Length Discipline (anmath-length-management)

When to trigger

• The paper has grown long and you cannot defend every section's necessity
• Background or preliminaries dwarf the actual new mathematics
• The same idea is explained several times in different words
• You feel pressure to "say it is easy to see" to shorten a hard step — stop

The principle: long is fine, bloated is not

Annals papers can be long — a deep result may require many pages. Length is not a defect. Bloat is: material that does not advance the proof, padding, repetition, or restating known background at textbook length. The test for every paragraph is necessity, not size.

Calibrating against verified Annals papers

There is no page limit to game at this venue. The verified landmarks in resources/exemplars/library.md span roughly 50 pages (Zhang 2014) to about 100 (Marques–Neves 2014; Wiles 1995); what they share is density — nearly every page is new mathematics or the minimum scaffolding a verifier needs. Because refereeing here is line-by-line verification, often over a year, every non-essential page taxes the one expert whose verdict decides the paper: cutting bloat shortens the verification critical path.

What to compress vs. what to keep

Compress / cut

• Textbook-level background that a citation handles
• Repeated re-statements of the strategy in slightly different words
• Over-long motivational prose that delays the mathematics
• Routine multi-page computations (summarize result in text, push detail to appendix)

Never cut for length

• The crux / new idea — it stays in full
• Any step a referee must check to believe the theorem
• Hypotheses, quantifiers, and constant dependence
• Precise statements of external results you invoke

Shortening must never create a gap. If a step is long because it is hard, that length is earned — keep it. The forbidden shortcut is replacing a hard step with "it is easy to see" (see anmath-writing-style).

Section-necessity pass

• For each section, write one sentence stating why it is necessary. If you cannot, the section is a candidate for merging, compressing, or appendixing.
• Check that preliminaries are proportionate to their use — recall only what the paper uses.
• Confirm appendices carry length relief, not hidden essentials.

Worked micro-example: compressing preliminaries without creating a gap

Before (textbook restatement, roughly a page): "We now recall the theory of X," followed by a dozen displayed definitions and three re-proved lemmas from a standard reference.

After (citation-anchored, six lines): "We use the theory of X as developed in [R, §2–3], and recall verbatim the two statements we invoke: Theorem 2.4 of [R] and Lemma 3.1 of [R], in the notation fixed above."

The cut keeps every statement the proof invokes and deletes only what the reader can retrieve from [R] — no checkable step removed, no gap created.

Checklist

• Every section has a one-line justification of necessity
• Length is calibrated against comparable verified Annals papers, not a page target
• Textbook background is replaced by precise citations
• No idea or strategy statement is repeated unnecessarily
• Routine long computations are summarized in text and detailed in an appendix
• No step was shortened in a way that introduces a gap
• The crux and all referee-checkable steps remain in full
• Overall length is defensible: every page earns its place

Anti-patterns

• Padding preliminaries with background the paper never uses
• Restating the proof strategy three times to fill space
• Cutting a hard step to "it is easy to see" to hit a length target (a fatal gap)
• A 90-page paper where 30 pages are recycled background
• Burying the new contribution under disproportionate motivational prose
• Trimming hypotheses or constant-dependence statements to save lines

Output format

【Total length】~N pages
【Sections without necessity justification】... → compress / merge / appendix
【Background to replace with citation】...
【Repetition removed】...
【Computations moved to appendix】... → anmath-supplementary
【Gap introduced by cutting?】no / FIX (return to anmath-writing-style)
【Next step】anmath-cover-letter