brycewang-stanford Stating the Main Theorem(s) (anmath-results-framing)
When to trigger
- Your main result is described in a paragraph but not stated as a precise theorem
- The introduction does not make the significance unmistakable in the first page
- Readers cannot tell which statement is the headline and which are corollaries
- You have not located your theorem relative to the prior state of the art
How to state the main theorem
A reader should be able to find the precise main theorem on or near the first page and understand exactly what is claimed without reading the proof.
- State it precisely and self-containedly. Every hypothesis explicit; every object defined or referenced. No "under suitable conditions" hand-waving.
- Quantify the advance. Make clear what is new: which hypothesis is removed, which bound is sharpened, which case is now covered, which conjecture is settled.
- Separate headline from consequences. One (or few) Main Theorem(s); corollaries and special cases stated separately so the contribution is unambiguous.
- Give the sharpest true statement. Do not under-claim out of caution or over-claim beyond what the proof delivers; the statement and the proof must match exactly.
Framing shapes from verified Annals landmarks
Benchmark against the verified papers in resources/exemplars/library.md; each makes
its advance quantifiable in one sentence, in one of three shapes:
- Settles a named conjecture (Wiles 1995; Marques–Neves 2014): name the conjecture, state the theorem that closes it.
- Breaks a quantitative barrier (Zhang 2014): a modest-sounding bound whose point is that none existed — say exactly which barrier fell.
- New method with reach (Green–Tao 2008; Bhargava–Shankar 2015): the theorem is the headline, but the transferable technique is named too.
A result fitting none of these shapes should go back through anmath-scope-fit first — framing cannot manufacture significance.
Introduction architecture (Annals style)
| Element | Purpose |
|---|---|
| Problem and history | Why this question matters and what was known |
| Precise statement of Main Theorem | The headline, fully rigorous, early |
| What is new vs. prior work | Named comparison to the closest prior results |
| Consequences / corollaries | Why the result has reach |
| Method in one paragraph | A pointer to the proof idea (detail belongs to anmath-methods) |
| Organization of the paper | Section-by-section roadmap |
Positioning against the literature
- Name the closest prior results and authors explicitly; state precisely what they proved and where your theorem goes beyond it (stronger hypothesis removed, sharper constant, new range, full generality).
- Engage carefully with priority: cite preprints/announcements you are aware of and state the relationship honestly. Claiming priority without engaging the record is a serious referee red flag.
- Do not rely on unpublished or unverifiable results for the core comparison; if a cited result is itself unpublished, say so and isolate your dependence on it.
Micro-example: from vague paragraph to precise theorem
Before: "We prove strong new bounds for the discrepancy of such sequences under mild conditions, improving earlier work."
After: "Theorem 1.1. Let (x_n) satisfy (H1)–(H2). Then D_N(x) ≤ C(α) N^{−1/2} log N for all N ≥ 2, where C(α) depends only on α." — plus one positioning sentence: "This removes the smoothness hypothesis of [Prior, Theorem 2] and sharpens the exponent from −1/3 to −1/2."
Named hypotheses, an explicit rate, stated constant dependence, a cited theorem precisely exceeded — the Annals headline register.
Checklist
- Main Theorem is stated precisely, with all hypotheses, near the first page
- The framing matches a landmark shape (conjecture settled / barrier broken / method with reach)
- It is clear which statement is the headline vs. corollaries
- The exact advance over prior work is quantified, not just asserted
- Closest prior results are named with authors and what they proved
- Priority relative to known preprints is engaged honestly
- The statement matches exactly what the proof delivers (no over/under-claim)
- MSC subject classification chosen to match the headline result
Anti-patterns
- Burying the main theorem on page 8 after long preliminaries
- "We prove strong results about X" with no precise statement up front
- Claiming generality the proof does not actually establish
- Vague positioning ("improving earlier work") without naming the earlier work
- Asserting priority while ignoring a known competing announcement
- Listing five "main" theorems so the actual contribution is unclear
Output format
【Main Theorem (precise)】...
【What is new vs. prior】removed-hypothesis / sharper-bound / settles-conjecture / ...
【Closest prior results】author (year): proved ...
【Corollaries / reach】...
【Priority note】no conflict / relationship to preprint X is ...
【MSC classes】primary ..., secondary ...
【Next step】anmath-methods (lay out the proof architecture)