Signposts to a proof in our research

This chapter is a guide (signposts to a proof). It lays the rows of the ledger of conditions on a chain diagram in two layers, our predecessors and this book, and marks what has closed and what remains open.

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1. The road to a proof is drawn on one page as a chain of the ledger rows (①–⑩) and the gaps outside the ledger (Out 1–4).
2. Each box separates what earlier work closed (source and year) from what this book added (with labels).
3. What is shown is "closed", what is shown under conditions is "conditional", what is not shown is "open".
4. The two core rows, ① and ⑦, are broken down in separate figures.
5. How to read: take the whole picture in Figure 1, go from the red boxes to Figures 2 and 3 and Table 1, and jump to the text from the chapter tags (for example 14-02).

closed — closed on paper or in Lean (within the conditions written in the row).
conditional — closed under conditions; the conditions remain as numbers (measured ranges, finite β) or readings (numerics, sketches).
open — not shown.
earlier work — the grey strip on top of a box: source and year of what earlier work closed (bibliography in a-03bibliography, 15-02sources, 9-07tools).

Difficulty is not counted in stars but read from the five labels of this book: Proved here = a paper proof placed in a chapter of this book / Cited = known, checked in the source / Computer-assisted = computer-assisted (interval arithmetic) / Reading = numerical reading / Assumption = assumption or sketch. For an open box, read the difficulty from the labels and from "what is needed to remove it" (14-02ledger Table 2). Thick frames mark the core (① and ⑦).

The boxes of Figure 1 are the ledger rows (①–⑩) and the gaps outside the ledger (Out 1–4). Arrows point from premise to conclusion, and every row has a larger number than its premises (the layer order of 14-02ledger §04). The grey strip on top of a box is what earlier work reached; the colour at the bottom is what this book reached. A strip reading "—" marks a condition stated in this book's own terms, with no statement of the same form in the sources. Clicking a box opens the chapter that treats the row.

Figure 1. The chain to a proof (2026-10-15). Thick frames: the core rows ① and ⑦. The dotted line between ⑨ and ⑩ means "miniature", not implication. Out 4 (verification) has every row as premise, so its arrows are omitted. On a narrow screen, scroll sideways.

What to read from Figure 1. (a) The strip of earlier work is thick at ⑤ (reflection positivity), at both ends of ⑦ (strong coupling and one loop) and at ① (Bałaban's series); this book adds parts that fit the group, the twist and the form of the measure there. (b) Among the parts added by this book, only ⑤ is closed; the rest is conditional or open. (c) The red boxes are ①, ⑦ and the gaps outside the ledger; the interior of the band in ⑦ (where non-commutativity acts) and the multi-step induction of ① are the wall shared by both roads.

Ultraviolet control ① is split by the decomposition of Part 15 (15-03decomposition) into "a class closed under finitely many parameters" and three remainders. Figure 2 colours the four cells by today's state. Earlier work: Bałaban's series (1983–1989, including steps the source describes as unpublished; 15-02sources), centred on small-field steps around flat fields.

Figure 2. The four cells of ① (2026-10-15); the coloured edge is the state. The flux-network theorems F, G and the row-of-four theorem U are in 15-05flux network and serve as parts of cells 2 and 3.

The renormalisation comparison ⑦ splits into five regions along the Wilson coupling β (13-06renormalisation). The two ends rest on earlier tools (strong-coupling expansion: Wilson 1974, Osterwalder–Seiler 1978; one loop: Gross–Wilczek, Politzer 1973), with the group and the twist fitted in by this book; the middle is closed by nobody's tools.

Figure 3. The β map of ⑦ and the direction of the one-step coupling shift (shape only, no scale; the numbers with lattice, seeds and errors are in 13-09record). SU(2) has β′<β across the band, U(1) flips sign at β_c≈1.01 Reading. The one-loop comparison β−β′=0.258 is Proved here at one-loop level (not a non-perturbative claim).

An argument closing the band must distinguish SU(2) from U(1): U(1) has a transition inside the band, where an inequality of the form ⑦ fails. The middle region ⑦c needs "one inequality that uses non-commutativity", with four faces (softness of the twist direction, mixing of box fluxes, cost of twisting a column, pressure margin); none is proved Assumption. A fifth candidate face (proposal SD) gives R=1.34(7) (4⁴/2⁴) and R=1.17(20) (6⁴/3⁴) in finite boxes, within resolution, so undecided Reading (15-04hierarchical).

The first road goes to the mass gap through the transfer matrix in the time direction. The chain of statements (10-15remaining distance, 10-12integrated table):

1. Construction line (10-01construction) — Langevin path measures on the lattice, and the one-sided bridge: the inverse squared gap bounds the Poincaré constant from below Proved here (published as a paper, a-07papers).
2. First step (10-05first step) — for the double-well path measure, the convex side (chain length ≤7) and uniqueness of the minimum are closed in the record; the section placing them in the book is a supplement Assumption.
3. ⑨ first-step remainder — a gap uniform in k and ℏ for chain length k≥8 (a uniform log-Sobolev inequality including non-convex regions) Assumption. A miniature of the band.
4. ⑩ four-dimensional target — the upper side ξ²↑ of the diffusive relaxation-time assumption (formerly hypothesis H↑) and the one-sided bridge, uniform as β→∞ Assumption.

Today ⑨ and ⑩ stand in conditional form; removing ⑨ does not move ⑩ (⑨ is a miniature, not an implication). ⑩ means crossing the band in the language of time, and it stops at the same place as ⑦ on the second road (13-07from time).

The same state is drawn in three views: the chain map (Figures 1–3), the classification by chapter (this section), and the dependencies of Part 10 (§05 and 10-12integrated table). This section sorts the chapters of the book into three classes and counts them by part.

A table classifying every chapter (grouped by ranges of chapters) and the vertical dependency diagram of Part 10 (conditions (H1)–(H4) and the two routes) will be added in the next version of this chapter. For now the dependencies of Part 10 are read in Figure 1 of 10-12integrated table and in 10-15remaining distance.

Five steps chosen from the table of questions (14-03questions) and the questions of Part 15 (15-09UV index), premises first. Each is a small question; settling it does not by itself close ① or ⑦.

1. ① cell 2: on a slab (many links), measure the difference in total variation between densities with and without one large-field region, and see whether (S_large)≤C′g² holds in the smallest system (F-UV12). 15-08four rows
2. ① cell 3: check whether the parity at the critical point holds for four plaquettes; if so, the background uniformity of (S_small) weakens (F-UV14). 15-08four rows15-07background
3. ① sources of observables: add a source to one loop in a row of four plaquettes and measure whether the coupling shift after one step is independent of β (F-UV13). 15-08four rows
4. ⑦a lower end: turn the finite-size criterion by mixing of box fluxes into a certificate with interval arithmetic. 13-06renormalisation15-04hierarchical
5. ⑦c middle: write one of the four faces uniformly in scale, in a form that fails for U(1); decide proposal SD in boxes of higher resolution. 13-06renormalisation13-10candidates

The lists of non-claims are in 14-04non-claims and 15-09UV index. Three are specific to this chapter. (1) Counting the colours in Figure 1 does not give a fraction of the way (the ledger is not a table that only shrinks, 14-02ledger §06). (2) The strips of earlier work summarise what the sources reached; they are not the statements of their theorems. (3) Even if every box turned green, that would not amount to a solution of the problem (Out 1–4 are gaps outside the ledger, shown in Figure 1 to make the distance visible).

Notes — Notation: ① ultraviolet control, (Cls) class, (S) one-step remainder, (Rem-shape) shape remainder, (C) continuity, (U3) the effective coupling stays small (15-03decomposition). ⑦a, ⑦b, ⑦c are the lower end, upper end and middle of ⑦ (13-06renormalisation). ξ²↑ is the upper side of the diffusive relaxation-time assumption (formerly hypothesis H↑, a-06assumptions). β_c≈1.01 is the U(1) transition inside the band.

What this chapter can say

Sources and reproduction

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