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".

Worked micro-example: paying the "clearly" debt

Before:

> Clearly f is uniformly continuous on K, so by a standard compactness argument the > bound (3.2) holds with a constant independent of ε.

After:

> Since K is compact and f is continuous, f is uniformly continuous on K. Covering K by > finitely many δ-balls with δ as in (3.1) and taking the maximum of the local bounds > yields (3.2) with C = C(K, f) independent of ε.

Two debts are paid for one extra line: "clearly" becomes the one-clause reason, and "a standard compactness argument" becomes the actual argument with the constant's dependence recorded.

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.

How the Annals referee reads your prose

Review at this journal is single-blind expert verification, frequently stretching past a year. Assume every softening word will be tested: the referee's default at "it is easy to see" is to try to see it, and each failure erodes trust in the steps they cannot check. A first "clearly" that conceals a subtlety invites a report that re-derives everything and finds the real gap you missed.

Checklist

• ] Every "clearly"/"easy to see"/"obviously" is either justified in a clause or removed
• ] Every softening word has been tested the way a verifying referee would test it
• ] "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