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.

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)