Theory of post-jamming rigidity in feedback-regulated cellular packings

Budding-cell packings jam before all buds are mechanically constrained. The post-jamming state is therefore set by both the pressure $P$ and the fraction $u$ of buds that remain unconstrained. We develop a mean-field theory for this regime. A modified Maxwell count predicts the post-jamming coordination. A depletion law gives the density at which the initially free-bud reservoir is exhausted, and a flux-partition argument shows how strong feedback can stiffen the packing while producing little pressure. The mechanism is that feedback shifts growth toward residual underconstrained bud modes. Depleting those modes raises the excess coordination without a comparable rise in prestress.