computo ergo sum

2026-10-15 · chapter Yang–Mills and the mass gapPart 14 Closing — conditional statementsentrance, promise, map of four chapters · Kiichi, Shiori and Rin

The whole book　Contents and reading orders
  1. Part 0 Entrance — how to read this book
  2. 0-01 How to read this book
  3. 0-02 One picture
  4. 0-03 Reading the problem statement word by word
  5. 0-04 Four dimensions in mathematics and four dimensions in physics
  6. Part 1 Mathematical preparation
  7. 1-01 Vector spaces and linear maps
  8. 1-02 What a group is
  9. 1-03 Lie groups and Lie algebras
  10. 1-04 SU(2) and SU(3)
  11. 1-05 Manifolds and tangent spaces
  12. 1-06 Differential forms and the exterior derivative
  13. 1-07 Vector bundles and connections
  14. 1-08 Probability and measure
  15. 1-09 Probability in infinite dimensions
  16. 1-10 Hilbert spaces and self-adjoint operators
  17. 1-11 Fourier analysis and distributions
  18. 1-12 Detour: what happens only in four dimensions
  19. Part 2 Physical preparation
  20. 2-01 Lagrangians and the action
  21. 2-02 The idea of a field
  22. 2-03 Electromagnetism is a gauge theory
  23. 2-04 Special relativity and spacetime
  24. 2-05 The bones of quantum mechanics
  25. 2-06 The harmonic oscillator and creation and annihilation
  26. 2-07 The idea of the path integral
  27. 2-08 Statistical mechanics and phase transitions
  28. 2-09 Detour: the gap and the correlation length in the Ising model
  29. 2-10 The relaxation time and the dynamical exponent z
  30. Part 3 Yang–Mills theory (classical)
  31. 3-01 The gauge principle
  32. 3-02 Non-abelian gauge fields
  33. 3-03 The action and the equations
  34. 3-04 Gauge theory as geometry
  35. 3-05 Instantons and topology
  36. 3-06 Detour: Donaldson and exotic R⁴
  37. 3-07 Yang–Mills inside the Standard Model
  38. Part 4 Quantisation
  39. 4-01 Canonical quantisation and the Hamiltonian
  40. 4-02 The path integral and Euclideanisation
  41. 4-03 Perturbation theory and Feynman diagrams
  42. 4-04 Divergences and renormalisation
  43. 4-05 The algebra of divergences
  44. 4-06 Asymptotic freedom
  45. 4-07 Dimensional transmutation and Λ
  46. 4-08 Gauge fixing and Faddeev–Popov
  47. 4-09 The axioms of quantum field theory
  48. 4-10 The Osterwalder–Schrader axioms and reconstruction
  49. 4-11 The definition of the mass gap
  50. 4-12 Detour: the perturbation series diverges
  51. Part 5 Lattice gauge theory
  52. 5-01 Wilson's lattice
  53. 5-02 The strong-coupling expansion
  54. 5-03 Reflection positivity and the transfer matrix
  55. 5-04 The infinite-volume limit and the cluster expansion
  56. 5-05 Weak coupling and the continuum limit
  57. 5-06 How U(1) differs from the non-abelian case
  58. 5-07 The Monte Carlo method
  59. 5-08 Measuring glueballs and the string tension
  60. 5-09 What to hold fixed when taking the limit
  61. 5-10 Lattice gauge theory with finite groups
  62. 5-11 Detour: Bakry–Émery and one-link integration
  63. Part 6 Constructive field theory
  64. 6-01 What constructive field theory is
  65. 6-02 Solvability in two dimensions and the Yang–Mills measure
  66. 6-03 Constructing scalar fields
  67. 6-04 The cluster expansion
  68. 6-05 The sequence of renormalisation group steps
  69. 6-06 Ultraviolet stability of three-dimensional Yang–Mills
  70. 6-07 Four dimensions
  71. 6-08 Triviality of φ⁴ in four dimensions
  72. 6-09 Stochastic quantisation and regularity structures
  73. 6-10 Where four dimensions stops
  74. 6-11 Obstacles other than divergence
  75. 6-12 Detour: "if 1/0 is given a symbol"
  76. Part 7 From the side of physics
  77. 7-01 How physics sees it
  78. 7-02 Mechanisms of confinement
  79. 7-03 The string picture
  80. 7-04 Large N
  81. 7-05 String theory and extra dimensions
  82. 7-06 How extra dimensions become invisible
  83. 7-07 Where gauge fields come from
  84. 7-08 Holography
  85. 7-09 The mass gap becomes geometry
  86. 7-10 Back to four dimensions
  87. 7-11 Supersymmetry and Seiberg–Witten
  88. 7-12 The sentence that plays the role of an equivalence principle
  89. 7-13 Where the way physics digs parts from mathematics
  90. Part 8 A dictionary of two languages — the physical and the mathematical view
  91. 8-01 How to read the dictionary
  92. 8-02 Dictionary A
  93. 8-03 Dictionary B
  94. 8-04 Dictionary C
  95. 8-05 The physical counterparts of our assumptions
  96. The map
  97. Part 9 Where things stand, and open tasks
  98. 9-01 How far the world has come
  99. 9-02 Where the two ways of digging part
  100. 9-03 Candidates for new concepts
  101. 9-04 The list of open tasks
  102. 9-05 Common misconceptions
  103. 9-06 The stages a proof must contain — the table of seven stages
  104. The first road — from the direction of time
  105. Part 10 The first road, first half — from the direction of time: the mass gap and the diffusive-relaxation-time assumption
  106. 10-00 Entrance — the staircase, and the random walk, the ruler and the clock
  107. 10-01 The line of construction and Assumption ξ²
  108. 10-02 Roads tried on the four-dimensional lattice, and walls
  109. 10-03 The four steps of the staircase — why we start from 0+1 dimensions
  110. 10-04 The path measure of the double well, and kinks
  111. 10-05 The statement of the first step — four conditions and Proposition 4.5
  112. 10-06 (H1) Local Poincaré within a phase
  113. 10-07 The map of (H2) — seven routes and the seam of the ring
  114. 10-08 What computer-assisted means — the conditions on the chain inside the window
  115. 10-09 The sector framework and the three strands of route 5
  116. 10-10 The list of detour plans — types of breakage
  117. 10-11 (H3)(H4) and the constant of the core
  118. 10-12 The integrated table and the dependency graph
  119. 10-13 Lean statements and the limits of machine checking
  120. 10-14 Reconciliation — the numbers of the record and the known numbers
  121. 10-15 What remains and its cost — the distance to a theorem
  122. 10-16 What is not claimed
  123. 10-17 Insights and small questions
  124. Part 11 The first road, second half — repairing the form of the assumption, making time continuous
  125. 11-00 Further considerations — entrance
  126. 11-01 One form that obstructs Assumption ξ² — a nearly conserved global quantity
  127. 11-02 Choosing the ruler and the coordinates again — the length at which conservation breaks
  128. 11-03 Two ways to repair Assumption ξ² — the sector Assumption ξ² and the local Assumption ξ²
  129. 11-04 Which Assumption ξ² the one-sided bridge needs
  130. 11-05 Making time continuous — the Hamiltonian lattice and the vacuum measure
  131. 11-06 Paying with rarity instead of convexity — renormalisation on groups and log-Sobolev
  132. 11-07 Because it is four dimensions — the dimensionless coupling and topology
  133. 11-08 Questions, and what is not claimed
  134. 11-A1 Table of notation for Part 11
  135. The second road — from the direction of space
  136. Part 12 The second road: from the direction of space — a small twisted T² and confinement
  137. 12-00 Entrance — space, not time
  138. 12-01 The twisted T² and 't Hooft flux
  139. 12-02 Vortices and fractional instantons
  140. 12-03 Containers — conditioning on the frame makes things independent
  141. 12-04 Slab confinement
  142. 12-05 Vortex-insertion identities
  143. 12-06 Driven insertion and the flow lemma
  144. 12-07 Why fixed maps fail
  145. 12-08 The measured record
  146. 12-09 Where this road stops — widening the T² is the band
  147. The crossover band — two roads reach the same wall
  148. Part 13 The crossover band — two roads reach the same wall
  149. 13-00 Entrance — confluence
  150. 13-01 The correct form of the band
  151. 13-02 Reduction to a single inequality
  152. 13-03 The only place where non-abelian enters
  153. 13-04 Structure alone cannot cross
  154. 13-05 The chessboard and simultaneous flips
  155. 13-06 Renormalisation comparison — a comparison that distinguishes groups
  156. 13-07 The band seen from the side of time
  157. 13-08 A map of the roads tried and stopped
  158. 13-09 The measured record
  159. 13-10 Candidate new viewpoints
  160. 13-11 Ultraviolet control — what it means to close at infinitely many steps
  161. Closing
  162. Part 14 Closing — conditional statements
  163. 14-00 Entrance — the promise of this part
  164. 14-01 The conditional statement — the mass gap and confinement, arrow by arrow
  165. 14-02 The ledger of conditions
  166. 14-03 The table of questions (a cross-index)
  167. 14-04 What is not claimed (a cross-index)
  168. Appendices
  169. A-01 Table of notation
  170. A-02 Glossary
  171. A-03 Guide to the literature
  172. A-04 Lean and machine checking
  173. A-05 How this book was made
  174. A-06 Index of assumptions
  175. A-07 Papers that came out of this book

Entrance — the promise of this part

What this chapter shows — The promise on how Part 14 is written, and the differences that remain even when the ledger is empty. The figure of the two roads meeting at the band, shown again, and a map of 14-01 to 14-04.

Prerequisite chapters — 9-06 (the table of seven stages). You can read this part without having finished Parts 10 to 13.

01

The promise of this part

Part 14 writes conditionally until the conditions disappear.

02

The two roads meet at the band (shown again)

Figure 1 9-06 Figure 1, shown again. On both roads, the arrow flowing into the band leaves only from the row of the four-dimensional goal. Stage 2 (ultraviolet) lies under both roads. Part 14 corresponds to the one box below the band
The two roads meet at the band Two roads branch off from the table of seven stages in Part 9. The first road (the direction of time, Parts 10 and 11) splits into two rows, what remains of the first step and the four-dimensional goal, and the arrow flowing into the band leaves only from the row of the four-dimensional goal. The second road (the direction of space, Part 12) has Stage 3, Stage 4 and the row of the goal, the slab tension uniform in L, and an arrow leaves the goal row for the band. An arrow leaves the band (Part 13) for Part 14. At the bottom sits the control of Stage 2, ultraviolet, which lies under both roads. Part 9 The map: seven stages of a proof (9-06 Table 1) The first road: from the direction of time Parts 10 and 11 — towards the mass gap What remains of step 1: uniform LSI for a non-convex path measure (0+1D, mini band) The 4D goal: ξ²↑ uniform in β, and the bridge (a statement that crosses the band) The second road: from the direction of space Part 12 — from a small twisted T² Stage 3: the gap on a small T² Stage 4: confinement on a small T² Goal: the slab tension uniform in L (a statement that crosses the band) Part 13 The crossover band (Stage 5) Crossing the scale L ~ 1/Λ with the same constants at every scale Part 14 Closing: conditional statement and ledger Foundation (under both roads): Stage 2, UV control — from the lattice spacing a to L, uniform in a
03

A map of the four chapters

Table 1 The chapters of Part 14
ChapterContents (one line)
14-01 The conditional statementUnder assumptions, to the mass gap and confinement. The label on each arrow, and the arrow still left blank (from the area law to the gap)
14-02 The ledger of conditionsThe conditions, one row each. The state in the record, the label, what is needed for it to disappear, the dependency diagram, and the differences that remain outside the ledger
14-03 The table of questionsA cross-index that looks up the questions of Parts 10 to 13 by chapter and item number
14-04 What is not claimedA cross-index that looks up "what is not claimed" in Parts 10 to 13 by chapter and item number

Sources

Figure 1 is the same figure as in 9-06. The grounds for the four items of the promise are in 14-02 §01, §05 and §06.

Next chapter to read: 14-01 The conditional statement.

← 13-11 Ultraviolet control — what it means to close at infinitely many stepsContents14-01 The conditional statement — the mass gap and confinement, arrow by arrow →

Revised 2026-10-15: first version.