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

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.

Checklist

• [ ] Every definition/lemma/proposition/theorem is stated before it is proved
• [ ] 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