / cited / computer-assisted / reading / assumption) (10-00 §01). The correspondence is roughly: Paper → proved here or assumption; Computed → computer-assisted (interval arithmetic) or reading (floating point, Monte Carlo); Known → cited; Physics → reading or assumption. Lean is a column outside the labels, and the theorem name and the assumptions are written on the same row.

The label most easily misunderstood is Lean. What machine checking guarantees is that the conclusion follows logically from the premises, that is, the arrow.

P_1\wedge P_2\wedge\cdots\wedge P_k\ \Longrightarrow\ Q \tag{3}

The truth of the premises $P_1,\dots,P_k$ is outside the label. If the premises contain measured values or a provisionally chosen form of a function, the conclusion is conditional too, and this book states those premises alongside (10-13). Conversely, even a value re-derived on this machine gets the Known label if the value itself is known (the one-loop coefficient; 4-06).

Example. In this site's paper (5-11), the labels divide stage by stage within a single paper. The chain from the lattice action to a certain lower bound (an inequality containing the constant 27) is Lean, and the theorem names are given in 5-11. The stage that goes from there to a mass gap on the lattice is Paper, and only a sketch at that. That the constant 27 cannot be made smaller than 21 has been checked by exact arithmetic on specific configurations Computed, and is not a Lean theorem. Terms such as what the lower bound means are defined in 5-11. Even within one paper, the label differs from sentence to sentence.

The form of a chapter — the three opening lines, numbering, notation, moving figures

Every chapter goes in this order: the three opening lines (what it shows, prerequisites, what is said first), the order of the chapter, numbered sections, and the two closing tables (what it can say, sources and reproduction). Within a section the order is words, formulas, examples. Formulas, tables and figures are numbered afresh in each chapter, and symbols are defined at first appearance and kept consistent with the table of notation (A-01). For example, the action is written this way in every chapter.

S[A]=\frac{1}{2g^2}\int_{\mathbb R^4}\operatorname{tr}\bigl(F_{\mu\nu}F_{\mu\nu}\bigr)\,d^4x \tag{4}

$A$ is the gauge field, $F_{\mu\nu}$ its strength, $g$ the coupling constant, and an index appearing twice is summed from 1 to 4 (3-03). (4) is the physicists' way of writing (the convention of writing $A$ and $F$ as Hermitian matrices). It is related to the curvature $F_{\text{geom}}$ of the geometers' way of writing by $F_{\text{geom}}=-iF$ , and written that way the sign of $\operatorname{tr}F^2$ changes (the two ways of writing are in A-01). From Part 3 on we take $\hbar=c=1$ , and say so at the top of the chapter.

Example. Two meanings of the same letter are also stated at the top. $\beta$ is the inverse coupling $\beta=2N/g^2$ on the lattice (5-01), and the inverse temperature $\beta=1/k_BT$ in statistical mechanics (2-08). Moving figures are still at first and move with "Play"; without JavaScript they can be read from the table below the figure. Terms are in A-02.

The map of prerequisites — how many chapters a given chapter needs

The "Prerequisites" column of each chapter in the contents lists the chapters to read first. Prerequisites have prerequisites of their own, so the whole set of chapters a chapter needs is the set collected by following the arrows backwards. We call this the transitive prerequisites. If the direct prerequisites of chapter $c$ are $\mathrm{Pre}(c)$ , the transitive prerequisites $\mathrm{Anc}(c)$ and the number of chapters in the longest chain $\ell(c)$ are determined by the following recursion (for a chapter with no prerequisites, $\varnothing$ and 1).

\mathrm{Anc}(c)=\bigcup_{p\in\mathrm{Pre}(c)}\bigl(\{p\}\cup\mathrm{Anc}(p)\bigr),\qquad \ell(c)=1+\max_{p\in\mathrm{Pre}(c)}\ell(p) \tag{5}

If the arrows form no cycle, this recursion terminates. ComputedAmong the 124 chapters there are 166 prerequisite arrows, and no cycle. There are 14 chapters with no prerequisites. The order of the contents is arranged to respect the direction of the arrows: for every chapter, its prerequisite chapters come earlier in the contents (checked on the contents of 2026-10-09). So if you read straight through in the order of this version of the contents, no prerequisite is missing.

Figure 1 Choosing a chapter with the slider colours its transitive prerequisites and draws the longest chain in a thick line. Rows are parts (the last row is the appendices), columns are the order of the contents within a part, and squares are detour chapters. "Play" steps through the chapters in turn

Chapter

4-11 The definition of the mass gap | direct prerequisites 4-10, 2-08 | transitive prerequisites 12 chapters | longest chain 7 chapters: 1-01 → 1-10 → 2-05 → 2-07 → 4-02 → 4-10 → 4-11 | chapters that have this chapter as a prerequisite 28

ComputedThe numbers in this figure were computed on this machine. The figure follows (5) on the spot from the table of the "Prerequisites" column, and agrees with Table 2 (contents of 2026-10-09).

Example. ComputedTable 2 is (5) computed for some chapters. The longest chain is 19 chapters, and leads to "Because it is four dimensions" in Part 11 (11-07). It passes through Parts 1, 2, 4 and 5 into Part 10, and from the statement of the first step (10-05) goes via the integrated table (10-12) and what remains (10-15) to reach Part 11. The longest within Part 10 is 18 chapters, leading to 10-15 and 10-14; in Part 9 it is 14 chapters, leading to the list of open tasks (9-04).

ComputedTable 2 Prerequisites of representative chapters.

Chapter	Direct prerequisites	Transitive prerequisites	Longest chain
4-11 The definition of the gap	4-10, 2-08	12	7
5-09 What to hold fixed	5-05, 4-11	26	10
6-10 Where four dimensions stops	6-07	16	10
7-10 Back to four dimensions	7-09	13	11
9-04 The list of open tasks	9-02	25	14
10-12 The integrated table and the dependency graph	10-06, 10-07, 10-09, 10-11	38	17
10-15 What remains and its cost	10-12	39	18
11-07 Because it is four dimensions	11-01, 11-02, 10-03, 10-15, 4-07, 6-07	49	19

Counting in reverse shows the foundational chapters. Computed1-01 (linear algebra) is a transitive prerequisite of 103 chapters, 2-01 (the action) of 90, and 1-10 (Hilbert spaces) of 77. On the other hand, the Prerequisites column lists only chapters needed as tools. Part 3 (classical Yang–Mills) is not among the 16 transitive prerequisites of 6-10. The Prerequisites column is a lower bound on the reading order, not every chapter worth reading.

How to skip detours — skipping them leaves no prerequisite missing

There are two kinds of detour. The 6 chapters whose titles carry "Detour" in the contents (1-12, 2-09, 3-06, 4-12, 5-11, 6-12), and subsections inside a chapter whose headings begin with "Detour:". Both are written on the understanding that you can skip them and still go on. A detour subsection inside a chapter appears with its "Detour:" title in the "Order of this chapter" at the top of that chapter, so you can recognise it there and skip it. If the set of detour chapters is $Y$ , then Computedthe main-line chapters whose transitive prerequisites contain a detour are only 2-10, which has the two-dimensional Ising model (2-09) as a prerequisite, the chapters of Parts 10 and 11 that have 2-10 as a prerequisite, and 6-12.

\mathrm{Anc}(c)\cap Y\ \subseteq\ \{\text{2-09},\,\text{4-12},\,\text{5-11}\},\qquad \text{2-09}\in\mathrm{Anc}(c)\iff \text{2-10}\in\mathrm{Anc}(c)\cup\{c\},\quad \text{4-12}\in\mathrm{Anc}(c)\iff c=\text{6-12},\quad \text{5-11}\in\mathrm{Anc}(c)\iff \text{10-02}\in\mathrm{Anc}(c)\cup\{c\} \tag{6}

Example. 1-12, 3-06 and 6-12 are prerequisites of no chapter. 4-12 is a direct prerequisite only of 6-12, and 5-11 only of 10-02, so you read 5-11 first only when going on to 10-02. 2-09 is a prerequisite of 2-10 (the relaxation time and the dynamical exponent), and many chapters of Parts 10 and 11 have 2-10 as a prerequisite, so when going on to Part 10 read 2-09 first as well.

Detour: how many orderings do not break the prerequisites

Detour

A subsection that may be skipped.

An ordering that respects the prerequisite arrows is called a linear extension of the partial order. The order of the contents is one of them. The number of ways to merge two chains of lengths $a$ and $b$ while keeping their orders is $\binom{a+b}{a}$ , given by which of the $a+b$ seats go to the first chain Known. Looking at Part 1 with only the arrows inside the part, branches come out of 1-01 (the chain 1-02 → 1-04 and 1-05 → 1-06 join at 1-07, and separately 1-10 → 1-11), and apart from this there are 1-08 → 1-09 and 1-12. Merging the three lumps (9, 2 and 1 chapters), merging the two chains before 1-07, and merging 1-10 → 1-11 into those 6 chapters gives

e=\frac{12!}{9!\,2!\,1!}\cdot\binom{5}{2}\cdot\binom{8}{2}=660\cdot10\cdot28=184800 \tag{7}

ways. Example. ComputedCounting by dynamic programming over subsets, Part 1 agrees with (7), about $3.9\times10^{-4}$ of 12!. Part 2 has 7560 ways and Part 3 has 20. In Parts 2 and 3 every chapter has at most one prerequisite within the part, so each is a forest, and the forest formula $e=n!/\prod_c h(c)$ ( $h(c)$ is the number of $c$ together with the chapters in the part that have $c$ as a transitive prerequisite) Known can be used. The values of this formula also agreed with the dynamic programming. Part 1 is not a forest, because 1-07 has two prerequisites, and so it is counted with (7). The $\binom{a+b}{a}$ for two chains is the case of the forest formula where $h$ becomes $1,\dots,a$ and $1,\dots,b$ . For the 21 chapters of Parts 1 and 2 there are 44437771778400 ways, about $8.7\times10^{-7}$ of 21!.

Road	Target chapters $G$	Chapters	Chapters to add when going on to 9-02
Entrance only	1-12, 9-05, 10-16	7	— (next, to one of the two roads)
From the mathematical side	1-11, 3-04, 4-11, 5-03, 5-09, 6-02, 6-06, 6-09, 6-11, 9-01, 9-03	48	7-04, 7-05, 7-08, 7-09, 7-10, 7-13
From the physical side	3-05, 3-07, 4-07, 5-08, 7-02, 7-03, 7-10, 7-11, 7-13	39	2-06, 4-09, 4-10, 6-01, 6-05, 6-07, 6-10, 6-11
The road that follows the study in Part 10	10-00 → 10-01 → 10-03 → 10-04 → 10-05 → 10-12 → 10-15 → 10-16 (and their transitive prerequisites)	45	—
The road that follows Part 11	11-00 → 11-01 → 11-03 → 11-04 → 11-05 → 11-08 (and their transitive prerequisites)	49	—

What this book does not say — the side that has been shown, and the side the problem needs

This book does not solve the problem. Here is one point that is easily confused. In lattice gauge theory (Part 5), with the lattice spacing $a$ fixed, much has been shown on the side where the inverse coupling $\beta$ is small (strong coupling). What the problem needs is the opposite side.

\underbrace{\beta\ \text{small (strong coupling)}}_{\text{the side shown on the lattice}}\qquad\longleftrightarrow\qquad\underbrace{\beta\to\infty,\ \ a\to0}_{\text{the side the problem needs}} \tag{9}

Here is one sentence of this book. A proof of the mass gap is a proof of a lower bound. Trial states and finite measurements give only upper bounds on the gap; a lower bound needs an argument about all states (the box "The asymmetry of upper and lower bounds" in 1-10).

On the road that enters from the lattice, what has to be shown is that $\Delta=m_{\mathrm{lat}}/a$ , the mass in lattice units $m_{\mathrm{lat}}$ divided by $a$ , stays positive as $a\to0$ , and further that the limiting theory satisfies the axioms and is non-trivial (5-09, 10-01, 0-03).

Example. On the strong-coupling lattice, the area law for Wilson loops has been shown (Osterwalder–Seiler 1978) Known. A Wilson loop is an observable made by multiplying the parallel transports of the gauge field along a closed path, and the area law means that its expectation decreases exponentially in the area enclosed by the path. It is taken as a sign of quark confinement Physics (5-02). But the area law holds at strong coupling in every lattice gauge theory, including those with the abelian group $U(1)$ , so on its own it is not evidence of confinement in the non-abelian theory (5-06) Known. What has been shown on the strong-coupling side about the infinite-volume limit is treated in 5-04. The paper in the example of section 03 is also a result on the strong-coupling side, and claims nothing about the continuum limit.

The part of study with Assumption H as its axis (10-00 onwards) sets out the range checked by computation, computer assistance and Lean, together with the assumption rows, and also contains ideas not yet verified (10-17), distinguished by labels. The part of further considerations (11-00 onwards) gives views other than Assumption H and new views that take H as a premise. What is not claimed is in 10-16; how to read Lean is in the Lean guide and the Lean record; the papers are in the list of papers.

What this chapter can say

What is said	What is not said (where it is treated)
ComputedWhat was counted from the "Prerequisites" column of the contents of 2026-10-09 (124 chapters): 166 arrows, no cycles, the order of the contents respects the arrows, Table 2, (6), the number of linear extensions, no missing prerequisites in the sequences of Table 3	The Prerequisites column records dependencies at the design stage and does not necessarily capture the full content of the chapters (the numbers change if the contents are revised). That the five roads are the best (they are suggestions, not an optimisation). That detours inside chapters can be skipped is a commitment, and has not been computed
KnownThe one sentence of the problem, and the date and place of the announcement. The area law on the strong-coupling lattice, and that it is a result on the opposite side from the continuum limit	The construction of the continuum limit and the proof of the mass gap (unsolved; 6-10, 9-01). The infinite-volume limit on the strong-coupling side (5-04)
The meaning of the labels, and that the Lean label does not guarantee the truth of the premises (a commitment of this book)	The premises and theorem names of individual Lean statements (10-13, A-04; for the paper, 5-11)

Sources and reproduction

Item	Kind	Source / tool
The one sentence of the problem	Known	A. Jaffe, E. Witten, “Quantum Yang–Mills Theory” (the official problem statement PDF of the Clay Mathematics Institute). The sentence in §4 was checked against the original. The translation is in 0-03
Date and place of the announcement	Known	J. Carlson, A. Jaffe, A. Wiles (eds.), The Millennium Prize Problems, CMI / AMS, preface (p. viii): 'The prize was announced at a meeting on May 24, 2000, at the Collège de France.'. Checked via the source table of 0-03 (the PDF of the book published by the Institute)
How the labels divide in the paper	LeanPaperComputed (by stage)	This site's paper (5-11). The labels by stage were checked against the manuscript of the paper in the review of this book's material
The area law on the strong-coupling lattice	Known	K. Osterwalder, E. Seiler, “Gauge field theories on a lattice”, Ann. Phys. 110 (1978) 440–471. Bibliographic data only (attributing the area law to this paper rests on Chatterjee §4 in the next row)
That the area law holds at strong coupling in every lattice gauge theory, including those with $U(1)$	Known	S. Chatterjee, “Yang–Mills for probabilists”, arXiv:1803.01950 (2018) §4. The survey of this book's material checked the text of §4: 'the area law holds for any lattice gauge theory at sufficiently small β. A rigorous proof was given by Osterwalder and Seiler … this also implied that the area law at small β cannot be evidence for quark confinement'
The forest formula, $\binom{a+b}{a}$	Known	D. E. Knuth, The Art of Computer Programming, Vol. 3, Addison–Wesley, 1973, p. 70. Bibliographic data only (attributing the formula for linear extensions of forests to this page rests on the text of S. C. Billey, J. P. Swanson, arXiv:2010.12701 §1.3 and §2.3). $\binom{a+b}{a}$ is the case of two chains
Table 2, Table 3, (6), (7), the numbers in the text, the figures	Computed	Computed with python3 from the "Prerequisites" column of the contents of 2026-10-09 (separators ",", "、", "・"; the table of notation for Part 11, 11-A1, is not counted as a chapter). Linear extensions by dynamic programming over subsets, cross-checked with the forest formula and (7). The figures are traced on the spot by JavaScript in the browser from the same table

Next chapter: 0-02 One picture.

Revised 2026-10-01: first version.
2026-10-09: to match the reorganisation of Part 10 and the addition of Part 11, the map of prerequisites and the reading orders were recounted over 124 chapters (166 arrows, longest chain 19 chapters). The rows of Figures 1 and 2 were split by part.