A controller designed in simulation may fail in the real world. The simulation is an approximation — it simplifies friction, linearizes dynamics, ignores disturbances. The gap between simulation and reality is acknowledged universally but rarely quantified formally. Engineers add safety margins, tune gains conservatively, test extensively. The gap remains a felt quantity, not a measured one.
Stochastic simulation-gap functions make the gap formal (arXiv:2603.20672). The approach treats a high-fidelity simulator as the nominal mathematical model plus bounded state-dependent and input-dependent disturbances. The disturbance bounds are estimated from data — repeated queries to the simulator at various states and inputs reveal how much the simulator's behavior differs from the mathematical model's predictions.
With the gap formally bounded, a controller designed for the mathematical model can be certified to work in the simulator with provable guarantees. The controller doesn't need to be redesigned — it needs to be robust enough to absorb the quantified gap. The formal bound tells you whether the controller's robustness margin exceeds the gap.
Validated on a TurtleBot and a pendulum in stochastic simulators, the method provides high-confidence certificates that the mathematical-model controller satisfies specifications in the more realistic simulator.
The structural insight: the sim-to-real problem is not a vague concern about “model mismatch.” It's a specific, measurable quantity — the maximum disturbance that the simplification introduces at each operating point. Once you measure it, you can decide whether your controller can tolerate it. The uncertainty is not about whether the gap exists but about whether the controller's robustness margin exceeds it. Quantifying the gap converts an existential anxiety into an engineering comparison: is the margin bigger than the gap? If yes, deploy. If no, improve the model or the controller. The question becomes answerable once the gap is a number.