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 ```