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name: anmath-referee-strategy description: Use when anticipating how an expert Annals of Mathematics referee will verify a proof — stress-testing the argument for gaps, checking external dependencies, and pre-empting the questions a careful reader will ask. Hardens the manuscript before submission; does not write the rebuttal (anmath-revision).

Referee Strategy (anmath-referee-strategy)

When to trigger

• The manuscript is essentially final and you want to pre-empt referee objections
• You want an adversarial read of your own proof before an expert does it for you
• You are unsure which steps a careful referee will distrust
• You want to decide what to make easier to verify before sending

How Annals referees read

Expect a referee who is an expert and verifies proofs in detail, often over a long period (a year or more is common). They do not skim; they reconstruct each step. Their implicit questions:

1. Is it important enough? Significance is judged again at the report stage.
2. Is every step actually valid? They will probe exactly the steps you find hardest.
3. Do the external results say what you claim? They will check the cited statement.
4. Does the theorem match the proof? Any over-claim is caught here.
5. Is anything hidden behind "clearly"? Softening words attract scrutiny, not less.

Self-adversarial pass

Probe	What to do before they do
Where is the crux?	Make sure it is fully spelled out and easy to locate
Which step is weakest?	Strengthen or expand it; do not hope it is overlooked
Every "clearly"/"standard"	Resolve or cite it (see anmath-writing-style)
Every external theorem	Re-read the cited source; confirm hypotheses match your use
Quantifiers / constants	Confirm order and dependence are stated and correct
Special / boundary cases	Check the degenerate cases the main argument might miss
Computer-assisted steps	Confirm code/data are archived and the claim is reproducible

Make verification easy

• A clear proof overview and named lemmas let a referee navigate and check piece by piece — this speeds a long review and reduces the chance of misreading.
• Precise citations (exact theorem number and statement) save the referee from hunting and remove a common source of objections.
• Isolate any dependence on unpublished work so the referee can see exactly what rests on it; reduce or remove such dependence where possible.

Suggested / opposed referees

• If the portal allows suggestions, propose genuine experts without conflicts of interest; do not pack the list with close collaborators.
• Declare conflicts honestly. Editors decide; transparency builds trust.

Checklist

• The crux is fully spelled out and easy to find
• The weakest step has been strengthened or expanded, not hidden
• Every "clearly"/"standard" is resolved or precisely cited
• Every external theorem's hypotheses match your use (re-checked at source)
• Quantifier order and constant dependence verified
• Degenerate/boundary cases handled
• Computer-assisted steps reproducible (archived code/data)
• Suggested referees are genuine, conflict-free experts; conflicts declared

Anti-patterns

• Hoping a referee will not check the step you yourself are unsure about
• Citing an external theorem without re-reading its exact hypotheses
• Leaving the crux hard to locate in a long, unstructured proof
• Suggesting only close collaborators as referees
• Treating significance as settled at submission — it is re-judged in the report
• Relying on "it is easy to see" to get past a careful reader (it will not)

Output format

【Crux locatable & complete】yes / fix: ...
【Weakest step】... → strengthened / expanded
【External-citation re-check】hypotheses match use: yes / fix: ...
【Boundary cases】covered / add: ...
【Computer-assisted reproducible】n/a / yes
【Suggested referees】names (conflict-free); conflicts declared: ...
【Residual risks to flag to editor】...
【Next step】submit, then await report → anmath-revision