Acute Multiscale Spacing Entropies and Quantitative Cluster Compression in Attractive Logarithmic Particle Flows

We study ordered particles on the real line evolving under the attractive negative-gradientflow of the logarithmic Vandermonde interaction. For each scale m, let Wm be the sumof the logarithms of all overlapping m-spacings. Our main structural result is an explicitpositive factorization of every pairwise gradient inner product ⟨∇Wr,∇Wm⟩. Hence the fullspacing hierarchy is pairwise acute, and every Wm is a strict Lyapunov functional for theVandermonde flow and, more generally, for every nontrivial nonnegative combination of thefixed-scale logarithmic flows. We then convert this qualitative structure into a quantitativemultiscale contraction law: the centered second moment decreases exactly linearly, whilethe geometric mean of the overlapping m-spacings contracts at an explicit scale-dependentrate. This yields a collision-or-cluster-compression certificate at every scale m. At thenearest-neighbor scale we further obtain an exact dissipation identity for reciprocal gaps.For an adjacent gap, the singular pair attraction separates from a nonnegative exteriorforcing, leading to an explicit lower collision-time bound. Weak back-pressure forces adefinite amount of local reciprocal-gap energy and, under a natural local-peak condition,a finite-window Parseval argument forces spectral mass at the inverse window scale. Theresults are finite-dimensional and deterministic and apply directly to simple real zeros ofpolynomial heat flow up to the first collision.