Start with a 3D ball of uniform sample density. A finite arrival-time window selects a thick region. Shrinking that window while dividing by its width exposes the instantaneous 2D level set and a finite response density.
The numerator is probability in a gate. Its shrinking value is divided by the shrinking width. The limit is a derivative, not a nonzero probability assigned to an exact time.
[F(τ+Δ/2) − F(τ−Δ/2)] / Δ → F′(τ)
A point source at the ball center and a far orthographic camera give excess path length τ(x)=|x|−z. The constant-arrival surface is a clipped paraboloid, not a concentric sphere. The right image integrates the selected support along camera rays.
Set R=1 and normalize the uniform ball density to K=3/(4πR³). This is a geometric probability model, not the physically attenuated point-light fluence. In the far-camera limit, subtract the common camera distance: τ=|x|−z, with 0≤τ≤2R. F(τ)=3τ/(4R)−τ³/(16R³); outside the interval it is 0 or 1. Therefore F′(τ)=3/(4R)−3τ²/(16R³).
Writing ρ²=x²+y², the sheet is z=(ρ²−τ²)/(2τ), clipped to the ball. Its projection has density K/|∂τ/∂z|=K|x|/τ at a valid ray crossing. The finite-window image uses the exact clipped z-interval; it tends to this density as Δ shrinks. Integrating the projected density over the image plane gives F′(τ). At τ=0 a boundary/axis degeneracy needs separate treatment; the slider stays away from it.
A finite time window leaves 3D thickness. A regular instantaneous level set has dimension two. Temporal width and a spatial photon kernel are distinct; both can soften a displayed image but their measures differ.
Analytic uniform-ball example with the source and orthographic camera legs explicitly included. No fitted visual curve substitutes for the finite-window or derivative calculations.