2026-09-26 (revised 2026-10-09) · book Millennium Problemsquantum field theorygauge theory

Yang–Mills and the mass gap — from the beginner to where things stand

This is the Clay Mathematics Institute's Millennium Prize Problem "Yang–Mills existence and mass gap", set out in the form of a single textbook. It starts where a second-year mathematics or physics student can read, and goes as far as where the world stands and the range measured on this machine. There are 124 chapters in all, and 124 can be read now. You do not need to read them in order. The intended way of reading is to jump from the contents to the chapter you need, and to go back to a prerequisite chapter when a tool you do not know appears.

What the problem is — On four-dimensional space, construct Yang–Mills theory for a compact simple group in a form that satisfies the axioms of quantum field theory, and show that the mass of the lightest excitation is positive (0-03).

Where this book stands — This problem has not been solved. This site has not solved it either. What is here is an exact copy of the shape of the problem, a map of how far the mathematical side and the physical side have each come and where they let go of each other, and a record of measuring what can be measured on the lattice.

Readers — If you know multivariable calculus, linear algebra, complex numbers, introductory mechanics and electromagnetism, and elementary probability, you can read from Part 1 and Part 2. Any further tools are prepared inside this book.

LeanMachine-checked (Lean 4, sorry 0, no native_decide, axioms a subset of the three standard ones) PaperA proof exists, but machine checking is not finished ComputedThe range checked on this machine. Not made into a claim for outside use KnownKnown theorems, restatements, and checks against outside literature PhysicsAccepted at the level of physics but not a theorem of mathematics (numerics, semiclassics, conjectures)

Chapters marked "Detour:" can be skipped without affecting the main line. Each chapter states its "Prerequisite chapters" at the top. Chapters "in preparation" will be added in turn.

Part 0 Entrance — how to read this book

Ch.	Title	Contents	Prereq.
0-01	How to read this book	Readers, prerequisite knowledge, the key to the labels, how to use the contents, how to skip detours, suggested reading orders (five)	—
0-02	One picture	What is asked, and where things let go (a short version of the entrance page)	—
0-03	Reading the problem statement word by word	Clay's text, the group, R⁴, existence, non-triviality, the gap, short distances, notes	—
0-04	Four dimensions in mathematics and four dimensions in physics	A bridge to the existing `four-dimensions.html` and `spacetime.html`	—

Part 1 Mathematical preparation

Ch.	Title	Contents	Prereq.
1-01	Vector spaces and linear maps	Eigenvalues, the matrix exponential, Hermitian and unitary	—
1-02	What a group is	Symmetry, the rotation group SO(3), SU(2) and the double cover, a first look at representations	1-01
1-03	Lie groups and Lie algebras	The exponential map, commutators, structure constants, the Killing form	1-02
1-04	SU(2) and SU(3)	Pauli, Gell-Mann, the adjoint representation, the list of compact simple groups	1-03
1-05	Manifolds and tangent spaces	Coordinate-free differentiation, vector fields	1-01
1-06	Differential forms and the exterior derivative	Stokes, closed and exact, a first look at de Rham	1-05
1-07	Vector bundles and connections	Parallel transport, curvature, holonomy, the geometry of gauge transformations	1-04, 1-06
1-08	Probability and measure	Probability as a measure, expectation, independence, Gaussian measures, the law of large numbers and the central limit theorem	—
1-09	Probability in infinite dimensions	The absence of Lebesgue measure, Gaussian measures and Cameron–Martin, white noise	1-08
1-10	Hilbert spaces and self-adjoint operators	Spectrum, the lowest eigenvalue and the variational principle, the Poincaré inequality	1-01
1-11	Fourier analysis and distributions	Test functions, propagators, singularities, the difficulty with products	1-10
1-12	Detour: what happens only in four dimensions	What happens only in four dimensions (a guide to the existing page, and the parts that matter in this book)	—

Part 2 Physical preparation

Ch.	Title	Contents	Prereq.
2-01	Lagrangians and the action	Least action, Euler–Lagrange, Noether (symmetries and conservation laws)	1-01
2-02	The idea of a field	From the vibrating string to the wave equation, energy, infinitely many degrees of freedom	2-01
2-03	Electromagnetism is a gauge theory	Potentials, gauge transformations, U(1), Aharonov–Bohm	2-02
2-04	Special relativity and spacetime	One sign differs (a bridge to the existing `spacetime.html`), Lorentz-invariant actions	2-01
2-05	The bones of quantum mechanics	States, the Hamiltonian, the spectrum, the ground state, what "gap" means	1-10
2-06	The harmonic oscillator and creation and annihilation	The smallest unit of quantum field theory, Fock space	2-05
2-07	The idea of the path integral	Feynman, imaginary time, the match with statistical mechanics, Gaussian integrals	2-05, 1-08
2-08	Statistical mechanics and phase transitions	The partition function, correlations, the correlation length, criticality, universality	1-08
2-09	Detour: the gap and the correlation length in the Ising model	The "gap" and the correlation length seen in the Ising model (with computations on this machine)	2-08
2-10	The relaxation time and the dynamical exponent z	Langevin and Glauber dynamics, relaxation time τ=ξ² for the Gaussian chain (exact), C_P/ξ² ∼ L^{z−2} on a small band of the two-dimensional Ising model (computed), η_dyn=2−z is fixed here, known examples where the type of Assumption H (C_P≍ξ²) fails (z≠2), the heat bath and Langevin are not identified	2-08, 2-09

Part 3 Yang–Mills theory (classical)

Ch.	Title	Contents	Prereq.
3-01	The gauge principle	Demanding a local symmetry produces a force	2-03, 1-04
3-02	Non-abelian gauge fields	Yang–Mills 1954, the covariant derivative, the field strength, self-interaction	3-01
3-03	The action and the equations	tr F², Bianchi, self-duality, energy	3-02
3-04	Gauge theory as geometry	The space of connections, the gauge orbit space, Singer's theorem (the impossibility of a global gauge fixing)	1-07, 3-02
3-05	Instantons and topology	The second Chern number, BPST, the θ term	3-03
3-06	Detour: Donaldson and exotic R⁴	Donaldson and exotic R⁴ — classical Yang–Mills sees the topology of four dimensions	3-05
3-07	Yang–Mills inside the Standard Model	SU(3)×SU(2)×U(1), quarks, Higgs, how it differs from pure YM	3-02

Part 4 Quantisation

Ch.	Title	Contents	Prereq.
4-01	Canonical quantisation and the Hamiltonian	The Gauss constraint, physical states	2-05, 3-03
4-02	The path integral and Euclideanisation	Wick rotation, field theory as a probability measure, "spacetime" becomes "space"	2-07, 2-04
4-03	Perturbation theory and Feynman diagrams	Propagators, vertices, loops	4-02
4-04	Divergences and renormalisation	Why infinities appear, counterterms, the renormalisation group equation	4-03
4-05	The algebra of divergences	BPHZ, Epstein–Glaser, dimensional regularisation, Connes–Kreimer, Costello	4-04
4-06	Asymptotic freedom	The β function, 11N/3, the history of the discovery, consistency with the lattice	4-04
4-07	Dimensional transmutation and Λ	One scale, exp(−1/g²), quantities invisible to perturbation theory	4-06
4-08	Gauge fixing and Faddeev–Popov	BRST, Gribov copies, the 0/0 on the lattice (Neuberger)	4-03, 3-04
4-09	The axioms of quantum field theory	Wightman, Haag's theorem, representations of infinitely many degrees of freedom	1-10, 2-06
4-10	The Osterwalder–Schrader axioms and reconstruction	Reflection positivity is the key	4-02, 4-09
4-11	The definition of the mass gap	The gap in the spectrum, how it differs from particle states, clustering, the correlation length	4-10, 2-08
4-12	Detour: the perturbation series diverges	Perturbation series diverge — Dyson, renormalons, Borel, transseries	4-04

Part 5 Lattice gauge theory

Ch.	Title	Contents	Prereq.
5-01	Wilson's lattice	Link variables, plaquettes, the action, gauge invariance kept exactly	3-03, 1-04
5-02	The strong-coupling expansion	Small β, the area law, confinement, convergence	5-01
5-03	Reflection positivity and the transfer matrix	Osterwalder–Seiler, the lattice Hamiltonian	5-01, 4-10
5-04	The infinite-volume limit and the cluster expansion	What can be shown on the strong-coupling side	5-02
5-05	Weak coupling and the continuum limit	a(β), scaling, the lattice version of asymptotic freedom, Symanzik	5-01, 4-06
5-06	How U(1) differs from the non-abelian case	Guth, Fröhlich–Spencer, whether there is a phase transition, the exact form of "because it is non-abelian"	5-02
5-07	The Monte Carlo method	Updates, detailed balance, autocorrelation, errors, critical slowing down	5-01, 1-08
5-08	Measuring glueballs and the string tension	The variational method, m/√σ, numbers from primary sources	5-07
5-09	What to hold fixed when taking the limit	Problem 5.1, dimensionless ratios, the misreading of "a uniform gap"	5-05, 4-11
5-10	Lattice gauge theory with finite groups	Exponential decay at weak coupling, the absence of confinement	5-02
5-11	Detour: Bakry–Émery and one-link integration	Bakry–Émery and one-link integration (this site's paper) — an example of a theorem on the strong-coupling side	5-02, 1-10

Part 6 Constructive field theory

Ch.	Title	Contents	Prereq.
6-01	What constructive field theory is	Six stages	4-10
6-02	Solvability in two dimensions and the Yang–Mills measure	Driver, Gross–King–Sengupta, Lévy, reconstruction by stochastic quantisation	5-01
6-03	Constructing scalar fields	φ⁴₂, φ⁴₃, the Glimm–Jaffe road	6-01
6-04	The cluster expansion	As a tool	6-03
6-05	The sequence of renormalisation group steps	The ideas behind Bałaban's method (blocking, large fields, local gauges)	6-01, 4-04
6-06	Ultraviolet stability of three-dimensional Yang–Mills	What has been shown, and what is not yet constructed	6-05
6-07	Four dimensions	Bałaban and Magnen–Rivasseau–Sénéor — what "the level of control" means	6-05
6-08	Triviality of φ⁴ in four dimensions	The triviality theorem (Aizenman–Duminil-Copin) and the requirements for non-triviality	6-03
6-09	Stochastic quantisation and regularity structures	Hairer, CCHS, Chevyrev–Shen, subcritical and critical	6-01, 1-09
6-10	Where four dimensions stops	Stage by stage, the words of the problem's authors	6-07
6-11	Obstacles other than divergence	Six of them	6-10
6-12	Detour: "if 1/0 is given a symbol"	"If 1/0 were given a symbol" — the algebra of divergences and exp(−1/g²)	4-05, 4-12

Part 7 From the side of physics

Ch.	Title	Contents	Prereq.
7-01	How physics sees it	Asymptotic freedom, dimensional transmutation, the picture of confinement, the premise that "the theory exists"	4-07
7-02	Mechanisms of confinement	Dual superconductivity, monopoles, centre vortices, Gribov–Zwanziger	7-01
7-03	The string picture	Nambu–Goto, the Lüscher term, fluctuations of the tube, the Wulff shape	7-01
7-04	Large N	't Hooft, planar diagrams, random surfaces	4-03
7-05	String theory and extra dimensions	Where 26, 10 and 11 dimensions come from	7-04
7-06	How extra dimensions become invisible	Kaluza–Klein, Calabi–Yau, branes, experimental lower bounds	7-05
7-07	Where gauge fields come from	D-branes, heterotic strings, geometric engineering	7-06
7-08	Holography	A first look at AdS/CFT, radius = scale, correspondences that can be written exactly	7-05
7-09	The mass gap becomes geometry	Witten 1998, the cigar, Klebanov–Strassler, glueball numbers	7-08
7-10	Back to four dimensions	The order of limits, the non-separation of KK, N→∞, λ→∞, the distance from the problem statement	7-09
7-11	Supersymmetry and Seiberg–Witten	An example where duality makes the gap visible (at the level of physics)	7-08
7-12	The sentence that plays the role of an equivalence principle	One scale, Assumption H, the three chains	7-01, 4-07
7-13	Where the way physics digs parts from mathematics		7-10

Part 8 A dictionary of two languages — the physical and the mathematical view

Ch.	Title	Contents	Prereq.
8-01	How to read the dictionary	The degree of certainty of each correspondence is given in two columns — (a) the correspondence as a definition (settled = the same as a theorem / conditional / unsettled) and (b) its reality in four-dimensional Yang–Mills (constructed / conditional / absent). Which language to take as primary, and who judges that two things are "the same" (a theorem or a convention)	0-03
8-02	Dictionary A	The theory itself — path integral / Feynman–Kac, functional integral / Euclidean measure, vacuum / ground state / uniqueness of the measure, particle / isolated eigenvalue of the mass operator, S-matrix / Haag–Ruelle, Wick rotation / OS reconstruction, unitarity / reflection positivity, Green functions / Wightman–Schwinger, gauge fixing / a section of the orbit space, spacetime / R⁴ and the lattice, regularisation / the lattice spacing, Hamiltonian / transfer matrix, physical states / reconstruction from gauge invariants, renormalisability / BPHZ, field / operator-valued distribution, Källén–Lehmann / the spectral measure of the transfer matrix, Elitzur's theorem	4-02, 4-10
8-03	Dictionary B	Mass and length — the mass gap / exponential decay of correlations, the correlation length / inverse mass, V(R) / Wilson loops, the string tension / the coefficient of the area law, three ways of saying confinement (V→∞, σ>0, centre symmetry), Λ and dimensional transmutation / one scale, glueballs / one-particle states (reality unsettled), two times (the Hamiltonian H_T of the transfer matrix and the Fokker–Planck generator L of Langevin, gaps 1/ξ and 1/C_P), autocorrelation / the Poincaré constant (equality only for reversible continuous-time dynamics, HMC is separate), the conventional theory of critical slowing down (van Hove, z=2−η_s) and the RG (Halperin–Hohenberg–Ma 1972), the lower bound z≥γ/ν, static η_s and dynamic η_dyn, m/√σ / the limit of dimensionless ratios (Problem 5.1), the Lüscher term / effective strings, roughening / Wulff, freezing of topological charge / bottlenecks in the state space	4-11, 5-08, 2-08
8-04	Dictionary C	Scale and limits — the renormalisation group / a sequence of steps, the β function / the flow of the effective coupling, the continuum limit / the scaling limit, universality / uniqueness of the limit, a(β) (the ε(β) of Problem 5.2 is existentially quantified over Wilson loops), the distinction between scaling and asymptotic scaling, Symanzik O(a²) (the expansion variable differs from that of δ(b)), the Δβ of MCRG and the scheme change of Λ (Dashen–Gross), Migdal–Kadanoff / hierarchical lattices, large N / planar graphs, the master field, the holographic radius / the RG scale, topological charge / the second Chern number (admissible), instantons / self-dual connections, Wilson flow / gradient flow, the KK tower / the first eigenvalue of the internal Laplacian, Polchinski's ERG / multiscale Bakry–Émery, the Gribov region / convexity of the positive domain of the FP operator, the clash between improved actions and reflection positivity, the trace anomaly (no counterpart word on the mathematical side)	4-06, 6-05, 7-08
8-05	The physical counterparts of our assumptions	as questions and analogies): Assumption H is the slowest relaxation of Langevin and is exact for the free field — does it have a name on the physical side? / Is H↓ the same thing as the lower bound "local dynamics cannot relax faster than the susceptibility" (z≥γ/ν = 2−η_s)? / Is H↑ ⟺ z≤2 a restatement under the dynamic scaling hypothesis (the HH hypothesis only states that z exists)? / The η of the record refers to two quantities (the static η_s of §1 and the dynamic η_dyn of §6.27), and 8-05 fixes it to one / Is the summability of δ(b) closer to the Δβ and scheme change of MCRG than to Symanzik? / Is (S4)⟺Σδ∧η=0 the combination of "mass is RG-invariant" and "z=2"? / Is "one scale ⟺ the extra dimensions are visible" an analogy with the non-separation of KK? / What is settled as known (the correction of direction = Problem 5.1, κ=4Nb₀, Migdal–Kadanoff, (S2)) is written as such. Our side's definitions are cited from 7-12 and 10-03 (10-03 after renumbering), and detailed derivations are left to later parts (this does not go against the order of the contents	8-03, 8-04, 7-12

Part 9 Where things stand, and open tasks

Ch.	Title	Contents	Prereq.
9-01	How far the world has come	A timeline, and a table by dimension	6-10
9-02	Where the two ways of digging part	The seven seams	6-11, 7-13
9-03	Candidates for new concepts	What plays the role of "Ricci flow"	6-09
9-04	The list of open tasks	What would move if what were known	9-02
9-05	Common misconceptions	"Infinities are the centre", "two and three dimensions are done", "because it is non-abelian", "the mass of a particle"	0-03

Part 10 Towards a proof of the mass gap — building the first step of the staircase

Ch.	Title	Contents	Prereq.
10-00	Entrance — the staircase, and the random walk, the ruler and the clock	A map of the part, the four steps of the staircase, the five quantities as metaphors, Assumption H in one sentence [Y2]	2-10, 4-11, 2-07
10-01	The line of construction and Assumption H	(S1)–(S8), the direction of the lattice version, the three C_P, Assumption H in six stages (the statement, what it gives, what it does not give, the one-sided bridge) [Y2]–[Y3]	10-00, 5-09
10-02	Roads tried on the four-dimensional lattice, and walls	One step of bookkeeping, the seventeen roads G0–G15′, the twenty-one walls GA1–GE4, why we go down to the first step [Y3]	10-01, 5-11
10-03	The four steps of the staircase — why we start from 0+1 dimensions	The definition of the four steps, the conditions on the model of the first step, the power form of H, what lifts [Y2]	10-01
10-04	The path measure of the double well, and kinks	The path measure, ξ and the tunnelling splitting, kinks and the diffusion coefficient, the lower bound C_P≥ξ² (Proposition L) [Y2]–[Y3]	10-03, 2-05, 2-07
10-05	The statement of the first step — four conditions and Proposition 4.5	The splitting of the variance, conditions (H1)–(H4), Proposition 4.5 as a single line, the dependency graph [Y3]	10-04
10-06	(H1) Local Poincaré within a phase	Proposition H1 (six nodes), the methods and costs between nodes [Res]	10-05
10-07	The map of (H2) — seven routes and the seam of the ring	The state of routes 1–7, the chain outside and the chain inside the ring [Y3]	10-05
10-08	What computer-assisted means — the conditions on the chain inside the window	Interval arithmetic, premises, an example of the chain outside the window, what is not said [Y2]–[Y3]	10-07
10-09	The sector framework and the three strands of route 5	Five stages, the conditions for closing, the three strands, the common wall [Res]	10-07
10-10	The list of detour plans — types of breakage	R1–R24, the line that killed it, seven types [Y3]–[Res]	10-09
10-11	(H3)(H4) and the constant of the core	Proposition H3, the product of boxes, Proposition H4, Proposition CORE3, the set of ends [Res]	10-05
10-12	The integrated table and the dependency graph	The 111-row table in the five grades, the numbering rules, that there is no ħ that fits all, the required set of leaves [Y3]	10-06, 10-07, 10-09, 10-11
10-13	Lean statements and the limits of machine checking	The list of bundles, the criterion of cleanliness, four limits, translating conventions [Y2]–[Y3]	10-12
10-14	Reconciliation — the numbers of the record and the known numbers	The distance from known theorems, checking the bibliography, positive controls, nineteen rows on the four-dimensional lattice [Y3]	10-12
10-15	What remains and its cost — the distance to a theorem	The rows and work needed for each of the two roads, estimates of cost, a ħ that fits all, the distance from the second step on [Y3]	10-12
10-16	What is not claimed	A table matching the fourteen items to the text, three meanings [Y2]	—
10-17	Insights and small questions	Twelve ideas and small questions, discarded hypotheses, Q1–Q17 [Y2]–[Y3]	10-12

Part 11 Further considerations — beyond Assumption H, and other views

Ch.	Title	Contents	Prereq.
11-00	Further considerations — entrance	Positioning, the sentences received from Part 10, a map of the nine chapters and what each chapter can say [Y2]	10-00, 10-01
11-01	One form that obstructs H — a nearly conserved global quantity	Five examples in one type, three strengths, two ways of counting the charge of a subregion [Y3]	10-01, 10-02, 2-10
11-02	Choosing the ruler and the coordinates again — the length at which conservation breaks	Three rulers, coordinates and Dirichlet forms, five points of the band for two-dimensional U(1), a two-dimensional test bed [Y3]	11-01, 10-01, 2-10
11-03	Two ways to repair H — the sector H and the local H	Definitions of the local H and the sector H, the ratio form and references, open boxes and closed boxes [Res]	11-01, 10-01, 10-11
11-04	Which H the one-sided bridge needs	Families local in time, the window bridge and Proposition W, thresholds and three references, a note on 0+1 [Res]	11-03, 10-01
11-05	Making time continuous — the Hamiltonian lattice and the vacuum measure	Proposition V (the gap = the Poincaré constant of the vacuum measure), three Poincaré constants, the Kogut–Susskind lattice [Y3]	11-04, 10-01, 2-05
11-06	Paying with rarity instead of convexity — renormalisation on groups and log-Sobolev	A lemma on the ceiling of convexity, the reversal of where things are convex and non-convex, tools for paying with rarity, a class of measures that pay with rarity (candidates), stochastic quantisation [Res]	11-02, 10-09, 10-10, 10-11
11-07	Because it is four dimensions — the dimensionless coupling and topology	Two points: the critical dimension and topology, the hurdles of the four steps, why the mathematics of the third step does not exist, physical mechanisms [Y3]	11-01, 11-02, 10-03, 10-15, 4-07, 6-07
11-08	Questions, and what is not claimed	The table of questions F1–F19, a list of candidate rules, the sixteen items that are not claimed [Y2]–[Y3]	10-16
11-A1	Table of notation for Part 11	Symbols, where they are defined, units, correspondence with Part 10 (pages of reference)	10-01, A-01

Appendices

Ch.	Title	Contents	Prereq.
A-01	Table of notation		—
A-02	Glossary	1–3 lines each, links to chapters	—
A-03	Guide to the literature	Primary sources, reviews, textbooks (with their availability)	—
A-04	Lean and machine checking	A guide to `../lean-guide/`, and the labels used in this book	—
A-05	How this book was made	The verification procedure (for readers; the internal history is not written)	—
A-06	Index of assumptions	H, H↑, H↓, H_band[normalised], (P), T(c,1), (S1)–(S8), the chains (i)(ii)(iii), η_dyn, θ, δ, κ, ℰ_ϑ / ℰ_A, in a table of symbol, one-line meaning, chapter where it is placed, and status (assumption / algebra closed in Lean / known). The table "eight readings of Assumption H" (10-03) is reproduced	—

Reading orders — five of them

I want to know first what the problem is

0-02 One picture → 0-03 Reading the problem statement word by word → 0-04 Four dimensions in mathematics and four dimensions in physics → 9-02 The seven seams → 9-05 Common misconceptions. No tools are needed. By this point the "shape" of the problem is visible.

I want to dig from the mathematical side

Part 1 (only where needed) → Part 3 → 4-02, 4-10, 4-11 of Part 4 → Part 5 → Part 6 → 9-03 Candidates for new concepts. This follows the road of those who construct existence, in order from regularisation to the axioms.

I want to dig from the physical side

Part 2 (only where needed) → Part 3 → 4-06, 4-07 of Part 4 → 5-07, 5-08 of Part 5 → Part 7 → 9-02 The seven seams. The road of those who measure what exists, the limit that brings extra dimensions back to four, and as far as the sentence that plays the role of an equivalence principle.

I want to follow the study in Part 10

10-00 Entrance → 10-01 The line of construction and Assumption H → 10-03 The four steps of the staircase → 10-04 The double well → 10-05 The statement of the first step → 10-12 The integrated table → 10-15 What remains and its cost → 10-16 What is not claimed. With Assumption H as the axis, this splits the claim of the first step into conditions and follows it in order up to what remains and what is not claimed. The prerequisites are 2-05, 2-07, 2-10 of Part 2 and 4-11, 5-09, 5-11 of Parts 4 and 5.

I want to follow Part 11

11-00 Entrance → 11-01 A nearly conserved global quantity → 11-03 The sector H and the local H → 11-04 Which H the one-sided bridge needs → 11-05 The Hamiltonian lattice and the vacuum measure → 11-08 Questions, and what is not claimed. Reading 10-00 and 10-01 of Part 10 first makes the meaning of the received sentences clear.

What happens only in four dimensions — a collection of cases for Euclidean R⁴. Referred to from 0-04 and 1-12 of this book
One sign differs — the four dimensions of spacetime. Referred to from 2-04
An explanation of Lean — what machine checking is. Appendix A-04
Papers — one-link integration for lattice Yang–Mills (5-11)
Natural philosophy index

Revised 2026-09-26: first version. Set out the frame of the contents and the reading orders.
2026-10-09: Part 10 reorganised into eighteen chapters (the old 14 pages are kept as guide pages); added the nine chapters of Part 11 "Further considerations" and the table of notation. Added two reading orders.