anmath-writing-style

Writing Style and Rigor (anmath-writing-style)

When to trigger

• The proof is complete and you are tightening the prose
• You wrote "clearly", "obviously", "it is easy to see", or "trivially" and are not certain
• Quantifiers, hypotheses, or "the constant" are stated loosely
• The English is uneven or the register drifts between sections

The cardinal rule: no hidden gaps

At Annals, an expert referee verifies the proof in detail. A logical gap — especially one hidden behind softening words — is fatal. Treat every "clearly" as a debt to be paid or deleted.

Phrase	What to do
"It is easy to see that ..."	Either show it in one line, or delete the claim if truly immediate
"Clearly / obviously ..."	Replace with the actual one-line reason, or cite the lemma
"A standard argument shows ..."	Name the standard argument and cite it precisely
"By a similar argument ..."	State exactly which prior argument and what changes
"It can be shown that ..."	Show it, or move it to a lemma with a proof
"Modulo routine modifications ..."	Spell out the modifications or do them

If a step really is immediate, a single clause giving the reason is better than "clearly".

Precision in statements and prose

• Quantifiers explicit. "For all ε > 0 there exists δ > 0" — never leave the order of quantifiers to the reader. State dependence of constants (e.g. "C depends only on n").
• Hypotheses carried, not assumed. Each lemma states its own hypotheses; do not rely on context the reader has to reconstruct.
• One name per object. Do not let a symbol mean two things; do not give one object two names across sections.
• Define before use. No symbol or term appears before it is introduced.
• Match statement to proof. The theorem proves exactly what it states — no more, no less.

Mathematical English

• Consistent tense and register; complete sentences around displayed equations.
• "We" for authorial voice is conventional and fine; keep it consistent.
• Punctuate display equations as parts of sentences.
• Use \eqref/\cref so references stay correct after edits.
• Light language polish is welcome for non-native authors, but never at the cost of precision.

Checklist

• Every "clearly"/"easy to see"/"obviously" is either justified in a clause or removed
• "Standard"/"similar" arguments are named and cited precisely
• Quantifier order is explicit everywhere it matters
• Constant dependence is stated (what each constant depends on)
• Each lemma carries its own hypotheses
• No symbol is overloaded; nothing is used before definition
• The theorem statement matches exactly what the proof establishes
• Displayed equations are punctuated and referenced consistently

Anti-patterns

• "It is easy to see" guarding a step that is not, in fact, easy — the classic fatal gap
• "By a similar argument" when the argument is not actually similar
• Loose quantifiers that hide an order-of-quantifiers error
• Constants introduced without saying what they depend on
• A lemma whose hypotheses live only in the surrounding prose
• Polishing language while a logical gap remains open

Output format

【"clearly"/"easy to see" instances】N found → resolved: justified / removed / made-lemma
【Standard/similar arguments】named & cited: ...
【Quantifier / constant fixes】...
【Statement–proof match】confirmed / fix: ...
【Remaining open gaps】none / list (BLOCKER — return to anmath-methods)
【Next step】anmath-length-management