Expository

Dependency order

Arrange definitions so that no definition uses a term that hasn't been defined yet. This is a topological sort of your concepts. If you find a circular dependency, it reveals a structural insight about the topic — note it.

Phase 2 — What We Know

List every known result (theorem, lemma, proposition) relevant to the topic. For each:

1. State it precisely. Full hypotheses, full conclusion.
2. Write the proof in your own words. Not a citation. Not "see [KTW03, Thm 2.1]." Actually write the proof. If you cannot write the proof, you do not understand the result, and that gap will bite you during research. Mark it clearly: % TODO: I don't understand this proof yet — need to work through it.
3. If the proof is long, break it into lemmas. Each lemma gets its own proof. The same "no idea used before proved" rule applies within proofs.
4. Give a concrete example of the theorem in action. Walk through a specific instance with specific numbers/objects. Show the theorem being true, not just state that it is.
5. Cite the source even though you've written the proof yourself. Future-you needs to know where this came from.

Arrangement

Order results so that no proof uses a result that hasn't been proved yet in this document. This is the pedagogical heart of the paper. If you need Theorem B to prove Theorem A, then Theorem B comes first. This ordering often reveals the logical structure of the theory in a way that reading papers in publication order does not.

Phase 3 — Connections

After definitions and results are written up, add a section on connections:

1. Which objects are "the same"? Known bijections, isomorphisms, equivalences. State precisely.
2. Which results are "the same"? Theorems that are equivalent, or that are special cases of a common generalization.
3. What's the picture? Draw a diagram showing how the main objects and results relate. This can be informal — a hand-drawn map of the territory.
4. What's missing? What questions are natural from the exposition that don't have known answers? These are potential research directions.

Phase 4 — Open Questions

List questions that arise naturally from the exposition. For each:

• State the question precisely
• Say what a positive answer would imply
• Say what a negative answer (counterexample) would look like
• Note any partial progress

How to Use This Document

When stuck on a proof

Come here. Don't work on the proof. Work on the expository paper instead. Pick a known result you haven't written up yet and write its proof in your own words. Or draw a better diagram. Or find a new example. The understanding you build will carry back to the proof.

When starting a new problem

Read the expository paper first. Everything you know is here, organized and proved. The tool you need might already be in your toolbox — you just need to see it.

When a result turns out to be wrong

Update the paper. The expository document must always reflect your current best understanding. Outdated results in the expository paper will poison future work.

Rules

• No hand-waving. If a step in a proof is "clear" or "follows easily," write it out anyway. You are writing for yourself on a bad day, when nothing feels clear.
• No proof by citation alone. You may cite sources for reference, but every proof must be written in your own words in this document. If you can't write it, you don't know it.
• Dependency order is sacred. If you catch yourself using a concept before defining it or a result before proving it, stop and restructure. The ordering IS the understanding.
• Diagrams are not optional. If an object can be drawn, draw it. Combinatorics is a visual subject. The diagram often contains the proof.
• Examples are not optional. Every definition and every theorem gets at least one concrete example with specific values. Abstract understanding without concrete instances is illusory.
• This document is never finished. It grows with your understanding. Return to it regularly. The best insights often come while writing up something you thought you already knew.