Cost convexity controls the architecture of resistance-optimal networks

A conducting network improves access to its interior only by spending material, so architectures compare only at equal budget. We redistribute conductance under a power-law resource law ${\text{C}}_{\alpha }={\sum }_e{a}_e{w}_e^{\alpha }$ and measure access by the reciprocal of the mean pairwise effective resistance. When the cost is strictly convex the allocation is unique and every admissible edge keeps positive conductance; at $\alpha =1$ uniqueness survives but an edge can be priced out. The marginal worth of an edge is then its squared biharmonic distance, a Murray-type balance whose balanced quantity is global rather than junction-local. The same optimum scores any weighted network at its own budget, separating an allocation factor from a topological one. Under concave cost the marginal price of an incipient channel diverges, the feasible set is nonconvex, and conductance condenses onto sparse backbones. The uniform star and the uniform complete graph cross at ${\alpha }_c(n)=\log (n\mathrm{/}2)\mathrm{/}\log (n-1)$, near which intermediate states beat both. Up to $n=64$ the best states found grow approximately linearly in $n$, so channels per node change much less than the active fraction does. Loops are governed instead by a collapse exponent ${\alpha }^{\ast }(n)$, resolved to $0.005$ at every size: it rises from $0.555$ at $n=8$ to $0.675$ at $n=64$, and the collapse across it is abrupt, the support at $n=64$ falling from $133$ active edges to the $63$ of a spanning tree. A fixed exponent therefore changes character with size by crossing this boundary. One prescribed source and sink instead gives a loopless optimum.