A useful local benchmark for Gaussoherence is ordinary Gaussian blur. Let ρ(X) = R(X)² be the configuration-space density, and let coarse-graining at scale Q₀ act by Gaussian convolution,
ρ_Q₀(X) = ∫ K(X, Y; Q₀) ρ(Y) dY, K(X, Y; Q₀) ∝ exp[−(X − Y)² / (4Q₀²)].
In this setting, the relevant quantity is the Fisher information of the density itself,
I[ρ] = ∫ (ρ′ / ρ)² ρ dX = 4 ∫ (R′)² dX,
so the question becomes how much Fisher structure survives under Gaussian smoothing.
This gives a precise version of Fisher leakage: fine-grained distinctions in ρ are not removed arbitrarily, but are progressively washed into the directions hidden by the kernel. The coarse-grained density ρ_Q₀ retains only the Fisher content accessible at resolution Q₀. For a pure Gaussian profile, the effect is exact and closed-form: if ρ has width σ, then ρ_Q₀ has width √(σ² + Q₀²), so the surviving Fisher information is reduced from 1/σ² to 1/(σ² + Q₀²). In that sense, Q₀ sets the leakage scale directly.
More generally, Gaussoherence acts like a Fisher-information filter. Structure on scales much smaller than Q₀ is strongly suppressed, while slowly varying envelope data survive into the stiff sector. The projection kernel does not destroy the wavefunctional’s geometry; it redistributes it across resolvable and unresolvable directions. Fisher leakage is therefore not a loss of all information, but a controlled transfer from sharp distinctions upstairs to coarse observables downstairs. The Gaussian kernel is the simplest solvable case of that mechanism, and it provides the cleanest local model for the more general Fisher heat-kernel flow used in the full theory.
