franklee16/academic-research-skills/anmath-methods

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.

1. 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.

1. Decompose into named results. Break the argument into Lemmas, Propositions, and

intermediate Theorems, each stated precisely and proved before it is used.

1. 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.

1. 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

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• Do not build an essential step on an unpublished or unverifiable claim; if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)