mathflow

name	mathflow
description	Use when mathematical research work needs stage-aware guidance before proceeding.

Use when

Use this skill when a task involves mathematical research, derivations, mathematical modeling, simulation, numerical experiments, theorem exploration, or conclusion-building across multiple stages of analysis.

Inputs

• The problem statement, known constraints, and current evidence.
• Any existing model, equations, assumptions, datasets, or intermediate results.
• The current stage if the user already knows it, otherwise enough context to assess the stage.

Outputs

• A brief stage assessment that identifies the current stage and why it fits.
• One required next stage by default.
• A short ranked candidate set only when stages genuinely overlap and there is real ambiguity.
• A clear handoff describing what to do in the selected next stage.
• mathflow selects the next needed stage skill on demand, one stage at a time.
• mathflow is not an eager loader for the whole skill family; load only the stage skill needed for the next step.

Hard rules

• Always assess the current stage before choosing the next step.
• Route mathematical research tasks into the correct stage instead of jumping directly to calculations or conclusions.
• Choose exactly one required next stage by default.
• Return a short ranked candidate set only when stages genuinely overlap.
• Start open-ended work in problem-analysis.
• Require mathematical-modeling before introducing new equations or assumptions.
• Require derivation-and-proof-checking when analytical claims depend on a derivation, proof attempt, or justification audit.
• Require research-planning before numerical work.
• Require numerical-experimentation for simulation, solver, sweep, or computational execution work.
• Require result-validation before making strong claims.
• Require result-validation before self-audit-loop; result-validation prepares work for audit.
• Require self-audit-loop as the final skepticism gate before report-writing.
• Route validated final writeups to report-writing only after self-audit-loop completes.

Behavior

1. Read the task and identify whether the work is open-ended, model-building, numerical, interpretive, or near-conclusion.
2. Produce a short stage assessment before naming any next action.
3. If the task is open-ended or underspecified, route it to problem-analysis.
4. If new equations, abstractions, or assumptions are needed, require mathematical-modeling first.
5. If the work depends on checking or extending a derivation, proof, or mathematical argument, route it to derivation-and-proof-checking.
6. If the user wants computation, simulation, estimation, or parameter sweeps, require research-planning first unless that stage is already complete.
7. After planning is complete, route execution work to numerical-experimentation.
8. Before endorsing a result, require result-validation.
9. In second-wave mode, require result-validation to prepare work for audit through special-case, limit-case, sensitivity, and consistency checks before any final takeaway.
10. Require self-audit-loop as the explicit final skepticism gate before report-writing.
11. After validation and self-audit support the claims strongly enough for communication, route final writeup work to report-writing.

Stages

• problem-analysis: clarify the question, constraints, success criteria, and missing information.
• mathematical-modeling: define variables, assumptions, structures, and candidate mathematical representations.
• derivation-and-proof-checking: review derivations, proofs, and mathematical arguments so justified results stay separated from heuristics, conjecture, or unsupported steps.
• research-planning: choose experiment objectives, baselines, metrics, stop rules, and fallback branches before numerical execution.
• numerical-experimentation: run reproducible simulations, sweeps, solvers, or computational studies while retaining anomalies and report-ready evidence.
• result-validation: test sensitivity, consistency, edge cases, special cases, limit cases, and alternative explanations so the evidence is prepared for audit.
• self-audit-loop: run the explicit final skepticism gate before report-writing by searching for failure regions, overreach, and unsupported stretches.
• report-writing: turn staged outputs into a final report that preserves claim strength, caveats, and evidence boundaries.

Routing and handoffs

• mathflow is a single-step router: it selects the next needed stage skill on demand instead of eager-loading every child skill.
• A typical second-wave path is problem-analysis -> mathematical-modeling -> derivation-and-proof-checking or research-planning -> numerical-experimentation -> result-validation -> self-audit-loop -> report-writing, but only the stages required by the actual task should be used.
• derivation-and-proof-checking may follow mathematical-modeling directly when the next need is analytical justification rather than numerical execution.
• research-planning feeds numerical-experimentation; do not collapse planning and execution into one stage.
• result-validation prepares work for audit; self-audit-loop is the final skepticism gate before report-writing.
• If self-audit-loop finds insufficient support, route back to mathematical-modeling, derivation-and-proof-checking, research-planning, or numerical-experimentation based on the source of the problem.
• Use report-writing only after the necessary validation and self-audit work is complete and the evidence is ready to be communicated honestly.