Connectivity Maintenance Cost: A Thermodynamic Threshold Model for the Failure Sequence of Sustained Biological Networks, with Application to Neurodegeneration

Version 4 adds Appendix B, a complete numerical validation of the model's three formal results (F1: failure threshold; F2: hub-first failure ordering; F3: closed-form decay and hysteresis). All simulations were reproduced by an independent reimplementation coded from the model specification alone.Appendix B states the exact edge-based form of the global underfunding threshold:B* = CMC · max_(i,j)∈E [ k_i k_j / (k_i + k_j) ]which corrects the heuristic bound kₘₐₓ·CMC/2 used in previous versions (exact only when the two highest-degree nodes are adjacent). The appendix also defines node failure events formally, reports the sharp failure-ordering transition at αγ = 1 (hub-first ordering holds if and only if αγ < 1), and specifies the precise origin of hysteresis: the rebuilding-cost assumption R_ij ≫ c₀τ, not the decay dynamics ẇ = −λw + κf, which are reversible (single stable fixed point w* = f).