spb/autoscholar

1---23mode: theoretical4discipline: mathematics5title: "<Working title: a theoretical result>"6---78# Research brief — <working title> (theoretical)910## 1. Research objective and scope1112- **Question/conjecture.** <Precise mathematical statement or model question.>13- **Setting.** <Objects, spaces, assumptions, notation conventions.>14- **Scope.** <Which generality is targeted; what is explicitly left open.>1516## 2. Domain, mode and methodology1718- Discipline: <mathematics/statistics/economic theory/theoretical CS/physics>. Primary mode: theoretical.19- Methods: <direct proof, induction, compactness argument, fixed-point theorem, variational argument, dynamic programming…>.20- Formal tooling (if practical): <sympy/numpy numeric checks of identities; Lean/Coq statements for key lemmas>.2122## 3. Contribution and novelty2324- <Expected theorem(s); the closest known results (cite); what is stronger/more general/simpler.>2526## 4. Literature and source rules2728- Indexes: arXiv + Crossref (+ OpenAlex, zbMATH via Crossref DOIs). Target ≥ 8 verified sources; must-read: <DOIs/arXiv ids>.2930## 5. Formal foundations3132- Definitions: <…>. Assumptions: A1 <…>, A2 <…>.33- Candidate propositions: P1 <…>, P2 <…>. Proof strategy for each.3435## 6. Procedure36371. Write `plan/formal-setup.md` with definitions, assumptions, statements and proof strategies.382. Derive each result step by step in `results/derivations.md`; every nontrivial step justified.393. Check algebra/identities numerically or symbolically in `code/` (logged); attempt formalization of <lemma> if feasible.404. Counterexample search: boundary cases, relaxed assumptions, random instances.4142## 7. Robustness and falsification4344- Which assumptions are necessary? Provide counterexamples when an assumption is dropped, or prove it is superfluous.45- Report any gap honestly; downgrade to a conjecture with partial results if a proof fails.4647## 8. Reproducibility and artifacts4849- `results/robustness.md` (counterexample search), symbolic/numeric check scripts, derivations.5051## 9. Manuscript5253- AutoScholar class with theorem/proposition/lemma/proof environments, numbered equations, complete proofs (appendix for long proofs), 6–15 pages.5455## 10. Review and revision5657- Reviewer focus: proof gaps, hidden assumptions, circularity, misuse of theorems, overclaimed generality.5859## 11. Limitations, ethics, legal6061- <Open questions, dependence on assumptions.>6263## 12. Acceptance criteria6465- Every stated theorem has a complete proof; symbolic/numeric checks pass; counterexample search documented; ≥ 8 verified citations.66