Exposition and Structure (anmath-figures)

When to trigger

• The paper is hard to follow even though the proof is correct
• Notation is inconsistent or introduced after first use
• Theorems and their proofs are interleaved confusingly
• A relation between objects would be far clearer as a commutative diagram
• You are deciding whether the rare case of an actual figure is warranted

> In pure mathematics, papers are theorem-and-proof and usually have no experiments and > few or no figures. "Figures" here means exposition and structure: a figure or diagram > is included only when it conveys structure more efficiently than prose.

Exposition principles (Annals readability)

1. Statements before proofs. State each definition, lemma, proposition, and theorem in

full before proving it. The reader should always know the target before the argument.

1. Notation introduced once, used consistently. Define every symbol at first use; keep

a fixed convention throughout; avoid overloading the same symbol for two things.

1. Logical sectioning. Preliminaries → key constructions → main lemmas → proof of the

Main Theorem → consequences. Each section has a clear job and a one-line opener.

1. Self-contained where reasonable. Recall the precise external statements you use so a

reader need not reconstruct them from memory.

1. Readable by an expert non-specialist. Someone strong in an adjacent area should be

able to follow the architecture; gloss the field-specific shorthand the first time.

Sectioning template

| Section | Contents | |---------|----------| | Introduction | Problem, Main Theorem, what is new, method sketch, organization | | Preliminaries / Notation | Conventions, recalled definitions, cited external results | | Constructions / setup | The objects the proof manipulates | | Key lemmas | The intermediate results, stated then proved | | Proof of Main Theorem | Assembling the lemmas into the headline result | | Consequences | Corollaries and remarks | | Appendices | Auxiliary/technical material (see anmath-supplementary) |

Numbering and cross-reference conventions

• Number theorem-like environments in one per-section sequence (Theorem 3.1,

Lemma 3.2, Corollary 3.3) so a referee deep in a long verification can locate any statement from its number alone.

• Give the headline result a stable early label — Theorem 1.1, or letters

(Theorem A, B, …) in a long paper — and use it everywhere.

• Number displayed equations only when referenced; an unnumbered-but-needed display

forces "the equation above," which breaks silently under revision.

• Open each section with what it proves and what it consumes: "In this section we

prove Proposition 3.1, using Lemma 2.4."

Diagrams and figures (when justified)

• Commutative diagrams: use tikz-cd (or amscd) when a chain of maps or an exact

sequence is clearer drawn than written. Keep arrows labeled and consistent.

• A genuine figure (a configuration, a region, a graph): include only when it removes

real ambiguity. Use vector output (PDF/EPS), label everything, and reference it in text.

• Tables: occasionally useful for case enumerations or notation summaries; keep clean.
• Most Annals papers have zero figures — do not add one for decoration.

Micro-example: when a diagram earns its place

Prose version: "The map φ factors through the quotient, and the induced map commutes with the two projections of Section 2."

As a tikz-cd square with labeled arrows, the same content is checkable at a glance. That is the admission test — the referee verifies commutativity faster from the picture than from the sentence. A picture that merely restates a relation the prose already makes precise is decoration.

Checklist

• ] Every definition/lemma/proposition/theorem is stated before it is proved
• ] Numbering is one per-section sequence; the headline theorem has a stable label used everywhere
• ] Every symbol is defined at first use and used consistently
• ] Sections are logically ordered with clear one-line openers
• ] External results used are recalled precisely (statement + citation)
• ] Commutative diagrams (if any) are typeset with labeled, consistent arrows
• ] Any figure is vector, labeled, referenced, and genuinely necessary
• ] An expert in an adjacent area could follow the overall architecture

Anti-patterns

• Using a symbol before defining it, or redefining a symbol mid-paper
• Interleaving proof fragments with statements so the target is unclear
• A wall of unsegmented text with no sectioning logic
• Adding a decorative figure that conveys nothing the prose does not
• Field jargon used without a single gloss for the adjacent-area reader
• Sloppy diagram arrows (unlabeled, inconsistent direction) that obscure the maps

Output format

【Section plan】1 Intro · 2 Prelim · 3 ... · n Appendix
【Notation issues fixed】...
【Statements-before-proofs】compliant / fix: ...
【Diagrams】none / commutative diagram in §... via tikz-cd
【Figure justification】none needed / figure in §... because ...
【Next step】anmath-supplementary (appendix triage) or anmath-writing-style