anmath-methods

Business

Use when laying out the proof strategy of a pure-mathematics manuscript for Annals of Mathematics — the architecture of the argument, the key lemmas and propositions, the novel technique, and where the difficulty lies. Designs and exposes the proof plan; does not check final correctness line-by-line (see anmath-referee-strategy).

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Proof Strategy and Architecture (anmath-methods)

When to trigger

The proof is essentially complete but its logical structure is not laid out for a reader
A referee would not be able to see the plan before drowning in the details
The key new idea is buried; it is not clear where the real difficulty is overcome
The argument is monolithic and should be decomposed into named lemmas/propositions

Architecture-first principle

For an Annals paper, an expert non-specialist should be able to read a proof overview and understand how the theorem is proved before verifying that it is. The architecture is part of the contribution.

Proof outline up front. After stating the Main Theorem, give a paragraph or short section sketching the strategy: the main steps, the key lemma(s), and the crux.
Decompose into named results. Break the argument into Lemmas, Propositions, and intermediate Theorems, each stated precisely and proved before it is used.
Isolate the new idea. Name explicitly which step is the genuinely new technique and why prior approaches failed there. This is what makes the paper publishable.
Locate the difficulty. Tell the reader where the hard part is and why it is hard; do not let the crux pass disguised as routine.

Decomposition guidance

Symptom	Action
A 10-page proof with no internal structure	Extract Lemmas/Propositions with clear statements
The same estimate reused three times	State it once as a Lemma and cite it
A self-contained technical computation interrupting the flow	Push to an appendix (anmath-supplementary)
Reliance on a deep external theorem	State it precisely with citation; do not paraphrase loosely
The crux step stated as "a calculation shows"	Expand fully — this is exactly what referees check

Handling the key technique

State the novel ingredient as its own result when possible (a key Lemma or Proposition), so others can cite and reuse it — methods with reach justify Annals.
Contrast with the standard approach: one or two sentences on why the obvious method does not work and how the new idea circumvents the obstruction.
If the method is borrowed and adapted, attribute it and state precisely what is new in your adaptation.

Dependence on external results

Every external theorem you invoke must be published and precisely cited; quote the exact statement you use, not a vague version.
Do not build an essential step on an unpublished or unverifiable claim; if unavoidable, isolate the dependence and flag it explicitly.

Checklist

A proof overview appears before the detailed argument
The argument is decomposed into precisely stated lemmas/propositions
Each auxiliary result is proved before it is used
The genuinely new idea is named and explained as the crux
Why the standard approach fails is stated explicitly
Every external result invoked is published and precisely cited
No essential step rests on an unpublished/unverifiable claim
The crux is proved in full, not waved through as "a calculation"

Anti-patterns

A monolithic proof with no roadmap — the referee cannot navigate it
Hiding the crux inside a step labeled "routine" or "standard"
Restating a known method as if it were new without attribution
Paraphrasing an external theorem loosely so the actual hypothesis is unclear
Reusing the same estimate inline three times instead of stating it once
Leaving the reader unable to say where the difficulty was overcome

Output format

【Proof strategy, one paragraph】...
【Key lemmas/propositions】L1: ...; P1: ...; ...
【The new idea (crux)】...
【Why the standard approach fails】...
【External results relied on】author (year), Thm X — exact statement used
【Steps to push to appendix】... → anmath-supplementary
【Next step】anmath-figures (exposition & structure)