2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerationsquestions, candidate rules, what is not claimed · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

Questions, and what is not claimed

What this chapter shows — We put the questions left by the chapters of Part 11 into one table (done, partly done, not yet, new; with what moves if it is settled and what does not move even if it is settled), list candidate rules in answer to the question "if new mathematics is to be made, what should be placed as rules?", and write what Part 11 does not claim as sixteen rows, matching each to the sentence of which chapter of the text supports it.

Prerequisite chapters — None. This chapter can be read from anywhere. The three meanings (not shown, not read strongly, not called false) are used as in 10-16 §01. Notation is in 11-A1.

Said first — The level is [Y2]–[Y3]. This chapter adds no new results. The table of questions is questions, and contains no claims. The candidate rules are candidates, and none of them promises a prospect of closing the third step (the continuum limit and the OS axioms). The "what is not claimed" of Part 10 (10-16) applies to this part as it is.

The order of this chapter

The table of questions — nineteen questions, their status, what does not move even if settled (Table 1)
The list of candidate rules — six candidates and three combinations, relation to the roots (Tables 2 and 3)
What is not claimed — sixteen items and the sentences of the text that support them (Table 4)
Relation to Part 10

The table of questions — questions that can be settled at the size of an undergraduate thesis

Table 1 lists the questions left by the chapters of Part 11. "Size" is the kind of work needed to settle it (paper, reading, computation, Lean), not an estimate of time. In each row we always wrote "what does not move even if it is settled". Even if a question is settled, the mass gap problem does not move. "Done" in the status means that the text of this part has answered the question (with conditions), and the label of the answer is in the chapter on the right.

Table 1 Questions of Part 11 (status: done, partly done, not yet, new [a question added while writing this part])
No.	Question	Size	Status	What moves if settled	What does not move even if settled	From which chapter
F1	For the free field, compute the comparison of excitation energies of observables supported in the box (assumption (Loc-Δ_c))	Paper	Done: in the moment form (Loc-Δ_c)′, it holds for the free field with $\gamma=d\pi^2$	The form of the coefficient of the local bridge	(Loc-Δ_c)′ for interacting models	11-04 §05
F2	Monotonicity of the window factor $\varphi(x)/x^2$	Elementary, Lean	Done: strictly decreasing for $x\gt0$ (Lean `wr_strictAntiOn`, no assumptions)	The explicit form of the window threshold	Nothing (only the form of the constant)	11-04 §03
F3	Whether the static ratio $\operatorname{Var}/\mathcal E$ catches the slowness, for the sign of the Polyakov loop on a small periodic box (a quantity of a slow sector whose answer is known)	Small computation	Not yet	A positive control for the method of static test functions	The verdict on the topological charge	11-01 §02
F4	For compact $U(1)$ on a two-dimensional layer, whether the gap keeps decreasing like $M^{-2}$ or saturates as the strip is lengthened (the non-linear value of the length at which conservation breaks)	Computation	Done: on the width-2 strip, outer boundary frozen, five points $\beta=2..6$ , $\ell_c\propto e^{0.916\beta}$ (a reading)	The non-linear value of the length at which conservation breaks	Four-dimensional $\ell_*$	11-02 §03
F5	A primary source on the autocorrelation of the topological charge in pure $SU(2)$ gauge theory (if none, a small measurement at fixed physical volume)	Reading, computation	Partly done: one primary source (HMC, small box, close to $a^{-2}$ ). The small measurement is not yet done	The label of the $SU(2)$ row (at the boundary, or clearly faster than $\xi^2$ )	The conjecture on the relation between HMC and Langevin	11-07 §07
F6	A many-family version of the lemma of two projections (the family of $2^d$ colours for a $d$ -dimensional box), and its condition on maximal correlation	Paper	Not yet	The node of reduction of the four-dimensional box	An estimate of maximal correlation uniform in $\beta$	11-03 §06
F7	On $SU(2)$ , the flow of $\inf\operatorname{Hess}V_t$ for the one-link Wilson weight convolved with the heat kernel, and of the weight of the bad set (one variable, class functions)	Paper + small computation	Not yet (the first step of the class of measures that pay with rarity)	The first numbers of the Polchinski flow on the group; whether the class closes in one step	Lattices that are not hierarchical; closing uniformly over many steps	11-06 §07
F8	For Wilson's transfer operator, show assumption (D) (the comparison of increments at neighbouring times with gradients, uniform in $\beta$ ) restricted to functions cut off at low energy	Paper	Not yet	Carrying the finite-volume window bridge to Wilson's lattice with discretised time	Infinite volume	11-04 §08
F9	For finite-state reversible Markov chains, close in Lean $\tau_{\rm int}\le C_P$ and the identity of the ground-state transformation (the algebra of Proposition V), defined from probabilistic objects	Lean	Not yet	An example of formalisation with the objects themselves as types (a check of a different form from 10-13)	Path measures, continuum fields	11-05 §01
F10	Reread the autocorrelations of local modes and topological modes in two-dimensional $CP^{N-1}$ through the eyes of the local H	Reading, computation	Partly done: the literature has been read. The test-bed measurement goes only as far as a conditional reading at one volume and one correlation length	Material for the test bed	Four dimensions	11-02 §07
F11	For two-dimensional $SU(2)$ (plaquettes independent), find $\lambda_W$ and $\Gamma$ of families of loops at finite $\beta$	Computation	New	The $\beta$ -dependence of the threshold for the colour singlet	Four-dimensional values	11-04 §06
F12	On a four-dimensional box, the values for families of loops the size of the span (including non-planar ones and products of lassos)	Computation	New	The four-dimensional value of the colour-singlet reference	The ratio for the target measure	11-04 §06
F13	For the 0+1 double well, sandwich the local ratio by a reduction to one dimension with the position of the kink as coordinate	Paper + small computation	New	The constant of the motion that crosses the window	The four-dimensional ratio	11-04 §07
F14	Add one point $\beta=7$ for the strip, and see whether the slope of the raw rate keeps matching $2-1/\beta$ and whether the slope of $\ell_c$ settles	Computation	New	The reliability of the slopes for the two-dimensional strip	Four dimensions	11-02 §05
F15	For fields whose ratio lies in the band $[K_*,1)$ between the threshold and 1, whether there is another argument that sees the mass while staying in the ratio form (outside Jensen and Dirichlet test functions)	Paper	New	Handling of the undecided band	Values of the ratio for interacting fields	11-04 §06
F16	On Wilson's lattice with discretised time, write the window version of the bridge (Proposition W and the lower bound in finite volume) under assumption (D)	Paper	New	The lattice version of the window bridge	Infinite volume	11-04 §03
F17	Whether the Kogut–Susskind Hamiltonian becomes frustration-free when it is split with terms grouped by sector	Paper	New	Whether local-to-global gap theorems apply	The gap at weak coupling	11-05 §04
F18	On the $CP^{N-1}$ test bed, add other random seeds and the remaining pairs of $N$ , and read the exponent of the slowness of the topological mode at a second correlation length	Computation	New	The reliability of the reading that freezing splits by $N$	Four dimensions	11-02 §07
F19	Measure the definition (2) of the length at which conservation breaks (the smallest $R$ at which the ratio reaches half its saturation value) with the charge of a four-dimensional subregion	Computation	New	The first value of the four-dimensional length at which conservation breaks	Whether H is true	11-02 §01

If any question is settled, the label of one row of one chapter moves. Whether the mass gap, Assumption H or H↑ is true does not move when any question is settled.

The list of candidate rules — "if new mathematics is to be made, what should be placed as rules?"

The considerations of Part 11 also ran into the question "if new mathematics is to be made, what should be made the rules of the mathematics?" (11-07 §06). Like Boolean algebra, which threw away the meaning of propositions and kept only the rules of equality, how could one assemble a system that places properties of renormalisation and the continuum limit as rules? Table 2 is the list of candidates. Every row is a candidate, and neither consistency nor the ability to restate existing results has been checked Estimate. The comparison with Boolean algebra is a metaphor for how to assemble, not a mathematical claim.

Every candidate can be written in the same form: remove one assumption of an existing framework, and place a replacement rule instead. We condensed what is removed and what is placed into one line per row, and added one question and what does not move even if that question is settled.

Table 2 Candidate rules (one line each, one question) Estimate
Symbol	Candidate rule (one line)	Question (one)	What does not move even if settled
N1	Axiomatising scale: a semigroup of coarse-grainings, locality, gauge covariance, contracting and marginal directions, weights for large fields	On the hierarchical $SU(2)$ model, which of the five rules apply and which do not	Non-hierarchical lattices, the continuum limit
N2	The algebra of loops: bounded, the Mandelstam identities, positivity and reflection positivity, the loop equations (throw away the connection, keep the values of loops)	On the two-dimensional $SU(2)$ lattice, whether the state satisfying the four rules is determined uniquely at finite $\beta$	Definition of continuum loops in three dimensions and above
N3	Conserved quantities and sectors: decomposition of the variance, the sandwich $C_{\rm trans}\le C_P\le\sup C_P^{\rm sec}+C_{\rm trans}$ , local windows, two lengths (11-03 §04)	Whether $C_{\rm trans}$ in the sandwich can be written as a quantity of the reduced chain between sectors	Construction (the third step)
N4	The class of measures that pay with rarity: local convexity on the good set, rarity of the bad set, a returning drift, the currency of entropy (11-06 §07)	How the parameters of the class are carried over when the one-link Wilson weight is convolved once with the heat kernel (F7)	Closing uniformly over many steps
N5	An axiom of asymptotic freedom: changes of scale, a monotone coupling, a free field in the ultraviolet, dimensional transmutation (nothing is placed in the infrared)	Whether it follows from the rule of approaching a free field in the ultraviolet that the local ratio approaches 1 at short distances	The infrared, the mass gap
N6	Rules for curvature: Bakry–Émery on the space of gauge orbits	On the orbit space of one plaquette, whether the curvature term of the quotient widens the range of strong coupling	Weak coupling
N1+N4	A rarity class closed under one step of coarse-graining	On the hierarchical $SU(2)$ model, how wide the coupling window is that remains between how the class closes on the ultraviolet side and the strong-coupling tools	The four-dimensional window
N3+N4	A rarity class with sectors (of Eyring–Kramers type, sandwiching by the larger of the constant within sectors and the rate of transitions between sectors)	On the two-dimensional $U(1)$ strip, whether the upper bound on the gap can be sandwiched by the larger of "within sectors" and "the rate of transitions"	The four-dimensional topological charge
N2+N3	Putting sectors into the algebra of loops (reading the decomposition of the set of states into extreme points as sectors)	Whether the non-uniqueness of solutions of the loop rules corresponds to there being several phases (sectors)	Uniqueness at weak coupling

Table 3 gives the relation to the three roots of the third step (11-07 Table 3: (R1) controlling infinitely many steps uniformly in the number of steps; (R2) that there is no expansion connecting the ultraviolet and the infrared; (R3) that there is no global gauge fixing and large fields cannot be controlled by convexity).

Table 3 Candidate rules and the three roots ("works" means the language of the rule can handle that root, not that it gets past the root) Estimate
Candidate	(R1) uniform in the number of steps	(R2) the ultraviolet and the infrared	(R3) gauge and large fields
N1 axiomatising scale	Works (it is the language itself)	Does not work (leaves the region when the coupling is $O(1)$ )	Half (the weights on the group are not yet formulated)
N2 the algebra of loops	Does not work (goes back to the definition of continuum loops)	Works in form (the same rules at every $\beta$ ). Whether the solution is unique is undecided	Works best (no gauge fixing appears)
N3 conserved quantities and sectors	Does not work	Does not work	Only gives a name
N4 the rarity class	Only when combined with N1	Does not work (the rate of rarity runs out)	Works head-on
N5 the axiom of asymptotic freedom	Only copies it out	Does not work (leaves the infrared empty)	Does not work
N6 rules for curvature	Does not work	Does not work (only on the strong-coupling side)	Gives a meaning

Reading the table Estimate. There is not a single "works" in the (R2) column. Only N2 has rules of the same form at every coupling, but whether the solution of the rules is unique has been replaced by the very content of (R2). The coupling window that appears with N1+N4 (11-06 §07) is this (R2) restated in the language of rules.

Where the Boolean-algebra type is possible Estimate. Replacing continuous objects by discrete rules is already half there at the level of the algebra of observables: the lattice (replacing a continuous field by group elements on edges), the Mandelstam identities and loop equations for loops, the integers of sectors, the one-variable recursion of hierarchical models. But the rules that decide the third step are inequalities, not equalities: positivity, rarity, convexity, estimates uniform in the number of steps. The loop equations (equalities) do not determine a unique solution; what narrows the solutions is positivity (infinitely many inequalities). Put as a metaphor, what the third step needs is closer to the type of measure theory (rules of inequality such as σ-additivity, and rules of limits) than to Boolean algebra.

What is not claimed about the candidate rules. (i) None of the candidates promises a prospect of closing the third step. (ii) The comparison with Boolean algebra is a metaphor, and "the rules of the third step are inequalities" is also an estimate. We have not shown that the rules of Yang–Mills cannot be written as equalities. (iii) The correspondences between candidates and existing results (for example, between the form of the stable manifold of N1 and ultraviolet stability in finite volume) are all restatements not checked in the texts of the originals. (iv) "The hierarchical model is a place where the three — the ultraviolet side, the window and the strong-coupling side — come together" is a way of putting it within the range searched, with no implication for four dimensions. (v) A candidate written as not reaching is not evidence of being false.

What is not claimed — sixteen items and the sentences of the text that support them

Table 4 writes, one row each, what Part 11 does not claim, and lists the places in the text that support each row. "Support" means that the sentence of that chapter already writes the conditions, assumptions and scope, and this row merely copies it into one line. The Kind column is the three meanings of 10-16 (not shown, not read strongly, not called false).

Table 4 What Part 11 does not claim, and the sentences of the text that support it
Item	What is not claimed	Kind	Supporting sentence of the text (summary)
1	Solution of the problem of the existence of Yang–Mills and the mass gap. The continuum limit, recovery of rotations, the OS axioms and non-triviality are not touched	Not shown	11-00 §05: what is not dealt with. 11-07 §04: the third and fourth steps are not reached
2	That Assumption H is false. The statement that H↑ does not hold on the four-dimensional periodic box is a statement on top of the numbers of the originals for $SU(3)$ and above, and a reading that extends a conjecture attached to observations with open boundaries (that HMC and Langevin are in the same dynamic universality class) to the topological slowness on the periodic box	Not called false	(a)(b) of 11-01 §06, 11-07 §07
3	The sector H and the local H are candidate definitions, not shown at any $\beta$ . Nor do we claim "the right way to fix" H	Not shown	11-03 §07
4	The one-sided bridge (including the local version) is an inequality on a lattice with continuous time, and on Wilson's lattice with discretised time assumption (D) remains. The local version further rests on assumption (Loc-Δ_c)′	Not read strongly	11-04 §08 (Table 4), 11-04 §09
5	"The gap = the Poincaré constant of the vacuum measure" is an identity, and we do not claim it has made the problem easier	Not read strongly	11-05 §01, 11-05 §07
6	The definition of the length $\ell_*$ at which conservation breaks, and the restatement of what to compare with, $C_P\asymp\max(\xi^2,\tau_{\rm glob})$ , are candidates, not shown in any model	Not read strongly	11-02 §06, 11-02 §08
7	The results for hierarchical models and two-dimensional models imply nothing for four-dimensional Yang–Mills	Not read strongly	(a) of 11-06 §09, 11-02 §08
8	The non-perturbative mechanisms of physics (confinement, instantons, centre vortices, duality) are not used as tools of proof in this part	Not shown	11-07 §08 (Table 4)
9	The two points of "because it is four dimensions" are estimates as properties of the critical dimension, and we do not claim they happen only in four dimensions	Not read strongly	11-07 §01, (d) of 11-07 §09
10	The "what is not claimed" of Part 10 applies to this part as it is	—	10-16 §02
11	The local H goes only as far as the definition and the ratio form. At no $\beta$ has the local H↑ been shown for the target gauge theory. What has been shown is properties of the definition and values for the free-field reference	Not shown	11-03 §03, 11-04 §04
12	The thresholds of the local bridge ( $0.342\kappa$ , $0.502\kappa$ , about $0.62\kappa$ ) depend on the choice of reference and on the family of test observables. For fields whose ratio lies between the threshold and 1, the bridge says nothing	Not read strongly	11-04 §06 (Table 2)
13	For the 0+1 double well, the local ratio is not 1 even for the shortest window. Taking "the ratio approaches 1 at short distances" in four dimensions is an assumption depending on asymptotic freedom	Not read strongly	11-04 §07, (7) of 11-03
14	The length at which conservation breaks was measured only on the width-2 strip of two-dimensional $U(1)$ (outer boundary frozen, $\beta=2..6$ ). Nothing is said about four dimensions, the non-abelian case, or the exponent as $\beta\to\infty$	Not read strongly	11-02 §03 (Table 2)
15	For $SU(2)$ , it is undecided whether the freezing of the topological charge breaks Assumption H. There is one primary source, and its numbers are on the side close to $a^{-2}$	Not called false	11-07 §07, (b) of 11-01 §06
16	The results of the two-dimensional $CP^{N-1}$ test bed imply nothing for four-dimensional Yang–Mills (the ratio is a ratio of conditional autocorrelations at one fixed volume, a different quantity from the static ratio)	Not read strongly	11-02 §07

Items 1–10 are the outline from the design stage of the part, and 11–16 were added from the descriptions in each chapter. For item 2 we corrected the sentence from the design stage: the word "conjecture" is, in the original, attached to observations with open boundaries, so its application to the periodic box is written as "an extended reading". What is not claimed about the candidate rules (§02) is placed at the end of §02.

Relation to Part 10

The fourteen items of "what is not claimed" of Part 10 (10-16) apply to this part as they are. We do not list them again here. In particular, two of them — that not reaching is not evidence of being false (items 8 and 12 of 10-16), and that conditions fall away in summaries (10-16 §06) — apply equally when reading the estimate sentences of this part (label [Estimate]).

What this chapter can say

What is said	What is not said
The list of questions of Part 11, and what does not move even if each is settled (Table 1)	The answer to any of the questions. The "size" of a question is the kind of work, not an estimate of time
The list of candidate rules, and the estimate of their relation to the three roots (Tables 2 and 3)	The consistency of any candidate, or its prospect for the third step
The sixteen items that Part 11 does not claim, and the places in the text that support them (Table 4)	Whether the proposition of each item is true

Sources and reproduction

Item	Kind	Source / tools
Each row of Table 1	This book	The chapter and section in the right-hand column
The candidates and roots of Tables 2 and 3	Estimate	11-06 §07, 11-07 §05, §06. The sources of the existing results the candidates draw on are in the source table of each chapter (the four properties of the weight from the book by Bauerschmidt–Brydges–Slade, the two-dimensional loop equations from the summary in §6 of Chatterjee's review, log-Sobolev at strong coupling from the abstract of Shen–Zhu–Zhu)
Each row of Table 4	This book	The chapter and section in the right-hand column
The three meanings	This book	10-16 §01

Terms used in this chapter — the status of a question (done, partly done, not yet, new), what does not move even if settled, candidate rules, combinations, the three roots, the coupling window, not shown / not read strongly / not called false. Their meanings are in A-02 Glossary and 11-A1.

Next page to read: 11-A1 Table of notation for Part 11. To go back to the map of Part 11, 11-00.

Revised 2026-10-09: first version.