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. 2. **Decompose into named results.** Break the argument into Lemmas, Propositions, and intermediate Theorems, each stated precisely and proved before it is used. 3. **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. 4. **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) ```