2026-10-01 · chapter Yang–Mills and the mass gapAppendixThe verification procedure

The whole book　Contents and reading orders

How this book was made — labels and source tables record how far a sentence was checked

What this chapter shows — The seven stages a sentence passes through before it appears in the text, and what each stage catches and does not catch (Figure 1).
The meaning of "text", "abstract" and "bibliographic data only" in the source tables, and that about half of the verdicts in the refutation check were corrections (with tables counted on this machine).
The convention for the label "Computed", the stage that aligns symbols and names across chapters, the mechanical checks of assembly, and what this procedure does not guarantee.

Prerequisite chapters — None. It is easier to read after looking at the legend of the labels (0-01). The procedure of machine checking with Lean is not dealt with in this appendix; it is in A-04.

Said first — What this appendix explains is a procedure for reducing errors, not a guarantee of correctness. This book has not been through expert peer review. Also, the existence and mass gap of four-dimensional Yang–Mills theory are unsolved, and no stage of this book shows them.

The order of this chapter

The role of this appendix — labels and source tables are a graduation for measuring how far one may believe; the procedure that marked that graduation
The road a sentence travels — seven stages, and what each catches and does not catch (Figure 1, Table 1)
Cross-checking against primary sources — three levels: text, abstract, bibliographic data only (Table 2); Detour: writing in the form of hearsay
Looking for refutations — another reader checks "is this too strong?"; the breakdown of verdicts and the types of correction (Tables 3 and 4)
Making computations reproducible — the label "Computed" covers a finite range; write the conventions; a scale that finishes in minutes
Attaching labels — a record of how far a sentence passed; the composition of labels by part (Figure 2, Table 5)
Aligning across chapters — five things: symbols, conventions, names, duplication, degree of certainty
Assembling and checking mechanically — pre-rendering of formulas, links, forbidden words (Table 6)
What this procedure does not guarantee — not peer review; texts not obtained; finite computations; an unsolved problem

The role of this appendix — so that readers can measure for themselves how far to believe

Sentences in textbooks tend to be written on the premise that readers will believe them as they are. This book does not do that. Each sentence carries a label (the legend in 0-01), the source tables write "whether the text was read", "only the abstract" or "only the bibliographic data", and computations are kept in a form that can be reproduced at hand. Labels and source tables are a graduation with which readers can measure for themselves "how far this sentence may be believed". Precisely, a label is a record of how far a sentence was checked, not a number expressing a degree of correctness.

A graduation cannot be read unless the procedure that marked it is known. For example, even for a sentence labelled "Known", the weight differs according to whether it was checked word for word against the text of a primary source or copied from the description in a review. This appendix explains that procedure from the reader's side. Stage by stage, it writes what was checked and what was not.

The flow in one line:

\text{research} \;\to\; \text{cross-check against primary sources} \;\to\; \text{look for refutations} \;\to\; \text{reproduce computations} \;\to\; \text{attach labels} \;\to\; \text{align across chapters} \;\to\; \text{assemble and check mechanically}

Each of the seven stages is placed to catch what the previous stage overlooks. But no stage guarantees "correctness". What is not guaranteed is collected in §09.

Example. The sentence of Part 5 that "the Wilson action has reflection positivity" carries the label "Known", and the source table lists the paper of Osterwalder–Seiler (5-03). The label "Known" shows that it is a known result checked in a primary source. The depth to which it was checked (text, abstract, bibliographic data only) is written in the source table, and the source table for this example says "abstract checked, text not checked". Whether it passed the refutation check (§04) is not distinguished by the label. If the source table says "bibliographic data only", the cross-checking stopped at the level of bibliographic data.

The procedure of machine checking with Lean is not dealt with in this appendix. It is in A-04. How labels are decided (one label per sentence; the strength of a chain is its weakest stage) also has its master copy in A-04 §07.

The road a sentence travels — seven stages

Figure 1 lines up the stages a sentence passes through before it appears in the text. Moving the knob shows below what that stage catches and what it does not.

The order of the stages has reasons. If refutation (stage 3) were placed before cross-checking (stage 2), the refuting side would not know what to compare against. Labels (stage 5) come after cross-checking and computation because a label is a record of "how far it passed". Consistency (stage 6) and assembly (stage 7) are stages for errors that cannot be caught by looking at one chapter at a time, and are run each time a part is complete (and finally for the whole book).

Figure 1 Choosing a stage with the knob shows below what that stage catches and does not catch. The frame of the chosen stage becomes thicker

Stage

Step 1/7 Research.

Table 1 has the same content as Figure 1 (the figure is schematic and contains no numbers).

Table 1 What the seven stages catch and do not catch
Stage	Name of the stage	What it catches	What it does not catch
1	Research	Literature on each topic, known results, candidate numbers	Whether the candidates are correct (here candidates are only collected)
2	Cross-check against primary sources	Errors in bibliographic data, over-paraphrased citations, misattribution	Errors in the primary sources themselves, the content of texts that could not be obtained
3	Look for refutations	Wording stronger than the source, promotion of the degree of certainty, omitted conditions	Cases where the cross-checking reader and the refuting reader share the same misunderstanding
4	Reproduce computations	Miscopied numbers, mixed-up conventions, counterexamples within a finite range	Whether it holds outside the range computed
5	Attach labels	Mixing of "proved", "conjecture" and "numerically supported"	Errors in the judgement of the label itself
6	Align across chapters	Discrepancies in symbols, names and conventions; the same name pointing to different sentences; duplication	All chapters being wrong in the same direction
7	Assemble and check mechanically	Formulas that cannot be rendered, broken links, forbidden words, unfilled placeholders	Errors of content (the machine does not read meaning)

Cross-checking against primary sources — three levels: text, abstract, bibliographic data

What this book calls a "primary source" is the paper that first stated a result, or the problem statement itself. Reviews and textbooks are used as guides to the primary sources, not as substitutes for them. Years and attributions (who showed what, when) are not written from memory but checked in the sources.

Cross-checking has three levels of depth, and the source table writes which level was reached.

Table 2 Depth of cross-checking, and how it is written in the source tables
Depth	What was read	How it is written in the source table	What can be written in the text
Text checked	The text of the paper (a public PDF or the like) was read, and the relevant passage checked word for word	Authors, year, journal or arXiv number, section	Exactly the strength stated by the source
Abstract checked	Only the abstract (the abstract in a literature database) was read	As above + "checked in the abstract", "abstract only" and so on (the wording differs slightly by chapter)	As far as is written in the abstract
Bibliographic data only	Only that the authors, title, journal and year exist was checked	"Bibliographic data only"	As far as the form in which another source reports that "… did …"

Word-for-word checking is the work of confirming by search that the original text enclosed in quotation marks exists literally in the source. Even when a translated sentence is placed in the text, the cross-checking is done on the original.

Example. The problem statement of the Clay Institute (Jaffe–Witten) was cross-checked in the text of the public PDF. The requirement of the problem, in the original, is as follows.

“Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R⁴ and has a mass gap Δ > 0.”

0-03 reads this sentence word by word. Because it was cross-checked in the text, 0-03 can use each of the words "existence", "non-trivial" and "mass gap" at the strength of the problem statement.

On the other hand, for papers of which only the abstract could be read, claims in the abstract are written as "checked in the abstract", and claims not in the abstract are not brought into the text. The same goes for section numbers: if the abstract does not mention sections, which section something is written in is not known until the text is read.

Example. The abstract of the review by Wilson–Kogut (Phys. Rep. 12 (1974) 75) is a single sentence and does not mention sections. So this book does not write that some topic "is discussed in §10 of Wilson–Kogut", and writes "§10 not checked" in the source table.

Detour: writing in the form of hearsay

Detour You may skip this. That the upper limit on the spacetime dimension in which supersymmetry can be realised with fields of spin at most 2 (supergravity, including the graviton) is 11 is widely known as a consequence of Nahm's 1978 classification. Yet the number 11 does not appear in the abstract of that paper. While the text cannot be obtained, this book writes "known as a consequence of Nahm's classification", and does not attribute the conclusion directly to Nahm's paper (7-05). It is not that the fact is doubtful. It is to show honestly, in the form of the sentence, how far this book checked. Precisely, the form of hearsay is not a mark that "it may be wrong" but a mark that "the cross-checking did not reach the text, and the abstract does not contain the number".

Looking for refutations — another reader checks "is this too strong?"

Material that has been cross-checked is handed to another reader. That reader's job is not confirmation but refutation. Going back to the primary sources once more, they look for "is this sentence stronger than what the source says?", "it was written as settled, but does it not need a condition?", "does this bibliographic entry exist?". There are three verdicts.

Confirmed (confirmed): against the source, the written strength is fine as it is.
Corrected (corrected): it needs to be weakened, given a condition, or have its bibliographic data fixed. The wording of the correction is attached.
Unverifiable (unverifiable): the primary source could not be reached, and neither can be said. It is lowered to the form of hearsay.

At the stage of writing chapters, the rule is to reflect every correction. Material that received a correction is not cited at its strength before the correction.

Table 3 counts the verdicts for the six materials used mainly by Parts 6 to 10 (and partly by chapters of Parts 0, 4 and 5). Many chapters of Parts 1 to 5 have not been through the materials of this refutation check.

Table 3 Verdicts of the refutation check by material (counting the materials as of 2026-10-01) Computed
Subject of the material	Confirmed	Corrected	Unverifiable	Fraction corrected
Dictionary of the physical and the mathematical vocabulary	25	20	0	44%
Extra dimensions and where gauge fields come from	16	12	1	41%
The physical view (confinement, lattice numerics, holography)	25	17	0	40%
Arrangement of the investigations on this machine	7	14	0	67%
Where things stand, dimension by dimension	25	8	0	24%
Where constructive field theory stands, and candidates for new concepts	16	29	0	64%
Total	114	100	1	47%

How to read it. About half of the verdicts are corrections. The highest fraction of corrections is for "arrangement of the investigations on this machine": in this material corrections were many, 14 of 21, and those that weakened wording stronger than the record, or added conditions that had been dropped, stood out. Conversely, materials close to a list of theorems, such as where things stand dimension by dimension, have fewer corrections.

Table 4 sorts the content of the corrections into four types by words. The sorting is mechanical, by words appearing in the wording of the correction, and one correction may fall into several types.

Table 4 Types of the 100 corrections (mechanical sorting by words; with overlaps) Computed
Type	Cue words	Count	Typical
Strength	too strong, promoted, showed, restricted	39	Lowering "showed" to "proposed"; "settled" to "conditional"
Level of the source	abstract, text, not obtained, not checked	34	Rewriting a claim not in the abstract as "text not obtained"
Numbers and signs	number, coefficient, sign, digit	15	Adding the convention for the coefficient (which $\beta$ )
Bibliographic data	bibliographic data, volume, page, author, journal	12	Corrections to the list of authors, such as a missing co-author
None of the above	—	30	Supplements, suggestions for rewording

Example. What §6.5 of the Clay problem statement says is that "with many regularisations, such as dimensional regularisation and Pauli–Villars, reflection positivity is lost, and there are few ways to recover it"; it does not reject continuum regularisations across the board. This book cites it at this strength, and does not write "continuum regularisations do not preserve reflection positivity". What the refutation stage catches is summaries that have become one step stronger than the source in this way (regularisation and reflection positivity are in 4-10 and 5-03).

Making computations reproducible — the content of the label "Computed"

Of the numbers in the text, those that can be recomputed on this machine are recomputed and given the label "Computed". All computations are short programs (Python), kept chapter by chapter. Their names are written in the "tools" column of the source tables, so running the same programs gives the same tables.

What the label "Computed" means is that it was checked on this machine, over the range written. As a formula, having checked a property $P(n)$ for finitely many $n$ is the distinction

P(n_1) \wedge P(n_2) \wedge \cdots \wedge P(n_k) \qquad\text{and not}\qquad \forall n\ P(n) \ \text{}

The left is a computation, the right a theorem. A sentence with the computed label is not cited as a theorem.

Computations have three conventions. The first is not to make numbers inside figures. The numbers in moving figures are limited to tables computed in advance and embedded, or simple formulas that can be computed on the spot (rotations, partial sums of series and so on). The second is to write the conventions. If the definition of the lattice coupling $\beta$ or the normalisation of the trace $\operatorname{tr}$ differs, the same quantity is off by several times, so the conventions are stated in the premises of a computation (conversion between conventions is in 5-01 and A-01). The third is to keep to a scale that finishes within a few minutes on an ordinary computer. A computation that readers cannot reproduce loses the meaning of the label.

Example. Tables 3, 4 and 5 of this appendix are also computations. We wrote one program that reads the materials and chapter pages and counts, and copied its output. Over the whole book, the places where computations are kept, chapter by chapter, number over 100, and there are several hundred programs.

Attaching labels — a record of how far a sentence passed

A label is a record that tells readers how far a sentence passed. A sentence that stopped at cross-checking gets "Known", a number reproduced at hand gets "Computed", something accepted in physics but without proof gets "Physics", and so on. The meaning of the five labels, and what the labels do not guarantee, are in the table of 0-01 §03.

There are two rules for attaching labels. One sentence gets one label. And for a claim connected through several stages, the label of the weakest stage is attached to the whole (A-04 §07). For example, if numbers are put as assumptions into an inequality checked in Lean and a conclusion is drawn, the label of the conclusion is "Computed" or "Physics". Labels are not pushed towards the strong side.

Figure 2 counts, by part, the labels appearing on the chapter pages of this book. The composition of labels differs clearly from part to part.

Figure 2 Number of labels by part. "View by fraction" scales each part to 100%. Choosing a part with the knob shows its breakdown below

Part

Part 10 selected.

ComputedThe numbers in this figure were computed on this machine (counting the chapter pages of 110 chapters as of 2026-10-01, excluding this appendix itself. Labels placed in legend tables are also counted. The numbers change as chapters are added).

Table 5 Number of labels, number of figures, and occurrences of "bibliographic data only" by part (as of 2026-10-01) Computed
Part	Chapters	Lean	Paper	Computed	Known	Physics	Figures	Bibliographic data only
Part 0 Entrance	4	3	3	50	97	14	8	11
Part 1 Mathematical preparation	12	0	3	165	229	18	40	40
Part 2 Physical preparation	10	0	0	123	166	53	28	41
Part 3 Classical theory	7	0	0	76	124	32	13	20
Part 4 Quantum theory	12	0	1	142	216	49	23	45
Part 5 Lattice	11	10	11	144	179	80	22	25
Part 6 Constructive field theory	12	0	7	130	200	30	24	51
Part 7 From the side of physics	13	3	0	154	150	169	28	60
Part 8 A dictionary of two languages	5	3	3	40	70	33	7	14
Part 9 Where things stand, and open tasks	5	0	1	44	106	55	10	14
Part 10 The record kept on this machine	14	93	38	136	130	67	25	47
Appendices (A-01, A-02, A-03, A-04, A-06)	5	21	7	37	27	18	10	20
Total	110	133	74	1241	1694	618	238	388

How to read it. Most labels are "Known" and "Computed". This is because much of this book is an exposition of known mathematics and physics, with numbers reproduced at hand added to it. "Physics" is most frequent in Part 7, because its centre is material accepted in physics but not theorems, such as mechanisms of confinement and the string picture. The "Lean" labels are concentrated in Part 10. However, many of those Lean theorems are statements about inequalities of real numbers or finite sums, and do not show the existence or the mass gap of four-dimensional Yang–Mills theory (10-16, A-04 §06).

Aligning across chapters — symbols, names, conventions

Checking chapters one at a time cannot catch discrepancies between chapters. For example, if one chapter writes $\beta = 2N/g^2$ and another $\beta = 1/g^2$ , each chapter may be correct on its own, but readers take them to be the same $\beta$ and get lost when the numbers do not match. So each time a part is complete, a stage of reading across parts is run.

This stage looks at five things.

Symbols: whether the same symbol is used for the same quantity. The standard is the table of notation A-01.
Conventions: when the coupling $\beta$ , the normalisation of the trace, or the lattice metric differs between chapters, whether a conversion is written (5-01).
Names: whether one name points to different sentences in different chapters. For example, Assumption H of Part 10 can be read in eight ways, according to which length it is measured in, which normalisation it is measured in, and whether it is stated as a power or a band. A name is assigned to each reading and fixed (calling the power level $H$ and the band statement $H_{\mathrm{band}}$ ; A-06), and other chapters link there and use the same name. The list of assumptions is in A-06.
Duplication: when the same explanation is in two chapters, make one the master copy, and reduce the other to a one-line summary and a link.
Degree of certainty: whether one chapter writes the same correspondence as "settled" and another as "conditional". In the dictionary part, correspondence as a definition (which can be settled) and actually holding in four-dimensional Yang–Mills theory (unsettled) are written in two separate columns (8-01).

When a discrepancy is found, it is aligned to one side or the other. Discrepancies that cannot be aligned (for example when conventions genuinely differ between papers) are written in the text as discrepancies.

Example. $C_P$ in Part 10 is a number that serves as a measure of the time it takes for fluctuations to be smoothed out (defined in 1-10 and 10-01). With the same symbol $C_P$ , it can point to different numbers according to which fluctuations are meant (of the field at one time, or of the whole path including the time direction). For the free field one is 4 times the square of the other, so mixing them up doubles the power. This is an example distinguished at this stage, and the whole book writes the two as different quantities (10-01).

Assembling and checking mechanically — what can be left to the machine

The last stage is done by machine. The tool that assembles the chapter pages does the following mechanically for all chapters, and reports what gets caught.

Table 6 Mechanical checks at the assembly stage
Check	What it looks at	What it does not look at
Rendering of formulas	Renders formulas written in TeX to static HTML with KaTeX, and reports formulas that cannot be rendered	Mathematical correctness of the formulas
Links and headings	Whether the files linked between chapters and the heading `id`s exist	Whether the link target fits the flow of the argument
Forbidden words	Whether real names, paths, internal numbers or unfilled-placeholder marks remain	The strength of wording (this is looked at by people and by the refutation stage)
Templates	Whether a class name not to be used for figure frames (`fig`) remains	The content of figures
PDF	Makes a PDF for printing for each part (moving figures go in as still images of their initial state)	Whether it is easy to read in print

Formulas are rendered at the assembly stage rather than in the reader's browser so that no JavaScript is needed on the reader's side, and so that they do not break when printed or made into PDF. Only moving figures use JavaScript, and even when JavaScript is disabled, the same content can be read in the table below the figure.

Mechanical checks do not read meaning. Passing this stage guarantees that the page is not broken, not its content. Judgements of the strength of wording, the choice of labels and the depth of sources are not left to the machine; people and the writers read them at the stages of cross-checking, refutation and consistency.

What this procedure does not guarantee

Errors can remain even after the seven stages. We write the limits of this book from the side of the procedure.

It has not been through expert peer review. The reader at the refutation stage is a different reader from the writer, but not a referee of an academic journal. Much of the research, drafting and refutation for this book was done by the AI writers of this site; people decide the policy and read before publication. Readers of the same kind can share the same misunderstanding. What the refutation stage cannot catch is exactly such errors (stage 3 of Table 1).

The content of texts that could not be obtained has not been checked. For works marked "bibliographic data only" or "checked in the abstract" in the source tables, this book does not know what is written in the text. The count of "bibliographic data only" in Table 5 is a measure of the size of that range.

Computations cover finite ranges. Numbers with the label "Computed" are what the computations at hand gave over the range written, and say nothing outside the range or about the continuum limit (§05).

The mass gap problem is unsolved. No stage of this book shows the existence or the mass gap of four-dimensional Yang–Mills theory. A list of what the records of this book do not claim is in 10-16.

If you notice an error, please let us know through the board or the contact page of this site. Corrections are recorded with dates in the "Revised" field at the end of each chapter.

What this chapter can say

What is said	What is not said
That the sentences of this book appear in the text after passing through the seven stages (research, cross-checking, refutation, computation, labels, consistency, assembly). What each stage catches (Table 1)	That sentences that passed the seven stages are correct. No stage guarantees correctness
That "text", "checked in the abstract" and "bibliographic data only" in the source tables are records of the depth of cross-checking (Table 2)	What is written in the texts of works marked "bibliographic data only" or "abstract"
That in the refutation check of the materials as of 2026-10-01, of 215 verdicts, 100 were corrections and 1 was unverifiable Computed (Table 3)	That the 114 judged confirmed contain no errors. If the refuting reader and the cross-checking reader share the same misunderstanding, this stage cannot catch it
The counts from sorting the wording of corrections by words Computed (Table 4)	That the sorting classifies the content of corrections correctly (it is a mechanical classification by words, with overlaps)
The numbers of labels, figures and "bibliographic data only" in the chapter pages of 110 chapters as of 2026-10-01 (excluding this appendix itself) Computed (Table 5, Figure 2)	That the number of labels represents the weight of claims. Labels placed in legends are also counted, and the numbers change as chapters are added
What the assembly stage looks at mechanically (Table 6)	That the content of pages that passed the mechanical checks is correct
That this book has not been through expert peer review, and that much of the research and refutation was done by AI writers	—
—	The existence and mass gap of four-dimensional Yang–Mills theory. This problem is unsolved, and no stage of this book shows it (10-16)

Sources and reproduction

Item	Kind	Source / tools
Tables 3 and 4 (verdicts of the refutation check and types of correction)	Computed	`tally.py` (counts the verdicts of this book's materials and sorts the wording of corrections by words; output `tally.json`)
Table 5 and Figure 2 (numbers of labels, figures and "bibliographic data only" by part)	Computed	`tally.py` (counts `class="label …"`, figure frames and the words "bibliographic data only" on the chapter pages, excluding the inside of figures' JavaScript)
Quotation of the Clay problem statement (§03)	Known	A. Jaffe, E. Witten, “Quantum Yang–Mills Theory” (the official problem statement of the Clay Mathematics Institute, public PDF). Checked in the text
What the problem statement says about regularisation and reflection positivity (the example in §04)	Known	Same, §6.5. Checked in the text (details in 4-10)
Reflection positivity of the Wilson action (the example in §01)	Known	K. Osterwalder, E. Seiler, “Gauge Field Theories on the Lattice”, Ann. Phys. 110 (1978) 440–471. Abstract checked, text not checked (how the text is handled is in the source table of 5-03)
An example where the abstract does not mention sections (§03)	Known	K. G. Wilson, J. Kogut, Phys. Rep. 12 (1974) 75–199. Abstract only (the abstract is a single sentence; §10 not checked)
The upper limit 11 on the dimension for supersymmetry (detour)	Known	W. Nahm, Nucl. Phys. B135 (1978) 149. Abstract only. "11 is the upper limit" is not in the abstract and not checked. The upper limit carries the condition of fields of spin at most 2 (7-05)
Pre-rendering of formulas	Tool	KaTeX (a JavaScript typesetting library that renders TeX formulas to static HTML. Used only at the assembly stage; it does not run on the reader's side)
Figure 1, Tables 1, 2 and 6 (description of the procedure)	—	A summary of this book's writing conventions and assembly procedure

Revised 2026-10-01: first version.