The conductance filtration: sink-relative persistence and exact edge sensitivity for heat transport in particle-filled composites

We formulate the thermal conductance $g_e$ of a particle pair as three branch types – contact, near-contact and bond – within a single construction, and we introduce the filtration that takes this $g_e$ as its axis (the conductance filtration) together with persistence taken relative to the sink boundary. We then show that the dissipation share is exactly the logarithmic sensitivity of the effective conductance, and that a finite change of a single edge is predicted exactly in closed form. The allocation of the dissipation among branch types, and the idealisation ratio of each mechanism (interface, matrix and particle), are thereby obtained as diagnostic quantities on one and the same graph representation.