/anmath-referee-strategy

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• **Isolate any dependence on unpublished work** so the referee can see exactly what rests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