Correlated Randomness in Diffusion Simulation of Full Collider Response

Simulating how a collider detector records the particles produced in high-energy collisions is a dominant computational cost of experimental particle physics, and the recorded response is intrinsically random. The physics lives in the structured of that randomness rather than in any single event. Per-channel hit counts are $2$ to $117$ times overdispersed relative to independently fluctuating pixels, the eight detector subsystems fluctuate together with a mean cross-channel correlation of $0.51$, and per-event activity has extreme tails. We study which parts of this structure a diffusion simulator must represent explicitly, generatingthe eight subsystem channels jointly as $125\times125$ images at $99.5\%$ sparsity. Plain denoising diffusion fails outright and activates essentially every pixel. We instead generate each event as three jointly diffused states. A value image sets pixel magnitudes, per-pixel sparsity bits select the active pixels, and a coarse count map sets the activity of each region of the image. Supervising subsets of the states in the same architecture shows that each state repairs a distinct failure of the generated randomness. The bits make events sparse and accurate for typical events, but their fluctuations sit at the bound of independent pixels, with overdispersion near $1$ on every sparse channel at every training seed. The count state acts as a low-dimensional correlated latent. It restores the dispersion to the real magnitude and fills $0.87$-$1.19$ of the correct total-energy tail mass beyond the reference 90th percentile on every channel at a quarter of the parameters of the enlarged baselines. Explicit noise structure, not capacity alone, determines whether a learned simulator reproduces the law of its data.