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.

1. State it precisely and self-containedly. Every hypothesis explicit; every object

defined or referenced. No "under suitable conditions" hand-waving.

1. 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.

1. Separate headline from consequences. One (or few) Main Theorem(s); corollaries and

special cases stated separately so the contribution is unambiguous.

1. 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.

Introduction architecture (Annals style)

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• Do not rely on unpublished or unverifiable results for the core comparison; if a

cited result is itself unpublished, say so and isolate your dependence on it.

Checklist

• [ ] Main Theorem is stated precisely, with all hypotheses, near the first page
• [ ] 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)