A density contrast measured by binning reports Poisson shot noise until you subtract it

Fact

Bin N points into C cells and the variance of the counts is not a property of the points' arrangement alone. Even for a perfectly uniform random field, each cell holds a Poisson count, so

sigma^2_measured  =  sigma^2_structure  +  1/mean       where mean = N / C

With 4096 particles in a 12x12x12 grid the mean is 2.4 and the noise floor is 1/sqrt(2.4) = 0.65. A freshly initialised simulation with no structure at all reports a density contrast of 0.65, which looks like a plausible measurement rather than a bug.

Why it matters

The number is usually shown as "how clumpy is this" and read as physics. It starts at the wrong value, and worse, it moves the right way as the simulation runs, so nothing prompts a second look. Changing the particle count to investigate makes it worse: more particles lower the floor, so the metric appears to depend on resolution, which sends the search into the solver.

The same term contaminates any binned count statistic: histogram variance, per-tile occupancy, request-rate jitter, heatmap contrast.

How to apply

• Subtract the floor rather than choosing a grid that hides it:

  const corrected = measured - 1 / mean;
  return corrected > 0 ? Math.sqrt(corrected) : 0;

• Size the grid from the sample, not from a constant, so the correction stays small: aim for roughly twenty per cell, cells = cbrt(n / 20).
• Sanity-check by shuffling the points uniformly and re-measuring. A correct metric reads zero on the shuffle; an uncorrected one reads the floor.
• Clamp at zero. The corrected variance goes slightly negative on a sub-Poisson arrangement such as a lattice, and sqrt of that is NaN.