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

| Material | Placement | |----------|-----------| | The Main Theorem and its proof's logical skeleton | Main text, always | | The crux / new idea | Main text, in full | | Key lemmas used by the main argument | Main text (state and prove) | | Long but routine computation verifying an estimate | Appendix, referenced from main text | | Standard background lemmas recalled for completeness | Appendix or Preliminaries | | Technical case analysis that would swamp the main flow | Appendix, with the conclusion stated in main text | | Machine computation (code, what was checked) | Appendix + archived code/data | | Anything a referee must check to believe the theorem | Main text — never hide it |

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