---

mode: theoretical
discipline: mathematics
title: "<Working title: a theoretical result>"
---

# Research brief — <working title> (theoretical)

## 1. Research objective and scope

- **Question/conjecture.** <Precise mathematical statement or model question.>
- **Setting.** <Objects, spaces, assumptions, notation conventions.>
- **Scope.** <Which generality is targeted; what is explicitly left open.>

## 2. Domain, mode and methodology

- Discipline: <mathematics/statistics/economic theory/theoretical CS/physics>. Primary mode: theoretical.
- Methods: <direct proof, induction, compactness argument, fixed-point theorem, variational argument, dynamic programming…>.
- Formal tooling (if practical): <sympy/numpy numeric checks of identities; Lean/Coq statements for key lemmas>.

## 3. Contribution and novelty

- <Expected theorem(s); the closest known results (cite); what is stronger/more general/simpler.>

## 4. Literature and source rules

- Indexes: arXiv + Crossref (+ OpenAlex, zbMATH via Crossref DOIs). Target ≥ 8 verified sources; must-read: <DOIs/arXiv ids>.

## 5. Formal foundations

- Definitions: <…>. Assumptions: A1 <…>, A2 <…>.
- Candidate propositions: P1 <…>, P2 <…>. Proof strategy for each.

## 6. Procedure

1. Write `plan/formal-setup.md` with definitions, assumptions, statements and proof strategies.
2. Derive each result step by step in `results/derivations.md`; every nontrivial step justified.
3. Check algebra/identities numerically or symbolically in `code/` (logged); attempt formalization of <lemma> if feasible.
4. Counterexample search: boundary cases, relaxed assumptions, random instances.

## 7. Robustness and falsification

- Which assumptions are necessary? Provide counterexamples when an assumption is dropped, or prove it is superfluous.
- Report any gap honestly; downgrade to a conjecture with partial results if a proof fails.

## 8. Reproducibility and artifacts

- `results/robustness.md` (counterexample search), symbolic/numeric check scripts, derivations.

## 9. Manuscript

- AutoScholar class with theorem/proposition/lemma/proof environments, numbered equations, complete proofs (appendix for long proofs), 6–15 pages.

## 10. Review and revision

- Reviewer focus: proof gaps, hidden assumptions, circularity, misuse of theorems, overclaimed generality.

## 11. Limitations, ethics, legal

- <Open questions, dependence on assumptions.>

## 12. Acceptance criteria

- Every stated theorem has a complete proof; symbolic/numeric checks pass; counterexample search documented; ≥ 8 verified citations.