An Amortized Efficiency Threshold for Comparing Neural and Heuristic Solvers in Combinatorial Optimization

A common critique of neural combinatorial-optimization solvers is that they are less energy-efficient than CPU metaheuristics, given the operational energy cost of training them on GPUs. This paper examines the inferential step from "training is expensive" to "neural solvers are net-inefficient", which is where the critique actually goes wrong. Training the network costs a large fixed amount of GPU energy; running the metaheuristic costs a small amount of CPU energy on every instance, repeated as long as the solver is deployed. The two are not commensurable until a deployment volume is fixed. We define the Amortized Efficiency Threshold (AET) as the deployment volume above which a neural solver breaks even with a heuristic baseline in total energy or carbon, under an explicit constraint on solution quality. We show that the cumulative-energy ratio between the two solvers tends to a constant strictly below one whenever the network wins per-instance, and that this limit does not depend on how the training cost was measured. An embodied-carbon term amortizes hardware fabrication symmetrically on both sides. We instantiate the framework on the Multi-Task VRP (MTVRP) environment at n=20 customers across 19 problem variants and five training seeds, with HGS via PyVRP as the heuristic baseline. The measured crossover sits near $1.58 \times 10^5$ deployed instances; the per-instance ratio is 0.41, reflecting the moderate size of the instances tested. The contribution is the framework, the open instrumentation, and the measurement protocol; structural convergence of the ratio at larger problem sizes is left to future empirical work.