2026-10-15 · chapter Yang–Mills and the mass gapPart 15 Ultraviolet controlWhat cannot be said, index of questions, Lean correspondence · 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
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

What cannot be said, the index of questions, the dependency diagram

What this chapter shows — what cannot be said across Part 15 as a whole (a summary), an index of the questions F-UV1–F-UV15 left by the chapters, the dependency diagram between chapters, the five grades of the main rows of each chapter, and the correspondence between the Lean bundle used in this part and the text.

Prerequisite chapters — 15-00 (entrance). The other chapters are reached from the rows of the tables on this page.

Said first — This chapter adds no new results. Questions are questions, not claims. No chapter of this part shows ① ultraviolet control. The Lean bundles of this part have passed re-verification by the main session (one file at a time on Shiori's machine), but the labels in the text do not use Lean (paper proofs stay [Proved here]); the bundles are only listed in Table 3. Attaching the bundle labels to the text is for the next version.

Order of this chapter

What cannot be said — eight items for the whole part, and the items of each chapter
The index of questions — F-UV1–F-UV15 (Table 1)
The dependency diagram of the chapters
Summary of the five grades (Table 2)
The Lean correspondence of this part (Table 3)
Connection to Part 14 and the glossary

What this chapter shows

What Part 15 does not say comes down to eight items. Each is there to stop readings that drop the conditions "smallest systems, leading term, one step, form of a statement" (§01).
There are fifteen questions, and each touches one component inside ① of the ledger. Settling them does not close ① (§02).
The dependencies run from the entrance (15-00), the difficulty (15-01) and the original sources (15-02) to the decomposition statement (15-03), branch into the computational line (15-04–15-06) and the wall line (15-07, 15-08), and return to 15-09 (§03).

What cannot be said

We gather what is not said across the whole part into eight items. The finer items of each chapter are in that chapter's "What cannot be said" section.

(a) ① ultraviolet control is not closed. Part 15 shows none of (U1)(U2)(U3), in four dimensions or in two. What it shows is statements of what would need to be shown, line-by-line computations, elementary lemmas (Lemma F, Lemma R, Proposition O, Lemma H, the flux network theorem), and numbers for the smallest systems.

(b) The arithmetic under the one-loop flow is not a theorem about the actual sequence of effective couplings. The divergence and finiteness of the sums in 15-01 are arithmetic that treats the one-loop flow as an exact equation.

(c) The numbers for the smallest systems are not the numbers of a real blocking step. The numbers of 15-05 and 15-06 are for the tree level restricted to commuting configurations (the leading term of $\beta\to\infty$ ), the one-loop level, one step seen through the marginal distribution of a single coarse plaquette, and small boxes (free boundary, free only on the outer rim, periodic); they are not numbers for a step that includes blocks embedded in the four-dimensional lattice, re-fixing of the gauge, and large fields.

(d) The results for the hierarchical model do not imply anything in four dimensions. The Migdal–Kadanoff map of 15-04 sends a single-plaquette weight back to a single-plaquette weight; multi-link couplings, re-fixing and the remainder from large fields are structurally absent, and it is a model in which confinement occurs even at $U(1)$ . The range checked by interval arithmetic is only a finite interval of $\beta$ , and the limit $a\to0$ is not closed.

(e) Lemma H is a statement only for one step, and only when the law of the fluctuations does not depend on the background. The true measure of the fluctuations in one step depends on the background, and the absorption of that dependence $\eta$ and the multi-step induction have not been written (15-07).

(f) The decomposition statement and T⁴-UV are statements. The decomposition of 15-03 and the statement T⁴-UV of 15-02 are in the form of [Assumption] and are not proved. Of the implications, only Lemma F and the arithmetic of geometric series were checked elementarily.

(g) Much of the checking against the original sources is at the level of abstracts and introductions. Stages 4 and 5 of the series and 119 are from abstracts, stages 1–3 from titles, and Theorems 1 and 2 from the abstract and introduction of the last paper. Appendix A.6 of BBS was read in the text itself (15-02).

(h) The proposals for the four rows that do not close are proposals. The proposals of 15-08 are ideas before verification, and none of them closes any of the four rows. We say neither "there is no action that closes with one parameter" nor "(Cls) closes with finitely many parameters".

The index of questions — F-UV1–F-UV15

Table 1 gathers into one the questions left by the chapters of Part 15. F-UV1–3 inherit the numbers of 13-11 (their content updated in 15-03), and from F-UV4 on, a band of numbers was assigned to each chapter. The status and size of each question, and "what does not move even if settled", are in the table of the chapter of origin and are not copied here. Whichever question is settled, ① of the ledger does not close.

Table 1 The questions of Part 15 (the sentence of the chapter of origin shortened to one line; the conditions are in the table of the chapter of origin)
Q.	Question (one line)	Components concerned	Origin
F-UV1	Does the one-step map contract transversally on the one-parameter class it draws towards? (i) Do $\lambda$ and the source stay finite for blocks embedded in the four-dimensional lattice? (ii) Is the constant $C$ of the one-step inequality independent of the step? (iii) Is the weight of the sign of the centre $\mathbb Z_2$ needed as a separate parameter?	(Cls), (Rem-shape)	15-03 §07
F-UV2	The smallest model in which (S) can be shown including large fields (a hierarchical $SU(2)$ ). Can absorption under a constraint for a single plaquette be restated at the true critical point for many plaquettes?	(Rem-flow), (S)	15-03 §07
F-UV3	On the twisted $M^4$ lattice, is the smallest eigenvalue of the operator of the adjoint fluctuations around an isolated flat connection at least $c/M^2$ ? Is the group version of property 3 unchanged by the twist?	(T-Z) of T⁴-UV	15-03 §07
F-UV4	Extend the tube and the attracting curve of the hierarchical model to $\beta\gt64$ (the seam between asymptotic estimates and interval arithmetic)	(Cls) of the hierarchical model	15-04 §05
F-UV5	Check the lower bound of the curvature outside the layer and the weight of the layer by interval arithmetic, and separate the flow coefficient $c^*=1/60$ on the curve from $1/48$	(Rem-flow) of the hierarchical model	15-04 §05
F-UV6	Along the sequence of coarse-plaquette sides $L$ , does the source $\cdot L^4$ (now $7.7$ – $8.5\times10^{-3}$ , transverse thickness 4) settle to a constant as the transverse directions are widened?	The sum of the source over steps	15-05
F-UV7	On the tree-level network with all coarse plaquettes fixed, can a local representative be built that gives an upper bound uniform in the volume? Does the operator norm of $\lambda$ in the coarse directions fall below 1?	The transverse contraction (matrix)	15-05 §07
F-UV8	Derive on paper the one-step correction to $O(1/B)$ for four in a row, giving the constant $C$ of the one-step inequality for (Rem-shape)	(Rem-shape)	15-06
F-UV9	Is the attracting class still uniquely determined when loop shapes other than four in a row (2×2, L-shaped, mixed quartic) are included? Does the fixed point stay put when the definition of the diagonal is changed?	The coordinates of (Cls)	15-06
F-UV10	For four in a row, split the joint law of $P_2,P_3$ into a product of local couplings × a density, and find the lowest degree in $\theta$ of the score of the density and the isotropy of the quadratic form	Absorption of $\eta$	15-07
F-UV11	Can the smallest multi-step statement be written in the hierarchical model or a two-dimensional model ((iv_G) and the contraction of the many-body norm)? The width of the band of analyticity of the coupling. Numbers for the smallest systems are the next test	The multi-step induction	15-07
F-UV12	Derive the Lipschitz constant of the read-off on paper, and measure on a slab the difference in total variation between the density with one large field inserted and without	Row 1 (S)	15-08
F-UV13	When a source $s\operatorname{Re}W_a$ is added to four in a row, is the shift of the coupling linear in $s$ ? On paper, the position-dependent coupling $\beta(x)$ for $U(1)$ Villain	Row 2 (T-Obs)	15-08
F-UV14	Does the evenness of the critical point hold for four plaquettes? (paired with F-UV10)	Row 3	15-08
F-UV15	Split the one-loop Hessian of a block with blocking factor 2 and measure the coefficient of the ghost $F^2$ (negative control: $U(1)$ ). Formalise the direct-product decomposition of the tree gauge for finite groups	Row 4	15-08

Reading Table 1. F-UV1, F-UV8 and F-UV9 are questions about the coordinates of (Cls) and (Rem-shape); F-UV2 and F-UV12 are questions about (S), corresponding to the content of Theorem 2; F-UV10, F-UV11 and F-UV14 are questions about row 3 of the core (absorption of $\eta$ and the multi-step). Questions that numbers for the smallest systems can answer are mixed with paper questions.

The dependency diagram of the chapters

We arrange the dependencies between chapters so that prerequisite chapters come above (the number of a prerequisite is smaller than the number of the conclusion). To the right of the arrow are the earlier chapters that the chapter uses.

[15-00] Entrance       problem, (U1)(U2)(U3), form that closes in one step, misreadings
[15-01] Difficulty     ⟸ [15-00]        dimensionless coupling, group-valued fields, gauge
[15-02] Sources        ⟸ [15-00]        series, Theorem 1/2, BBS chain, T⁴-UV
[15-03] Decomposition  ⟸ [15-01][15-02] (Cls)(Rem-flow)(Rem-shape)(C), Lemma F, bookkeeping
[15-04] Heat kernel    ⟸ [15-03]        telescoping sum, invariant curve, tube (β∈[2.46,64])
[15-05] Flux network   ⟸ [15-03][15-04] Theorem F, inequality, table of boxes, Theorem G
[15-06] Parameters     ⟸ [15-03][15-05] couplings by loop shape, attracting class
[15-07] Background η   ⟸ [15-01][15-02][15-03]  Lemma R, Lemma H, shift of local couplings (proposal)
[15-08] Four open rows ⟸ [15-05][15-06][15-07]  statements and proposals for four rows, dependencies (row 3 is the core)
[15-09] Index          ⟸ [15-00]–[15-08]

To check it, see that the number of every chapter to the right of "⟸" is smaller than the one on the left. There are no cycles. No node of this diagram is a node saying that ① has been shown.

Summary of the five grades

Table 2 picks only the main rows from the five grades table of each chapter. The full statements, grounds and ranges of the rows are in the table of each chapter.

Table 2 The main rows of each chapter (labels as in each chapter's table; only some of the conditions in the range column are copied)
Chapter	Main row	Grade	Range (excerpt)
15-00	Number of steps $K\to\infty$ , lower bound for the product of losses	Proved here	$0\le\delta_k\le\frac12$
15-00	(U1)(U2)(U3) and the form that closes in one step	Assumption	From $a$ to $L_0$ , constants independent of the step
15-01	Logarithmic divergence of $\sum g_k^2$ , the sum of $g^{2+\delta}$ is finite, Proposition O	Proved here	Under the one-loop flow (arithmetic)
15-02	Theorem 1 with assumption (U3), Theorem 2 unpublished, Lemma A.6.1 of BBS	Cited	Abstract and introduction / text of Appendix A.6
15-02	Statement T⁴-UV	Assumption	$SU(2)$ , $n_{12}=1$ , finite volume
15-03	Lemma F, the bookkeeping of the transverse contraction	Proved here	Under (S)(C) / assuming the one-step inequality
15-03	The decomposition statement ((Cls) is the one-parameter attracting class + a contracting form)	Assumption	Form of the statement only
15-04	Saddle-point estimates for MK (1/16, 1/3600, $c^*=1/60$ )	Proved here	Leading term at the saddle
15-04	Invariance of the tube, pointwise contraction, attracting curve	Computer-assisted	MK map, single plaquette of $SU(2)$ , $\beta\in[2.46,64]$ (curve $[15.26,64]$ )
15-05	Theorem F, $1/(16d-8)\le\lambda\le1/16$ , inequality for coarse-plaquette side $L$ , Theorem G	Proved here	Tree level restricted to commuting configurations, 4 coarse plaquettes symmetric (lower bound)
15-05	Table of boxes, type of the boundary, source $\approx8\times10^{-3}/L^4$ , $\lambda\gt1/16$ in the coarse directions	Reading	Small boxes, one loop (starting from the heat kernel, $\beta\to\infty$ )
15-06	$D=\frac25+\frac{24}5\frac qB$ , the tree map and 9/89, $\gamma^*=37/76$ , $c=2/3$	Proved here	Four in a row, one-loop level / leading term of Laplace
15-06	Fundamental + adjoint root $\rho^*=-\frac14-0.84/\beta$ , no root for the prefactor family	Reading	$SU(2)$ , quadratic, $\beta\ge12$ (root) / $\beta\in\{4,8,16,32\}$
15-07	Lemma R, Lemma H, the quaternion identity	Proved here	Lemma H for one step, when the law of the fluctuations does not depend on the background
15-07	The group version of properties 3 and 4, the proposal of the shift of local couplings	Assumption	Statement and proposal
15-08	(U2) is a Chebyshev corollary of (iii_G), the tree gauge has no Jacobian	Proved here	Assuming (iii_G) / axial gauge
15-08	Statements of the four rows and their dependencies (row 3 is the core)	Assumption, Reading	Statements, estimates

The Lean correspondence of this part

Table 3 lists the bundles written in Lean 4 for the elementary parts of the text of this part, as finite sums, real algebra and trigonometric identities. None of them contains sorry, and the axioms printed at the end were only the three standard ones (propext, Classical.choice, Quot.sound). The re-verification by the main session (2026-10-15, one file at a time with lake env lean) also gave rc=0, sorry 0, native_decide 0 and the three standard axioms for all seven. The labels in the text stay [Proved here]; the Lean labels will be attached in the next version. No bundle contains the lattice, the group, the measure or the minimisation itself. What they contain is single algebraic lines from inside the paper proofs.

Table 3 The Lean bundles of this part (re-verification: all done (main session, 2026-10-15); assumptions on the same line as each theorem; the numbers of theorems are 6, 10, 11, 6, 9, 8, 12 in order)
Bundle name	Main theorems and assumptions	Place in the text	Not included
LatticeGauge.ClassResidualTelescoping	`telescoping_residual` (increment of the deviation at each step $\le r_k$ ), `telescoping_residual_multi` (same assumptions, n parameters), `inv_sq_sum_le` ( $a,c\gt0$ ), `uniform_deviation` ( $0\le q\lt1$ , increment $\le C/(a+ck)^2+Dq^k$ )	15-03 §04, 15-04	Measure, curvature, map
LatticeGauge.HierarchicalShapeFlow	`quarticStep_fixed` (fixed point 1/3600 of $\sigma\mapsto\sigma/16+1/3840$ ), `shapeStep_contract` ( $k\ge2$ ), `flow_coefficient` ( $1/48-15/3600=1/60$ ), `affine_contraction` ( $0\le\lambda\lt1$ , each step $d_{k+1}\le\lambda d_k+\varepsilon$ ), `tube_iterate`	15-04	The interval-arithmetic tube, the derivation of the leading term at the saddle
LatticeGauge.FluxNetwork	`ratio_le_max` (each $f_i^2\le M$ , $\sum f^2\gt0$ ), `ratio_ge_of_unit_flux` (unit flux), `energy_split` ( $\sum f\psi=0$ ), `thomson_lower`, `thomson_attained`, `minimizer_unique` (the set is closed under midpoints), `quad_lower` ( $\beta\gt0$ ), `chain_one_step` (five inequalities)	15-05 §02, 15-07 Lemma H	Projection, the Dirichlet principle, interchange of differentiation in the Taylor seminorm, invariance of Haar measure
LatticeGauge.AllFacesBlocking	`fine_energy_lower` (the map to sets of four plaquettes is injective), `block_energy_lower`, `smooth_ratio` ( $n\ne0,\theta\ne0$ ), `equal_split_ratio`	15-05 §07	Pseudo-inverse, coarse Bianchi, Fourier symbols
LatticeGauge.RatioTwoCriterion	`ratio_two_iff`, `one_loop_D` ( $B\ne0$ , in the form with the Gaussian moments substituted), `gamma_family_one_loop_lower` ( $c\lt3$ ), `fund_adj_one_loop_root` ( $\beta\ne0$ , $1+2\rho\ne0$ ), `no_pointwise_sqrt_one_loop`	15-06	Haar measure, moments, integration by parts, saddle-point expansion
LatticeGauge.FundAdjointRefit	`refit_solution` ( $\beta\ne0$ ), `tree_quadratic_split`, `tree_value_at_saddle`, `image_rho`, `image_gap` ( $-53/356+1/4=9/89$ ), `image_one_loop_D`	15-06	Convolution, reduction to the tree, the envelope argument
LatticeGauge.PrefactorFlowFixedPoint	`step_k2`, `step_k2_fixed_iff` (fixed point 37/76), `step_k2_contract`, `fixed_point_general`, `transverse_hessian` ( $\sin a,\sin b\ne0$ ), `convolution_prefactor`, `rem_shape_geometric` ( $0\le\lambda\lt5$ , assuming the one-step inequality), `ratio_coefficient`	15-06, 15-03 §04	Convolution on the group, the Laplace method, one-loop formulas

Connection to Part 14 and the glossary

The row ① ultraviolet control of the ledger of conditions (14-02) is split into smaller pieces by Table 1 of this part (questions) and 15-08 Table 1 (the four rows that do not close). Its label and state (Assumption, open) do not change. The cross index of questions (14-03) cites F-UV1–F-UV15 of this part as a fifth origin. The words added in this part (the invariant curve, the transverse contraction, the source, the flux network, couplings by loop shape, the attracting class, the shift of local couplings) are in the A-02 glossary.

What this chapter can say

Can say	Cannot say
Index: questions, dependencies, summary of the five grades, correspondence of the Lean bundle (this chapter adds no new results)	① ultraviolet control. The settlement of any of the questions
That the Lean bundle depends only on the three standard axioms (printed for each bundle)	Formalisation of the lattice, the group, the measure and the minimisation (re-verification by the main session is done)

Sources and reproduction

Item	Kind	Source, tool
What cannot be said, questions, five grades	This book's study	The tables and sections of each chapter 15-00–15-08
The Lean bundle	This book's study	The record of the formalisation of this book's study (Lean 4, with the axioms printed at the end of each bundle). Re-verification by the main session done (2026-10-15)

Words used in this chapter — what cannot be said, Question F-UV, the dependency diagram, the five grades, the Lean bundle. Their meanings are in the A-02 glossary.

Next chapter to read: 14-00 Entrance — the promise of this part (the whole conditional statement, including ① of the ledger).

Revised 2026-10-15: first version. Placed as the index after §13 and §14 of 13-11 were divided among the chapters.
2026-10-15 (same day): the seven Lean bundles passed re-verification by the main session, so "not yet" was changed to "done" in Table 3 and in the table of scope.