Tunneling nanotubes (TNTs) are actin-supported membrane conduits that carry both rescuing mitochondria and pathology-driving proteins, yet no quantitative link connects nanoscale tube physics to population-level network behavior. Combining membrane-tube mechanics, cable theory, stochastic transport, adaptive-network dynamics, and percolation theory, we build a falsifiable framework with three core results. First, classical tube mechanics predicts a tether force of 24.3 pN and a diameter of 155 nm without free fitted parameters, matching measurements (10—40 pN; 50—200 nm); rupture resolves molecularly as necking instability plus actin—cortex detachment, predicting a median lifetime of 60—62 min (95% CI 37—99 min) within the observed 30—120 min. Second, the contact network is electrotonically compact (λ ≈ 158 µm ≫ L), and finite-size scaling in two and three dimensions constrains percolation thresholds to p_c = 0.257 ± 0.002 (2D) and ≈ 0.169 (3D; geometry-dependent), cross-validated by invariant scaling; whether resting networks actually sit above threshold remains a calibration question (P1). Third, competing cargo flows—protective transport linear under an adapter bottleneck, pathological transport superlinear—yield an optimal connectivity p* = a/2b that falls from 1.0 (healthy) to 0.32 (late pathology); the downshift is robust whenever pathological transport is superlinear. A molecular loading bottleneck explains the few-long-lived-TNT strategy. Five falsifiable experiments are mapped, including the embryonic-heart TNT—Notch axis.