# mode: theoretical discipline: mathematics title: ""

# Research brief — (theoretical)

# 1. Research objective and scope

• Question/conjecture.
• 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 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.