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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 ","title":"An Amortized Efficiency Threshold for Comparing Neural and Heuristic Solvers in Combinatorial Optimization","url":"https://arxiv.org/abs/2605.14624","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.14624v1 Announce Type: cross \nAbstract: 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. 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