Scale Graphs II: Cheeger Dichotomy, (t,s)-Transport, and Tail Drainage

This paper develops the dynamical part of the Onsager-Cheeger scale-graph framework introduced in Scale Graphs I. The first paper established the finite-graph algebra: nonnegative edgewise Onsager production, Prigogine square completion, and an exact minimum-production Lyapunov identity under decimation. Here the algebra is coupled to dissipative scale dynamics on a global attractor. Under shellwise parabolic damping, controlled forcing tails, finite-window Onsager scale graphs, and a Cheeger alternative, we prove a conditional two-channel tail-control theorem. In coercive windows, the Prigogine defect budget controls chemical-potential oscillation through weighted Poincare and discrete BMO estimates; in bottleneck windows, Onsager positivity suppresses flux across a small Cheeger cut. Both channels feed a square-root-influx versus linear-damping tail inequality. The comparison estimate is stated in its sharp safe form. From dE/dt ≤ b√E + F − γE one obtains a comparison in √E. Consequently the robust energy estimate inherits an e^(−γt/2) factor at the square-root level; this is the estimate used throughout the paper. The numerical part is a theorem-level audit of the mechanisms in the proof. The accompanying Figshare dataset contains the frozen reproduction dossier v6.2; its master command regenerates every paper-facing numerical table from the shipped object. The paper-facing numerical outputs are: a regenerated Branch-B bottleneck certificate; an E1 terminal-band balance audit; an E2 stationary K⁴-plateau dichotomy, which is the main numerical diagnostic; a source-canonical Branch-A weighted-gap regrade; and an E3a direct-BMO diagnostic on regenerated canonical mu-fields. The claim levels are explicit in the manuscript and in the dataset certificate. This gives a reproducible numerical audit of the scale-graph mechanisms while keeping the mathematical theorem conditional on the stated hypotheses. Numerical dataset (frozen dossier v6.2): https://doi.org/10.6084/m9.figshare.33001511Paper I of the series: https://doi.org/10.6084/m9.figshare.32942279