Background: A connectome specifies the structural substrate available to a nervous system, but it does not indicate how much of it a particular computation requires. Reported compact sufficient circuits are point estimates—one subset, one task, one fidelity threshold, one parametrisation.
Methods: We formalise computational sufficiency as a cost‑indexed family of neuron subsets conditioned on a task and a tolerance pair (permitted degradation $\epsilon$, permitted failure fraction $\rho$), separating task adequacy from phenotype fidelity. Candidates are certified on held‑out parameter draws with one‑sided bounds rather than reporting a single minimum. We measure three quantities a single circuit cannot express: non‑decomposability, degeneracy, and an intervention‑derived role taxonomy.
Results: On a released walking simulator, the published 148‑neuron circuit is certified sufficient with mean $D_{\text{task}} = 0.017$ over 32 held‑out draws and no failures. No 146‑ or 147‑neuron subset is sufficient, and removing any of its three interneurons destroys adequacy. Complement splitting recovers the published set exactly in three of seven seeds and reaches its size in five, using 66‑97 evaluations, whereas the bundle hierarchy over 221 neurons at the same budget requires more evaluations. Changing only the task changes the required neurons: our best solutions retain 155 and 149 neurons respectively, sharing only 3 of the 11 searchable neurons. Relaxing the tolerance reveals alternative sufficient circuits; the whole‑CNS interface fails every replicate, while 93 search‑chosen neurons yield a certified solution.
Conclusions: Sufficiency is set‑valued and conditioned jointly on task and tolerance; no single search strategy dominates across regimes. A frontier is interpretable only alongside its interface‑only baseline, its tolerance, and its budget. All results are model‑relative.
Review: This work provides a systematic framework for assessing neuronal subset sufficiency under connectome constraints, highlighting how task definitions and tolerance levels shape circuit size and offering a nuanced perspective on the interpretability of neural computations.