Horizon-optimal allocation of finite thermodynamic support in redundantly supported constrained Markov networks

Horizon-optimal allocation of finite thermodynamic support in redundantly supported constrained Markov networks. When multiple physical support channels can realize the same present active-face requirement while drawing from different finite reservoirs, the allocation of throughput among channels becomes a horizon optimization problem distinct from present-task satisfaction. We study the allocation question that emerges under redundancy: how to route support consumption to maximize the time a maintained operating point can be held before resource exhaustion. In a four-state continuous-time Markov network with bounded locally detailed-balanced support channels, redundancy creates a one-dimensional family of exact-maintenance current vectors. Reallocation along this family leaves the focal state and active-face support unchanged, and the total reservoir drain is policy-independent, but the partition of drain between finite reservoirs is not. We prove that the exact-maintenance horizon is maximized by allocating preferentially to the substitutable reservoir at every instant and transferring support only as its channel saturates. The optimal policy deliberately depletes the substitutable reservoir below a threshold terminal in the nonredundant case, because redundancy converts that threshold into a switching surface. An instantaneous-effectiveness-greedy allocator instead balances reservoir depletion and is strictly suboptimal. We compute the horizon gap exactly: 10% of extension attributable to redundant architecture under balancing, 8.91% from horizon-aware routing, 18.91% combined. We also identify the resource-visible part of allocation freedom as ker(H)/(ker(H)∩ker(Ξ)), where H maps support currents to present active-face support and Ξ maps them to reservoir expenditure. The result separates present-task equivalence from finite-horizon equivalence in a thermodynamically consistent realization.