Simulation Foundations

Use this contract for the concrete task. Apply a short relevant check for a small change; expand investigation when the failure, risk or requested artifact warrants it. Resolve facts from the repository and runtime before imposing a process. Delegation is optional and should answer a bounded unresolved question.

Mathematical and numerical contract

Identify state, units, transitions/rates, time scale, boundaries and the behavior the simulation must preserve. Choose continuous, discrete-event or hybrid modeling from the quantities and workload; an ODE is not mandatory for every game system.

• State conservation, positivity, boundedness, equilibrium and event-order invariants relevant to the model.
• Select an integrator using stiffness, step size, error/energy behavior and runtime budget. A locally accurate method may drift or destabilize over the required horizon.
• Distinguish continuous-model stability from discrete-update stability. Linearization/eigenvalues have local assumptions and inconclusive cases; they do not prove all nonlinear trajectories safe.
• Test boundaries, large time steps, long horizons and sensitivity to parameters/initial state. Mathematical predictions and empirical traces should agree within the declared tolerance.
• Define determinism scope: same process, platform or cross-machine. Control RNG streams and arithmetic/order as required; fixed timestep or shared seed alone does not prove replay equivalence.
• For controllers, record target, delay, saturation and disturbance assumptions. Validate overshoot, windup and response under actual update timing.
• For random systems, distinguish the intended distribution from player-perceived fairness constraints such as pity bounds. Test both statistical behavior and deterministic replay where required.

Evidence and output

Produce the model and assumptions, selected method/step policy, invariant/error budget and validation traces. State what is analytically established, locally approximated or observed. Do not promise zero desyncs, no extinction or no overshoot from reading a recipe.