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Abstract — 140 words
Transport in disordered media is usually described through spectra, eigenmodes, and localization lengths. Yet finite systems are probed point to point: a source excites a specific realization at a specific frequency, and transmission follows sparse, heterogeneous corridors. We show that the resolvent response defines a source-conditioned communication geometry directly on the medium. Its logarithm acts as a transport potential whose basins, ridges, and saddles organize propagation. Within this geometry, attenuation is controlled not by the cumulative cost along a path, but by the minimax saddle: the lowest pass at which source and target first become dynamically connected. This yields a finite-scale constitutive closure for transmission, with subleading corridor and entropic corrections. Tests on disordered graphene and protein elastic networks show that the saddle systematically outperforms path-based predictors, revealing transport as a threshold-controlled, geometric process in finite disordered matter.