brycewang-stanford 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- Do not build an essential step on an unpublished or unverifiable claim; if unavoidable, isolate the dependence and flag it explicitly.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)