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.

| Question | If "no" → | |----------|-----------| | Does this paragraph advance the proof or its understanding? | Cut or compress it | | Is this background needed to read this paper specifically? | Replace with a precise citation | | Have I already said this elsewhere? | Delete the repetition | | Could this routine computation be summarized + appendixed? | Move to appendix (anmath-supplementary) | | Is this section's job stated in one line at its top? | If not, the section may be unfocused |

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