Tails and Walls: Reality's Two Honest Priors

<title>Abstract</title> <p>The maximally ignorant priors of a regular statistical model come in two dual branches: a heavy branch [1 + αν Iδ]−1/ν (2002; Student-t for scalar location models, up to Cauchy; multivariate for ν &lt; 2/d) and a compact-support droplet [1 − αν Iδ]1/ν + (2026), with Iδ an information deviation, 0 ≤ ν ≤ 1 the tail index, and δ ∈ [0, 1] a robustness (loss) parameter. In infinite dimensions there is no canonical volume (no locally finite translation-invariant measure), so densities exist only against a reference; but on a finite-dimensional model the Information Volume dV = √ det I dθ is canonical. Writing a measure in its δ-coordinates ℓδ(p) = pδ/δ—a canonical vector embedding into L1/δ, self-dual Hilbert at δ = 1 2—the pairing of a coordinate against its dual returns the same metric I for every δ, so dV is deformation-independent and computable (I = J⊤J; Cauchy–Binet and Hutchinson/Lanczos probes, the operator the companion diffusion paper builds). Extremizing the same action over the truth-guess gives t⋆, the models’ barycenter in these coordinates—the Bayesian predictive at δ = 1, Robbins’s empirical Bayes—so the empirical marginal one measures is t⋆, and its branch is the prior’s: a heavy prior stays heavy, while a droplet’s wall survives a bounded observation but washes under an unbounded one. The branches map onto the extreme-value trichotomy (heavy ξ = ν/(2−ν) ∈ (0, 1], droplet ξ = −ν/(1 + ν) ∈ [−1 2 , 0), meeting at Gauss), so the tail shape ξ classifies an observable as droplet- or heavy-type within the footprint ξ ∈ [−1 2 , 1]. In QM9 the molecular ensemble hosts both branches (C–O walled on both sides, the hydrogen-bonded O–H heavy); sound is heavy, speech relaxes to Gauss, and 8-bit images are droplets whose wall a pre-registered test attributes to the representation, not the radiance field. Every sign-stable marginal but one lands inside the footprint—the exception, HDR linear luminance, is a super-Cauchy stress test we leave open. MSC Classification: 62B11 , 60G70 , 62F15</p>