Complementary to the Wasserstein/heat-flow description, Fisher–Rao geometry suggests a possible effective relaxation closure for the pushed density ρpush(x) = ∫ |Ψ[X]|² KQ₀⁽ᴳ⁾(X,x) dX. Over an adiabatic interval in which G and Q₀ vary slowly, let ρ denote the corresponding quasi-stationary pushed distribution. If relaxation toward ρ is effectively described by a Fisher–Rao gradient flow, then for classes of f-divergences satisfying the corresponding Fisher–Rao functional inequalities, 𝓕[ρpush ∥ ρ] obeys d𝓕/dt ≤ −ceff𝓕 and hence 𝓕(t) ≤ e^(−ceff t)𝓕(0). Such Fisher–Rao bounds can be independent of target log-concavity, making this route particularly attractive for multimodal pushed distributions. Carillo arXiv:2407.15693. In sPNP, the nontrivial step is to derive ceff from the geometry generating the push. A natural hypothesis is that ceff is controlled by Q₀, low-order spectral data of ΔG[R], and finite moments of ρpush. A concrete route is to linearize the effective pushed-density dynamics about ρ and determine whether its lowest nonzero relaxation rate is controlled by those Gaussoherence spectral data. This defines τrelax = ceff⁻¹ and gives the quantitative consistency condition τrelax ≪ τbackreaction. Establishing this dependence analytically, or estimating ceff from representative Fisher–Rao flows, would test whether Gaussoherence admits a rapidly relaxing effective channel before appreciable geometric backreaction occurs.
KL mismatch and finite-resolution Gaussoherence Stramigioli’s Entropic Physics (zenodo preprint) provides a complementary finite-to-local information hierarchy. He distinguishes finite KL mismatch relative to a reference state from incremental KL between neighboring information states, and shows that the latter reduces at second order to the Fisher metric governing local information change. This suggests a direct interpretation in sPNP. Because the Gaussoherence heat semigroup ΠQ₀ is a Markov coarse-graining channel, its action contracts relative entropy:
DKL(ΠQ₀ρ ∥ ΠQ₀σ) ≤ DKL(ρ ∥ σ). Thus Gaussoherence cannot increase the KL distinguishability of two global densities available through the coarse channel. The finite quantity ΔKL(Q₀; ρ, σ) ≔ DKL(ρ ∥ σ) − DKL(ΠQ₀ρ ∥ ΠQ₀σ) ≥ 0 measures the portion of their KL distinguishability rendered inaccessible at resolution Q₀. This finite statement has a local Fisher limit. For σ = ρ + εδρ with ∫δρ dμG = 0, DKL(ρ + εδρ ∥ ρ) = ½ε² ∫ (δρ)²/ρ dμG + o(ε²). Applying the Gaussoherence contraction before taking ε → 0 gives ∫ (ΠQ₀δρ)²/(ΠQ₀ρ) dμG ≤ ∫ (δρ)²/ρ dμ_G, so the Fisher–Rao norm of an infinitesimal statistical perturbation is itself contracted by the same coarse-graining channel. Gaussoherence therefore supplies a concrete resolution parameter to Stramigioli’s KL-to-Fisher hierarchy: KL measures finite distinguishability lost across a coarse-graining step, while Fisher–Rao geometry measures the corresponding infinitesimal susceptibility of neighboring densities. This provides a natural information-theoretic bridge between the heat-kernel coarse-graining of sPNP and the effective Fisher–Rao relaxation closure above.