anmath-writing-style

name	anmath-writing-style
description	Use when polishing the prose and rigor of a pure-mathematics manuscript for Annals of Mathematics — eliminating gaps, removing "clearly"/"it is easy to see", precise quantifiers, and consistent mathematical English. Late-stage polish; run only after the proof and architecture are fixed.

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