Two problems for threshold cascades of interacting diffusions on unimodular random trees: front propagation with a Bramson correction, and the continuous-type limit theory

Companion to arXiv:2608.XXXXX, which reduces threshold cascades of coupled Ornstein-Uhlenbeck diffusions on graphs converging Benjamini-Schramm to a unimodular Galton-Watson tree, unconditionally in a dissipative regime, to a finite-type Galton-Watson process. Two problems are left open there; we formulate both precisely, supply the analytic framework, and prove partial theorems. First, front propagation. We show the generation-indexed front admits a genuine branching random walk comparison, prove the front depth grows ballistically with an explicit speed $c_{\ast }$ given by the Perron root of a tilted mean matrix (unconditional in the dissipative regime), and reduce the conjectured Bramson delay $c_{\ast }t-\frac{3}{2c_{\ast }}\log t$ to the uniform integrability of a derivative martingale, which we construct at the linearised level. The front central limit theorem and the Bramson correction are stated as conjectures with a proof strategy. Second, the sensitive regime, where the type is a continuous failure strength and the offspring law is a Cox mixture. We prove the mean offspring operator $K$ on $L^2$ of the strength variable is quasi-compact with a spectral gap, the mechanism being Hilbert-Schmidt smoothing of the Ornstein-Uhlenbeck kernel governing strength inheritance. This yields a general-state-space Kesten-Stigum theorem, extends the $n^{-3\mathrm{/}2}$ total-progeny law to continuous types with an explicit constant, and gives a strength-resolved central limit theorem. The continuous-type CRT scaling limit is reduced to a multitype invariance principle on a Polish type space, conjectured with all hypotheses verified modulo one tightness estimate. Neither problem is fully closed; each is given a framework, first-order theorems, and a precisely delimited remaining step.