Why it is a question
THEORY-478, narrowed and promoted on 2026-10-19 under ADR-047 v2, makes a conditional statement. Suppose the diffusion densities’ Hellinger distances obey a Hölder bound with exponent θ in d_T / t^(1/β), and a separation bound once d_T exceeds a multiple of t^(1/β). Then for 0 < α < θ/β the multiscale hyperbolic diffusion distance of LIT-870 is two-sided equivalent to min{1, d_T
Two things are left unsettled.
- The tree heat kernel. LIT-870 takes β = 2 and θ = 1 for the heat kernel of a metric tree. It cites the Gaussian and Hölder kernel bounds (Lemmas C.4–C.6) without proof. Its passage to a Hellinger bound for close points (Proposition C.5) holds as written only for d_T ≥ 1, and NOTE-675’s repair gives θ = 1/2. So the range of α for which the theorem applies to trees is not fixed.
- The data bridge. That the diffusion operator of a sampled point cloud with a hidden hierarchy approximates a tree’s heat kernel is assumed, by analogy with the manifold limit of diffusion maps.
It bears on QUESTION-025. Whether hierarchy shows in co-occurrence statistics through a multiscale distance needs the second bridge.
What would answer it
The condition formerly on THEORY-478, unchanged. A complete proof that the heat kernel on a metric tree meets the Gaussian upper, lower and Hölder bounds the theorem uses, with the Hölder step of Proposition C.5 repaired for close points. Also a stated condition under which the diffusion operator of sampled data approximates a tree’s heat kernel. The equivalence to a thresholded snowflake would then be a theorem on its stated range of α, and the bridge from data to it an assumption with content. Better benchmark scores, or more datasets where the method has the best mean average precision, cannot settle it. Neither can the experiments at α = 1/2, which lie outside the theorem’s range.