Inductive Bias Is Not Free: Lifecycle-Aware Evaluation of Topological Priors Under Data and Compute Constraints

Explicit structure can reduce what a model must learn, but constructing and serving that structure also consumes resources. We test whether a topological inductive bias pays for itself on binary MNIST (0 versus 8) under seven labeled-data budgets. A convolutional network is augmented either with exact one-dimensional persistent-homology features or with a cheap seven-dimensional morphology proxy; both are compared with a parameter-matched control over ten paired seeds. We report clean performance, four deterministic corruptions, preprocessing, training, inference, memory, and cold lifecycle time. Exact topology does not improve clean accuracy: relative to the control, it loses 0.205--0.420 percentage points at 25--100 labels per class and shows no detectable advantage thereafter. It also increases full-data cold lifecycle time by 29.5 times and deployment latency by roughly 15,300 times. The proxy reduces topology cost by about 33 times but offers no dependable accuracy or robustness gain. The result is a caution and a protocol: under constrained budgets, an inductive bias should be evaluated as a lifecycle intervention, not as a cost-free input.