Imported from haohan-sun/shumo-quanliucheng (.agents/skills/mm-mathematical-derivation/SKILL.md). Install upstream with npx skills add haohan-sun/shumo-quanliucheng --skill mm-mathematical-derivation. Copyright stays with the author.

Mathematical Model Derivation

Translate one selected modeling route into a concise, verification-ready mathematical derivation. Write the public derivation and its checks, not a transcript of private reasoning.

Inputs and boundary

Use the problem analysis, evidence passport, data-audit contract, selected route, and recorded assumptions. Preserve user-established notation when it is unambiguous. Never choose the final route, invent missing laws or parameter values, silently strengthen assumptions, or treat a symbolic simplification as empirical validation.

The canonical output is 03_建模工作区/model/MODEL_DERIVATION.md. mm-model-spec consumes this artifact; implementation and solver work remain downstream.

Derivation workflow

1. State the modeled scope, requested outputs, spatial/temporal scale, and what is deliberately outside the model.
2. Build one symbol table covering sets, indices, variables, parameters, functions, domains, units, and provenance. Resolve collisions before writing equations.
3. Separate problem facts, definitions, modeling assumptions, empirical estimates, and mathematical consequences. Every load-bearing assumption must identify where it enters.
4. Choose the governing structure appropriate to the route:
   • mechanistic/dynamic: balances, rates, constitutive relations, initial/boundary conditions;
   • optimization: decisions, feasible set, objective, hard/soft constraints, coupling and scale;
   • probabilistic/statistical: sample space, conditional assumptions, likelihood/loss, priors or regularization, estimand and identifiability;
   • network/discrete: nodes, edges, flows, conservation, state transitions and integrality.
5. Derive the formulation in numbered steps. For each non-trivial transformation, cite its basis as given fact, definition, assumption, law/theorem, algebraic consequence, or empirical fit, and record the conditions under which it is reversible or valid.
6. Present the final formulation as a closed contract: inputs, outputs, objective or governing equations, constraints, parameter sources, initial/boundary conditions, and applicable range.
7. Run independent checks before handoff. Use deterministic algebra/numeric tools when available; retain exact expressions until approximation is necessary and record the check performed.

Required checks

• dimensional and unit consistency for every additive equation and objective term;
• domain, sign, range, normalization and denominator checks;
• equation/unknown count, identifiability, feasibility or well-posedness as applicable;
• initial, terminal and boundary condition coverage;
• limiting, degenerate and special-case behavior;
• conservation laws, invariants, monotonicity or symmetry when the model claims them;
• at least one independent check such as back-substitution, symbolic equivalence, finite-difference comparison, numerical spot check, or a small exactly solvable instance;
• full coverage of each selected-route requirement, with unresolved gaps explicitly blocking freeze.

Output contract

MODEL_DERIVATION.md must contain:

1. scope and source boundary;
2. canonical symbols, domains and units;
3. assumptions with evidence and impact;
4. numbered derivation with validity conditions;
5. final mathematical formulation;
6. check table with method, result and residual/diagnostic where meaningful;
7. unresolved gaps and a field-by-field handoff to MODEL_SPEC.md.

Read references/methodology-sources.md when provenance for the workflow or a deterministic verification approach is needed. The reference records concepts and licenses; do not copy third-party prose or code into the derivation.