Appendices and Auxiliary Results (anmath-supplementary)

When to trigger

• A long, self-contained computation interrupts the main line of argument
• You have auxiliary lemmas that are needed but not conceptually central
• The proof depends on a machine computation that must be documented
• You are tempted to "move detail to supplement" the way science papers do — pause first

Important: pure-mathematics journals generally do not have a science-style "supplemental material" section. Everything essential to the proof stays in the main text and must be fully present and verifiable there. Appendices hold material that is necessary but would break the flow if kept inline.

What belongs where

Appendix discipline

• Each appendix is self-contained and clearly scoped ("Appendix A. Proof of the estimate in Lemma 3.4"). State in the main text exactly what the appendix establishes.
• The main text must stand on its logic with appendices referenced, not required to be read in parallel to follow the argument.
• Do not use an appendix to hide a gap: if a step is essential and hard, it stays visible. Appendices are for length relief, not for sweeping difficulty out of sight.

When an appendix is not enough: the companion-paper option

If auxiliary material grows into a self-contained contribution — its own theorem and proof, plausible independent use — a companion paper is the honest structure, not a hundred-page appendix. The canonical precedent, verified in resources/exemplars/library.md: Wiles's 1995 modularity paper ran alongside the Taylor–Wiles companion in the same Annals issue, citing it for the input it needs. The discipline carries over: state precisely what you import, and prove it fully where it lives. A companion is a second refereed paper, never a place to park an unproved step.

Appendix mechanics

• Use letter appendices (Appendix A, B, …) with results numbered A.1, A.2, so main-text citations stay unambiguous.
• Each main-text use reads like an import: "By Proposition A.2 (proved in Appendix A), estimate (4.7) holds with C = C(n)" — so the reader can defer the appendix and still audit the logic.
• Order appendices by first citation from the main text, not by topic.
• A referee should be able to read all appendices last, in one pass; an appendix that must be consulted mid-argument belongs in the main text.

Computer-assisted proofs

• If a result depends on machine computation, describe precisely what was computed, the software and version, and the exact claim verified.
• Make the code and data available (archive, repository, or supplement permitted by the journal) so a referee can reproduce or audit it.
• The human-readable argument must make clear how the computation closes the proof; the computation is evidence within a proof, not a substitute for exposition.

Checklist

• Everything essential to the theorem is in the main text, not an appendix
• The crux/new idea is fully in the main text
• Each appendix is self-contained with a clear scope statement
• The main text states what each appendix establishes and references it
• No proof gap is concealed by relegation to an appendix
• Any machine computation is documented (software, version, exact claim)
• Code/data for computer-assisted steps are archived and citable

Anti-patterns

• Treating an appendix as a science-style "supplement" and shipping essentials there
• Pushing the hard step to an appendix to make the main text look clean
• A machine computation with no description of what was actually checked
• Unavailable / unreproducible code behind a computer-assisted claim
• Appendices the reader must consult continuously just to follow the main argument
• An appendix that restates main-text material instead of carrying new detail

Output format

【Stays in main text】Main Thm, crux, key lemmas: ...
【To appendix】App A: ...; App B: ...
【Reason each is appendix-not-main】length relief, not gap-hiding: ...
【Computer-assisted?】no / yes — software, version, what was checked, archive location
【Gap check】no essential step hidden / fix: ...
【Next step】anmath-writing-style