thedixitjain 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/\crefso 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
Source: brycewang-stanford/Awesome-Journal-Skills → Annals-of-Mathematics-Skills/skills/anmath-writing-style/SKILL.md