Smartphone Patch Gap 2D: N-Day Exposure Gas Simulator

This is the 2D counterpart to the 3D smartphone patch-gap fleet simulator, built as an independent model rather than a re-rendered scene. The fleet is a real 2D gas of particles undergoing momentum- and energy-conserving elastic collisions, while each particle's exposure state is driven by the same OEM-delay-plus-adoption-rate statistics that govern real mobile device management (MDM) patch compliance. What sets this version apart is a live analytic overlay: a closed-form probability curve for the expected patched fraction, computed by numerically integrating the exact jitter-plus-geometric-adoption law, plotted alongside the empirical curve traced out by the particles on screen — a real-time check that the simulated randomness matches the theory. Drag the exploit-release day and watch both the simulated and analytically-predicted cumulative "vulnerable device-days" track each other as the rollout unfolds.