computo ergo sum

2026-10-15 · chapter Yang–Mills and the mass gapPart 14 Closing — conditional statementsthe ledger, the dependency diagram, the differences that remain outside the ledger · 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

The ledger of conditions

What this chapter shows — A ledger that lists, one row at a time, the conditions placed by the conditional statement of Part 14 (14-01). For each row: the mark of its state in the record, its label in the book, what is needed for it to disappear, the rows it presupposes, and the chapters that treat it. A diagram of the dependencies between rows. And, in a frame separate from the ledger, what remains as a difference from the problem statement even when every row of the ledger has disappeared.

Prerequisite chapters — 9-06 (the table of seven stages, and the figure of the two roads). The rows of the first road are taken from 10-15 Table 1 and 10-15 §07.

Said first — The ledger is a list of conditions, not a gauge of progress. We do not count its rows and read off "this much left". This chapter adds no new results. The proofs and counterexamples for the contents of the rows are in the chapters in the right-hand column (including chapters still being written).

Order of this chapter
  1. The definition of "a row disappears" — the rule placed at the head of the ledger
  2. A copy of the table of stages — the marks of 9-06 Table 1, and the rows of the ledger (Table 1)
  3. The ledger — ten rows and their breakdown (Table 2)
  4. The diagram of dependencies between rows — the numbers of premises are smaller than those of conclusions
  5. The differences that remain outside the ledger — what remains even when every row has disappeared (Table 3)
  6. The day rows increase — the ledger is not a table that only shrinks
01

The definition of "a row disappears" — the rule placed at the head of the ledger

RuleA row of the ledger disappears when, and only when, the following two are both in place. (1) A paper proof of the statement of the row has been placed in some chapter of this book (a proof existing in the record is not enough). (2) If the row has a Lean entry, the theorems written in that entry have passed a re-check on the side of this book that reads the axioms again.

The point of this rule is to separate the state of the record from the state of the book. Even a row that is closed in the record (mark ■) is not removed from the ledger until its paper proof has been placed in a chapter of the book. For example, ⑤, reflection positivity in z, is closed in the record, but the chapter that is to hold its paper proof (13-04) has not yet been written, so the row remains.

One more thing. When a row disappears, the rows that use it as a premise do not disappear automatically. For example, even if ④, ⑥ and ⑦ all disappear, ⑧ disappears only when the paper for the reduction from ④, ⑥ and ⑦ to ⑧ has been placed in the book (13-02). And even if ⑧ disappears, not one of the differences outside the ledger (§05) moves.

02

A copy of the table of stages — the marks of 9-06 Table 1, and the rows of the ledger

Table 1 copies the column of marks of 9-06 Table 1 and lists which rows of the ledger bear on each stage. The marks mean the same as in 9-06: they mark the state in the record, and are not a gauge of how far things are proved (9-06 §02). The contents of the stages and the column of what is known are kept only in 9-06 and are not repeated here.

Table 1 The marks of the seven stages (a copy of 9-06 Table 1) and the rows of the ledger (■ closed in the record / ◐ partial / □ open)
StageMarkLedger rows
1 Regulator◐⑤ (writing out the assembly of the regulator is not set up as a row. §06)
2 Ultraviolet□①
3 Gap on a small T²◐②
4 Confinement on a small T²◐③, ④ (and the form of ⑥ with L fixed)
5 The band□⑥, ⑦, ⑧. In the language of the side of time, ⑨ (in miniature) and ⑩
6 R⁴◐Outside the ledger: Outside 1, Outside 2
7 General groups□Outside the ledger: Outside 3
03

The ledger — ten rows and their breakdown

Table 2 is the ledger. ①–⑧ are the rows of the second road (from the direction of space), and ⑨ and ⑩ are the rows of the first road (from the direction of time). We put the ledgers of the two roads into one because both roads stop at the same band (9-06 Figure 1). The rows are numbered in an order in which premises come before conclusions (§04). We write for the case of the group SU(2), the Wilson action on the lattice, and the 't Hooft twist n12=1n_{12}=1 put on T².

Table 2 The ledger of conditions (to the right of the row name is the mark = the state in the record. The label is the label in the book, for each claim within the row. "What is needed for it to disappear" is what is needed in addition to (1)(2) of the rule in §01. The five rows under ⑦ are its breakdown)
RowWhat is to be shown (with its conditions in the same row)LabelWhat is needed for it to disappear, premises, chapters
① Ultraviolet control □A multiscale analysis uniform in aa from the lattice spacing aa up to a small physical scale L0L_0 (locality and analyticity of the small-field effective action, rarity of large fields, the effective coupling staying small). On a small T²AssumptionCompleting the written-out steps of Bałaban (including the steps the original sources call unpublished), and verifying them. An assumption that depends on asymptotic freedom, which does not apply to U(1) or ℤ₂
Premises: —
Chapters: 6-07, 12-00, 13-11 (the statement of a class that closes in one step, Lemma R, Lemma F, Statement T⁴-UV)
② Gap on a small T² ◐The twist removes the zero modes, and the gap of the transfer matrix ≥c/L\ge c/L is uniform in aaProved here the twist part (uniqueness of the vacuum, the gap 4sin⁡2(π/2N)4\sin^2(\pi/2N) of the two-dimensional lattice Laplacian, Smin⁡>0S_{\min}\gt0 in the (1,1) sector; NN fixed) / Assumption the form uniform in aaUnder ①, writing the lower bound on the gap uniformly in aa
Premises: ①
Chapters: 12-01
③ Uniform acceptance of the insertion □Under the small-field frame, the mean of tanh⁡(W/2)\tanh(W/2) for the dynamically driven vortex insertion is at most 1−δ1-\delta, uniformly in aa (WW is the work of the insertion)Assumption / Reading making the procedure longer lowers the work (measurement conditions in 12-08)An estimate of the distribution of the work at the one-loop (determinant) level
Premises: ①
Chapters: 12-06, 12-08
④ The base point on a small T² ◐At a small physical length L0L_0, θ(L0)≤1−ε0\theta(L_0)\le1-\varepsilon_0, ε0∝(L0Λ)11/3\varepsilon_0\propto(L_0\Lambda)^{11/3}Cited the form of the string tension (Tanizaki–Ünsal, arXiv:2201.06166 eq. (2.37), dilute regime, one loop) / Assumption deriving it on the lattice from ① and ③A proof under ① and ③. The lattice version is the same semiclassical picture as ⑦b
Premises: ③
Chapters: 12-02
⑤ Reflection positivity in z ■The ensemble summed over twists is positive under reflections through lattice points in directions 3 and 4. As a corollary, the chessboard estimateProved here / Lean: ReflPos (ReflPos.refl_pos, ReflPos.refl_cs) (the algebra of the core on non-negative weights)Placing the paper proof in the book, with, in the same row, the fact that transverse reflections are not positive because of the twist
Premises: —
Chapters: 13-04, 13-05
⑥ The uniform slab gap (frame decorrelation) □The correlations between the frames of containers decay uniformly in LLAssumption. With LL fixed, ⑤ and the chessboard replace it by a single density κcb\kappa_{\rm cb} (Proved here, LL fixed)Comparing κcb(L)\kappa_{\rm cb}(L) between scales. Split into columns, it becomes the same problem as one of the faces of ⑦c (the cost of twisting a column)
Premises: ⑤
Chapters: 13-05
⑦ The renormalisation comparison □The vortex free energy at side 2N2N and coupling β\beta is at most a constant times that at side NN and coupling β′\beta', with β′<β\beta'\lt\beta. Non-perturbatively, for every β\betaFive rows of breakdownAn estimate with the error kept within a constant factor, for every β\beta. An argument that closes the band must be able to distinguish U(1) (U(1) has a transition inside the band, at βc≈1.01\beta_c\approx1.01)
Premises: ⑤, ⑥
Chapters: 13-06
Strong-coupling end ■The range β≲10−79\beta\lesssim10^{-79}Assumption (there is a proof in the record. The constants are crude and the bookkeeping is a sketch, so it stays Assumption until the paper is placed in the book)Writing out the steps, and improving the constants
Chapters: 13-06
⑦a Lower end □Pushing confinement and the comparison up to β≈0.3\beta\approx0.3–11AssumptionA finite-size criterion by mixing of the box flux, and computer assistance (a certificate with interval arithmetic)
Chapters: 13-06
⑦c Middle □One inequality that uses non-commutativity. Any one of the four faces (softness in the direction of the twist, mixing of the box flux, the cost of twisting a column, the margin of the pressure), uniformly in the scaleAssumption (none of the faces is proved)A proof in a form that fails for U(1)
Chapters: 13-06, 13-10
⑦b Upper end ◐Semiclassics at fixed NN and large β\betaAssumption the confinement direction (Theorem A of the draft paper: at fixed NN, −ln⁡θ≥c β−ae−β(S∗+ε)M2-\ln\theta\ge c\,\beta^{-a}e^{-\beta(S_*+\varepsilon)}M^2. No proof is placed in this book. A proof with citations is in the paper: the general chessboard theorem and the Morse–Bott argument are cited) / Assumption the converse uniform in MM, and S∗=Smin⁡S_*=S_{\min}A uniform bound in the converse direction (already reduced to a single positivity lemma), and agreement of the minimal actions
Chapters: 12-02, 13-06
One loop ■With the comparison that reshapes the fluctuations, β−β′=0.258\beta-\beta'=0.258. With the comparison that keeps the shape of the fluctuations, β′≥β+0.023\beta'\ge\beta+0.023 (the anti-asymptotically-free direction)Proved here (at the one-loop level; not a non-perturbative claim) / Reading the direction of the numbers (SU(2) has β′<β\beta'\lt\beta throughout the band; for U(1) the sign flips at βc≈1.01\beta_c\approx1.01. Lattices, seeds and errors in 13-09)Making it non-perturbative is ① and ⑦c
Chapters: 12-07, 13-09
⑧ The band □ln⁡(1/θ(L,L))≥cL2\ln(1/\theta(L,L))\ge cL^2, with cc independent of LL (the 't Hooft string tension is positive)Proved here the reduction from ④, ⑥ and ⑦ (the algebra of the iteration in Lean: Doubling (Doubling.iterate)) / Assumption the band itself④, ⑥ and ⑦ disappearing, and the paper for the reduction being placed in the book. Intermediate milestones: (a) ruling out a non-confining phase, (b) ln⁡(1/θ)≥cL2/ln⁡L\ln(1/\theta)\ge cL^2/\ln L
Premises: ④, ⑥, ⑦
Chapters: 13-01, 13-02
⑨ What remains of the first step □For the path measure of the 0+1-dimensional double well, a gap uniform in kk and ℏ\hbar for chain length k≥8k\ge8 (a uniform logarithmic Sobolev inequality for a measure with non-convex regions). The band in miniatureAssumption. The convex side (length ≤7\le7) and the uniqueness of the minimum are closed in the record (the section that places them in the book is a supplement to 10-15)Closing the rows of 10-15 Table 1 in a form uniform as ℏ→0\hbar\to0, or stitching finite blocks together with strong mixing
Premises: —
Chapters: 10-09 §06, 10-15, 13-07
⑩ The four-dimensional goal (side of time) □The upper side ξ²↑ (CP≤Kξ2C_P\le K\xi^2) of the diffusive-relaxation-time assumption and the one-sided bridge, uniform as β→∞\beta\to\infty (level (T2) of 10-15 Table 5)Assumption (formerly Assumption H↑)Crossing the band in the language of time. Even if ⑨ disappears, ⑩ does not move (⑨ is a miniature, not an implication)
Premises: ①
Chapters: 10-01, 10-15 §07, 13-07

What to read from Table 2. (a) The rows marked ■ in the record are only ⑤ and two rows of the breakdown of ⑦ (the strong-coupling end and one loop), and by the rule of §01 none of them has disappeared yet. (b) Non-abelian dynamics enters at ⑦c (and at ⑥ and ⑧, which gather into it), and this row has a form that correctly fails for U(1) and ℤ₂. The other rows are structural facts, or rows that depend on the ultraviolet wall ①. (c) The weak form of ⑧ (the weak form of the lateral-widening inequality, the uniform slab tension) has almost the same content as the band. We read the reduction not as having made the problem easier but as having pinned down ⑦c as the place where the non-abelian input is needed.

04

The diagram of dependencies between rows — the numbers of premises are smaller than those of conclusions

Figure 1 writes the dependencies between the rows of the ledger and the rows outside it, split into layers. Arrows point from premise to conclusion, and every row has a larger number than its premise rows. The rows of the ledger are written in forms whose statements do not contain the quantities of the conclusion (the gap on R4\mathbb R^4, the area law on R4\mathbb R^4) (the assumption rows ①, ③, ⑥, ⑦c, ⑩). The assumptions in forms that contain the quantities of the conclusion (the assumption of G1 and the forms of the band) are marked explicitly as "restatements" in 14-01 (assumptions of arrows that restate uniform decay of correlations as a gap, and uniform bias and tension as an area law). ⑧ is the band itself and contains a quantity close to the conclusion of the conditional statement. We put ⑧ in the ledger to show, in one row, the road to removing ⑧ once the reduction (from ④, ⑥ and ⑦ to ⑧) is on paper.

Layer 0   ① ultraviolet control □          ⑤ reflection positivity in z ■
Layer 1   ② gap on a small T² ◐ (← ①)
          ③ uniform acceptance of the insertion □ (← ①)
          ⑥ uniform slab gap □ (← ⑤)
Layer 2   ④ base point on a small T² ◐ (← ③)
          ⑦ renormalisation comparison □ (← ⑤, ⑥)
Layer 3   ⑧ the band □ (← ④, ⑥, ⑦)
──────── below this line: outside the ledger (§05) ────────
Layer 4   Out 1  limit to R⁴ ◐ (← ⑧)
Layer 5   Out 2  joining the area law and the gap □ (← ②, Out 1)
          Out 3  all compact simple groups □ (← Out 1)
Layer 6   Out 4  verification □ (← all rows)

First road  ⑩ the four-dimensional goal □ (← ①). ⑩ is the entrance to Out 2 from the side of time
            ⑨ what remains of the first step □ is ⑩ in miniature (not an arrow of implication)

Figure 1 Dependencies between rows (the premises are to the right of ←). ④ and ⑦b rest on the same semiclassical picture, but that is not an arrow of implication, so it is not in the figure.

05

The differences that remain outside the ledger — what remains even when every row has disappeared

Separate frame

Even if ledger rows ①–⑩ all disappear, a difference from the requirements of the problem statement (Jaffe–Witten 2000, 0-03) remains. An empty ledger does not mean that the problem is solved.

Table 3 The differences that remain outside the ledger (the correspondence with the tasks of 9-04 is this book's arrangement)
RowDifferenceMarkPresent state9-04
Outside 1The limit to R4\mathbb R^4: the infinite-volume limit, the OS axioms, the recovery of rotation invariance, uniqueness of the limit, non-triviality◐Assumption Only the outline that, under ⑧, most of it is a standard argument. The infrared side of rotations, uniqueness and non-triviality are openK2, K4, K5
Outside 2From the area law to the mass gap: the gap of the transfer matrix is positive in the theory on R4\mathbb R^4□The area law alone does not give the gap (9-06 §03). Even the statement of what should be added is not settled. The only form that can be written now is "carry the gap of ② through the band to R4\mathbb R^4 with the same constants"K3 (the premise of rule R3)
Outside 3All compact simple groups□For SU(N), 't Hooft flux is expected to be usable as it is Reading (physics). For groups without a centre (G₂, F₄, E₈), neither the vortex tools nor the arrows that use the centre can be used Cited (9-04 §07)9-04 §07
Outside 4Verification□Writing out the whole, Lean (as far as it reaches), expert refereeing, publication. The rules of the prize require refereed publication followed by a period of acceptance—

Outside 1–4 are not put in the ledger because they are not conditions of the conditional statement (14-01) but the distance between the conclusion of the statement and the requirements of the problem statement. The rows of the ledger are "work of removing assumptions"; the rows outside are "work of carrying the conclusion as far as the problem statement". If the two were mixed in one table, then on the day the ledger became empty it would read as if the problem had been solved.

06

The day rows increase — the ledger is not a table that only shrinks

As research advances, rows disappear, but they can also increase. For example, when the reduction of the band was worked out, the lateral-widening inequality split into two rows, a strong form and a weak form, and it turned out that the strong form is stronger than confinement (it contains monotonicity of the slab tension). When the renormalisation comparison was worked out, two new rows appeared, the strong-coupling end and the one-loop end, and the middle row split into four faces. Writing out the assembly of the regulator (Stage 1) has no row for now, but a row may appear at the stage of writing it out.

So the number of rows of this ledger, laid out over time, is not a gauge of progress. What to read is the content of each row, and what is needed to remove it.

What this chapter can say

What is saidWhat is not said
The list of conditions of the conditional statement, and for each row its state in the record, its label, and what is needed for it to disappear (Table 2)The proof of any row. This chapter adds no new results
Proved here the reduction to ⑧, ⑤, the twist part of ②, the confinement direction of ⑦b, the one-loop level of ⑦ (within the conditions written in the rows)⑧ itself, the non-perturbative form of ⑦, the ultraviolet control of ①
That there are four differences that remain even when every row of the ledger has disappeared (Table 3)That an empty ledger would mean the problem is solved (this cannot be said; Outside 1–4 remain)
The arithmetic of the theorem names in the Lean column (the algebra of finite sums and the inequality of the iteration)That the Lean column checks the statement of the row itself

Sources and reproduction

ItemKindSource / tools
The rows of the ledger, the breakdown, the dependenciesThis bookThe material of Parts 12 and 13 (the list of tasks up to a proof with its dependency diagram, and the list of assumptions), renumbered so that premises come first. The marks are copies of the state in the record
Rows ⑨ and ⑩ of the first roadThis book10-15 Table 1, Table 5, 10-09 §06
The form of the string tension in ④CitedY. Tanizaki, M. Ünsal, arXiv:2201.06166 eq. (2.37) (bibliographic data and the equation in the source table of 12-02)
Theorem A of ⑦bAssumptionDraft paper 'Fractional instantons confine on the lattice: a rigorous semiclassical bound for SU(2) gauge theory on a twisted small torus' (2026-10, A-07). The general chessboard theorem and the Morse–Bott argument are cited. A proof with citations is in the paper, and no proof is placed in the book, so the label is Assumption
The reduction of ⑧Proved hereThe conditional theorem of the draft paper 'Where the twisted-slab route to confinement stops: a corrected window target, a reduction to transverse widening, and SU(2) versus U(1) numerics' (2026-10, A-07), and 13-02
The tasks of Outside 1–3This book9-04 Table 1 and §07, 9-06
The problem statementCitedA. Jaffe, E. Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute (2000)
LeanLeanReflPos (ReflPos.refl_pos, ReflPos.refl_cs), Doubling (Doubling.iterate). What the bundles guarantee is only the algebra of finite sums and the inequality of a real iteration (A-04)

Terms used in this chapter — the ledger, a row disappears, the marks of the state in the record (■◐□), the differences that remain outside the ledger, the renormalisation comparison, the four faces, the band in miniature, the diffusive-relaxation-time assumption (formerly Assumption H). Their meanings are in the A-02 Glossary and the A-06 Index of assumptions.

Next chapter to read: 14-03 The table of questions. The conditional statement itself is in 14-01.

← 14-01 The conditional statement — the mass gap and confinement, arrow by arrowContents14-03 The table of questions (a cross-index) →

Revised 2026-10-15: first version.