Branching Cascades in Hierarchical Coordination Systems: Criticality Conditions, Finite-Depth Effects, and Tests of Geometric Scaling

Abstract

Hierarchical coordination can support branching cascades, but regulated load does not establish critical reproduction. The paper distinguishes independently measured load, offspring transmission, boundaries and observation rules. Classical critical Galton-Watson theory supplies the 3/2 mass exponent under its explicit conditions. Exact finite-progeny and finite-depth laws separate off-critical suppression, nonextinction and physical termination. Extinction selection can hide the sign of criticality distance; distinct roots and propagation mixtures change burden under specified alternatives. Loading-dependent branching, logistic failure links, boundary feedback and immigration have close antecedents. The proposed contribution is a narrower restriction: independently measured coordination coefficients must constrain several intervention effects through one bridge and then predict offspring, family size and boundary occupation together. Four prospective tests compare that restriction with unconstrained covariate, redistribution, heterogeneous-offspring and external-drive models. A conditional joint test predicts temporary cascade attenuation followed by recovery after an activity-controlled capacity step. Primary neural, retweet and outage studies constrain numerical mappings; deterministic and synthetic checks verify calculations. No raw-data GGT calibration, independent empirical confirmation or priority certification is claimed. Note on Version 2.0: this version revises the registered v1.0, titled "Cascade Universality in Hierarchical Coordination Systems" (about 5,200 to 11,000 words). The earlier structural-consistency explanation of the 3/2 cascade exponent is restated: the exponent is attributed to classical critical Galton-Watson theory under explicit conditions, finite-progeny and finite-depth effects are separated, and the remaining claim is a narrower calibration restriction tested against rival models. The recorded claim history is preserved in Appendix B. Files: the v2.0 manuscript and a supplement archive with integrity and reproduction instructions, source ledgers, a manifest and SHA-256 checksums. Series Paper II of the Governance Geometry Theory (GGT) papers within the author's Deficit-Fractal Governance (DFG) framework.