Math Model

竞赛建模

用途

Turn the locked problem brief into a problem-driven model route: variables, constraints, assumptions, units, objective/mechanism, validation, robustness plan, result schema, and code handoff.

Read math-hub/references/quality-contract.md only when promoting a route to paper-ready, classifying blocked status, preparing a final gate, or writing a cross-module handoff. For exploratory model sketches, keep the boundary local and do not load the shared contract by default.

Do not choose an algorithm because it sounds advanced. First state what mathematical problem the subquestion is, then choose the simplest route that can answer the required deliverable.

Required Inputs

• problem_brief.md and deliverable_matrix.csv, or an equivalent current lock from math-problem-reader/math-hub;
• active subquestion, input data, assumptions, limits, and expected output;
• known units, source files, and official constraints;
• research_brief.md or method_source_matrix.csv when the route was literature-informed or when multiple source-backed candidate routes were scouted;
• prior model version if this is a revision.

If these are missing, return to math-problem-reader or math-hub instead of inventing the task.

Modeling Flow

1. Restate the subquestion as inputs, limits, decision object, and outputs.
2. Name the mechanism or mathematical family before naming software or algorithms.
3. Define variables, domains, units, assumptions, objective, constraints, and boundary conditions.
4. Select one main route; keep secondary methods subordinate.
5. If prior research produced candidates, compare their task fit, data fit, validation burden, and citation boundary before selecting the main route.
6. Add a baseline or reference scheme before advanced claims.
7. Define validation tier: direct check, scenario check, robustness-required, or diagnostic-only.
8. Write the result schema and paper figure/table plan.
9. Prepare model_handoff.md for math-code and math_verification.csv expectations for math-verifier.

Record method choice rationale with evidence id, baseline defect, and boundary: do not copy AI suggestions or write only "works better"; name the data, PoC, baseline failure, and condition under which the chosen route is valid.

Route Triage

• Discrete decisions with coupled binary logic may use MILP/MIP, but never as a generic wrapper.
• Flow, assignment, scheduling, and staged decisions should consider network flow, DP, or greedy/baseline logic before generic MILP.
• Continuous resource allocation should consider LP/QP/convex/nonlinear programming according to structure.
• Geometry/physics tasks need derivation, direct numerical evaluation, or simulation according to the mechanism.
• Ranking/evaluation tasks need indicator design, weighting rationale, and sensitivity.
• Uncertainty tasks need scenario, robust, stochastic, or sensitivity analysis.
• Simulation is appropriate only when time evolution, events, queues, Monte Carlo uncertainty, agents, or spatial/temporal discretization are part of the problem mechanism.

For simulation, include state variables, initialization, boundary conditions, event rules, seeds/replications when stochastic, calibration/back-test when data allow it, and a downgrade rule when sensitivity fails.

Robustness Plan

Every subquestion needs a validation tier:

• direct-check: unit, balance, accounting, boundary, or formula substitution is enough.
• scenario-check: compare named scenarios, stress cases, or baselines.
• robustness-required: conclusion depends on weights, thresholds, fitted parameters, solver tolerances, stochastic search, or fragile assumptions.
• diagnostic-only: check reduces uncertainty but cannot support final wording.

A robustness plan must name varied inputs, tested range, baseline, stability metric, pass/fail criterion, and downgrade rule.

PoC Gate

Before a computational route can support paper_ready evidence, require a poc_registry.csv row for each main or baseline candidate: a <=30-line PoC script or command run on a real data slice or real attachment-derived sample, with input path, command, candidate id, output number or failure verdict, and status. A synthetic-data PoC may guide modeling, but it stays diagnostic-only and cannot support paper_ready promotion.

If a candidate failed its PoC or is rejected after comparison, keep poc_registry.csv.status as passed/failed/blocked/diagnostic-only, record failure_reason, and mark the deprecated route in model_handoff.md; any remaining reference to it is diagnostic-only. A rejected method must not remain as a backup main conclusion; paper wording is blocked until the surviving route has passed real-data evidence.

Code Handoff

The code handoff in model_handoff.md is blocked if implementation would need to invent a parameter, threshold, unit, solver tolerance, output schema, feasibility label, or missing constraint. Keep the route inside math-model until these are explicit.

Minimum handoff content:

• active subquestion and deliverable;
• variables, constraints, units, assumptions, and objective/mechanism;
• input files and trusted columns;
• PoC ids from poc_registry.csv, including which candidate passed, failed, or stayed diagnostic-only;
• algorithm or solver route with parameters and tolerances;
• output tables/figures and expected filenames;
• validation and robustness checks;
• paper-ready versus diagnostic-only boundaries.

When a route is replaced, publish a new model_handoff.md version with changed assumptions/variables/constraints and which prior claims or runs are no longer paper-ready.

Verification Handoff

Before paper-ready promotion, provide enough structure for math-verifier:

• formula list, units, domains, and boundary cases;
• feasibility wording and hard constraints;
• assumptions that may fail;
• expected rows for math_verification.csv.

If these are missing, keep the route blocked or diagnostic-only.

Output Contract

Return:

• model route and why it fits the subquestion;
• source-backed route rationale and adaptation boundary when prior research was used;
• variables, constraints, units, assumptions, and objective/mechanism;
• validation tier and robustness plan;
• result schema and figure/table plan;
• model_handoff.md content or edits;
• blockers and the next owner module.

Return to hub: math-hub.

Red Lines

• Do not model before deliverables and attachments are understood.
• Do not use a method name as the model.
• Do not treat a literature idea as the selected model until it is adapted to current deliverables, data, units, constraints, and validation.
• Do not omit units, domains, hard constraints, or feasibility conditions.
• Do not hand code a model that requires guessing.
• Do not promote simulation, optimization, ranking, or fitted results without validation and downgrade rules.
• Do not strengthen another subquestion to hide an uncovered deliverable.