brycewang-stanford 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