2026-10-09 · chapter Yang–Mills and the mass gapPart 11 Further considerationsentrance, map of nine chapters · Kiichi, Shiori and Rin

The whole book　Contents and reading orders

Further considerations — entrance — what is taken from Part 10

What this chapter shows — Where Part 11, "Further considerations", stands; the sentences it takes from Part 10; the map of its nine chapters (title, one line, level, and what each chapter can say); the words fixed in this part; and what this part does not treat.

Prerequisite chapters — 10-00 (the entrance to Part 10), 10-01 §05 (Assumption H), 10-01 §06 (what H gives and does not give).

Said first — The level is [Y2]. This page is a map and proves nothing. No chapter of Part 11 shows the mass gap. We write neither that Assumption H was wrong nor that it was the right road.

The order of this chapter

Where this part stands — what to think about after the study of Part 10
Sentences taken from Part 10 — five sentences and their labels (Table 1)
Map of the part — the nine chapters, and what each can say (Figure 1, Table 2, Table 3)
Words fixed in this part — the local H and the ratio form, the threshold, the length at which conservation breaks
What is not treated, the table of notation, returning to Part 10

Where this part stands — what to think about after the study of Part 10

Part 10 restated the problem of the mass gap as a relation between quantities that can be computed on the lattice and are within reach of machine checking — a power relation between the relaxation time $C_P$ of Langevin dynamics and the correlation length $\xi$ (Assumption H) — and used that restatement as a foothold for its study. Setting up H as a sharp sentence made it possible to name where H fails. Which test functions obstruct H↑, the upper half of H, came together as a single pattern. This is not "H was wrong"; the order of events is "because H was set up, the nature of how it breaks became visible".

Part 11 is the "further considerations" that start from there. First we see that what obstructs H comes together in one form — a global quantity not reflected in the gap slows relaxation on a length other than the correlation length (11-01, 11-02). Next, keeping H as a premise, we define two candidates that restate H so as to avoid that form (H for each topological sector, and the local H with windows or boxes limited to a few times the ruler), and examine which part of them the one-sided bridge needs for the mass gap (11-03, 11-04). From there we go on to two views that do not pass through H: that on a lattice with continuous time the gap is exactly the Poincaré constant of the vacuum measure (11-05), and that on groups one needs tools that pay with rarity instead of convexity (11-06). Finally we state where being four-dimensional matters, limited to the two points "the dimensionless coupling" and "topology" (11-07), and close with questions and what is not claimed (11-08).

Many sentences of this part are estimates, and we distinguish them by adding one estimate label, Estimate, to the five labels of Part 10 (Proved here, Cited, Computer-assisted, Reading, Assumption). An estimate is an opinion offered as material for judgement; it carries neither a proof nor numbers.

Sentences taken from Part 10

Table 1 lists the five sentences that Part 11 takes from Part 10 and uses. Labels and statements remain as in Part 10; this part only cites equation numbers and sections, and does not copy the proofs.

Table 1 Sentences taken from Part 10
Sentence	Label	Place in Part 10	Chapters that use it
The trade of the restatement: the quantity of the problem statement is the gap in Euclidean time; the quantity of H is the relaxation $C_P$ in Langevin time (a quantity defined by a supremum, bounded from below by a single test function, and whose upper bound can be built from parts). The two are different quantities	Convention	10-01 §04 (the three $C_P$ )	11-01, 11-04, 11-05
Assumption H is the power statement $\lim\ln C_P/\ln\xi=2$ . The halves of the band statement are written H↑ and H↓	Assumption	10-01 §05	All chapters
The one-sided bridge: on a lattice with continuous time, $C_P\ge\kappa\hbar^2/\Delta^2$ . Under the assumption of the bridge, H↑ is a stronger sentence than the gap of the lattice in physical units (there is also a version in the form of a paper, before external checking)	Proved here (continuous time), Assumption (assumption (D) on Wilson's lattice with discretised time)	10-01 §06 (Lean `hup_is_gap_positivity_of_lower`, assumptions: $0\lt m$ , $\kappa/m^2\le C_P$ , $C_P\le K\xi^2$ )	11-04, 11-05
The converse is false in general: for a free particle on a circle, $\Delta\gt0$ holds while $C_P=\infty$ (Proposition C)	Proved here	10-01 §06	11-01, 11-04
In the four-dimensional periodic box, H↑ reads as broken by the freezing of the topological charge (on original sources for $SU(3)$ and above, and a reading that extends a conjecture). The note on topological sectors and the order of volume	Assumption, Reading	the note in 10-01 §05, 10-02 §05	11-01, 11-03, 11-07

Map of the part — the nine chapters, and what each can say

The nine chapters of Part 11 and the table of notation

11-00 Entrance — what is taken from Part 10 [Y2] (this page)
11-01 One form that obstructs H — a nearly conserved global quantity [Y3]
11-02 Choosing the ruler and the coordinates again — the length at which conservation breaks [Y3]
11-03 Two ways to repair H — the sector H and the local H [Res] (summary at [Y3])
11-04 Which H the one-sided bridge needs [Res] (summary at [Y3])
11-05 Making time continuous — the Hamiltonian lattice and the vacuum measure [Y3]
11-06 Paying with rarity instead of convexity — renormalisation on groups and log-Sobolev [Res] (summary at [Y3])
11-07 Because it is four dimensions — the dimensionless coupling and topology [Y3]
11-08 Questions, and what is not claimed [Y2]–[Y3]

11-A1 Table of notation for Part 11 (a reference page)

The order of the chapters is: the form of the breakage (11-01, 11-02), restatements that keep H as a premise (11-03, 11-04), views that do not pass through H (11-05, 11-06), what is specific to four dimensions (11-07), and questions (11-08). From 11-01 to 11-05 is a single chain in which the conclusion of each chapter becomes the question of the next. 11-06 is not a consequence of the chain but a chapter of tools needed to show any of the statements of 11-03 to 11-05, and on the line of the argument it is placed as a branch.

Figure 1 The routes of Part 11 (an arrow means "the conclusion
of the previous chapter becomes the question of the next";
it is not a theorem of implication)

  11-01 One form ──→ 11-02 Ruler, coordinates ──→ 11-03 Sector H,
                       │                          local H
                       │                            │
                       │              (ratio form, threshold)
                       │                            ↓
                       │           11-04 Which H the bridge needs
                       │                            │
                       │                            ↓
                       │                11-05 The vacuum measure
                       │
     Two-dimensional test beds (the U(1) strip, CP^(N−1))
        ── a branch reaching 11-02 and 11-07

  Tool branch: 11-06 Paying with rarity instead of convexity
               (needed by every one of 11-03 to 11-05)
  Specific to four dimensions:
               11-07 The dimensionless coupling and topology
  Closing: 11-08 Questions, and what is not claimed

Table 2 One line for each chapter, and what that chapter can say
Chapter	One line	What this chapter can say
11-01	Puts in one table, together with their strength, that the number of kinks, the winding number, the flux, the topological charge and the flux through the box are all of the same pattern: "a global quantity not reflected in the gap slows relaxation on another scale"	That the five examples are of the same pattern is a classification (an estimate). The numbers carry the label of each row
11-02	Splits the length to compare with $C_P$ into three rulers, and sets out that the type of relaxation changes with the choice of coordinates (the Dirichlet form), together with numbers for the two-dimensional strip and the test beds	The length at which conservation breaks is a reading on the two-dimensional $U(1)$ strip. Restating what to compare with is a candidate. $CP^{N-1}$ is a conditional reading at one volume and one correlation length
11-03	Defines H conditioned on topological sectors, and the local H with windows or boxes limited to a few times the ruler, and states them in the ratio form against a massless reference	The two forms of H have definitions, and the local H has content only in the ratio form. There is no inequality shown at any $\beta$
11-04	States that the one-sided bridge uses only the part of H↑ that is local in time, and sets out the bridge on a window, the threshold of the ratio form and three references	The threshold of the bridge depends on the reference and the family, and the band between the threshold and 1 is undecided. Everything rests on assumption (Loc-Δ_c)′ and the assumptions of Proposition B
11-05	Sets out that on a lattice with continuous time the gap is tied, as the exact reciprocal, to the Poincaré constant of the vacuum measure at a single time, and why the tools needed there do not apply	The gap is the Poincaré constant of the vacuum measure itself, and the identity does not make the problem easier
11-06	States that on groups the tools of convexity cannot, in principle, stand at weak coupling, that the non-convex places are pushed into places of small probability, and the tools that pay with rarity and candidate rules	A chapter of tools. The ceiling of convexity has a proof; the class of measures that pay with rarity is a candidate. Results on hierarchical models imply nothing for four dimensions
11-07	Limits what matters in four dimensions to two points, the dimensionless coupling and topology, and sets out the four hurdles and, from what primary sources say, why the mathematics of the third hurdle does not yet exist	The two points are an estimate of a property of the critical dimension. Freezing for $SU(2)$ is undecided. Results in two dimensions imply nothing for four dimensions
11-08	The table of questions, the list of candidate rules, the sixteen items that are not claimed	Questions, and what is not claimed. Candidate rules are candidates

The list of material that this page used to hold (titles and one line each) is now taken up as body text by the chapters of Table 3.

Table 3 Material of the former entrance, and the chapters that take it up
Material	Related in Part 10	Chapter that takes it up
The table of other roads (the renormalisation group and log-Sobolev, stochastic quantisation, the Hamiltonian lattice, hierarchical models, and so on)	10-02 §08	11-05, 11-06
H↑ and the one-sided bridge to the mass gap	10-01 §06	11-04
Hierarchical gauge models and the problem of coordinates	10-17 Insight 12	11-02 §02, 11-06 §05
Conservation of flux and "the length at which conservation breaks"	10-17 Q17	11-01, 11-02
The study of Assumption H on the four-dimensional lattice (freezing of the topological charge, and how to correct it into a sentence per sector or a local sentence)	the note in 10-01 §05, 10-02 §05	11-01, 11-03

Words fixed in this part

The local H and the ratio form. The sentence that states the Poincaré constant $C_P^{\rm loc}(c)$ , with test functions limited to those supported in a window (or box) of $c$ times the ruler $\ell$ , by its ratio $K_{\rm rel}(c)$ to the massless reference on the same window with the same graduation (11-03 §03). The form "bounded by some constant" also holds for the free field and is empty, so we use the ratio form.

The threshold $K_*$ . The value of the threshold such that, if the ratio is below it, the one-sided bridge gives a lower bound on the mass under the assumptions. Depending on the choice of reference it changes: $0.342\kappa$ (scalar), $0.502\kappa$ (gauge quotient), about $0.62\kappa$ (colour singlet of $SU(2)$ ) (11-04 §06; the numbers are readings for the free-field reference).

The length at which conservation breaks $\ell_*$ . The length at which a global quantity not reflected in the gap begins to relax through the breaking of conservation rather than through local transport. It is a ruler different from the static correlation length, and the values measured on the two-dimensional $U(1)$ strip are not proportional to the correlation length (11-02 §01, §03).

What is not treated, the table of notation, returning to Part 10

What is not treated. Part 11 does not touch the ultraviolet side — the existence of the continuum limit, the recovery of rotations, the OS axioms, non-triviality ((S3)(S6)(S7)(S8) of Part 10, 10-01 §02). Only 11-07 §05 sets out, from primary sources, how far the third hurdle has been reached as a difficulty specific to four dimensions, and even there construction is not treated.

Notation. The notation of Part 11, and its correspondence with symbols of the same letter in Part 10, are in 11-A1 Table of notation for Part 11. In particular $\Delta$ is reserved for the gap of the Hamiltonian $E_1-E_0$ , and the mass in physical units is written $m_{\rm phys}$ .

Returning to Part 10. The map of Part 10 is in 10-00 §01, what is not claimed in 10-16, and the small questions in 10-17.

What this page can say

Said	Not said
The order of the chapters of Part 11, and one line on what each chapter can say (Table 2). The sentences taken from Part 10 and their labels (Table 1)	No chapter shows the mass gap. Nor does any show whether Assumption H is true
The meaning of the words fixed in this part (§04)	Theorems about the local H, the threshold, or the length at which conservation breaks (we go as far as definitions, the ratio form, and numbers for the free field and in two dimensions)

Sources and reproduction

Item	Kind	Source / tools
The sentences of Table 1	This book	The sections and source tables of 10-01 and 10-02
The rows of Table 2 and §04	This book	"What this chapter can say" in each chapter of Part 11

Terms used on this page — further considerations, the trade of the restatement, the one-sided bridge, the estimate label Estimate, the local H, the ratio form, the threshold, the length at which conservation breaks, the tool branch. Their meanings are in the A-02 Glossary and 11-A1.

Next chapter to read: 11-01 One form that obstructs H — a nearly conserved global quantity.

Revised 2026-10-09: rewrote the page listing material as an entrance with body text.