COARSE-GRAINING RECURSIVE COMPUTATION: CONTINUATION-RELATIVE INTERFACES AND FAILURE MODES

A learned substitute can replace a run of blocks or iterations in a deep or recursive network at lower cost. Replacement methods fit the substitute to reproduce the output of what it replaces and accept it on that fidelity. Fidelity, however, is measured at the replacement boundary, and usefulness is decided after it, by the continuation: the computation that runs on the substitute’s output, and whether it still solves the task at lower total cost. We give an evaluation contract that judges a replacement by its continuation. The contract separates boundary-state agreement, path statistics, executed computation, and task loss, and states what agreement at each level certifies about the next; it decomposes the continuation gap exactly into a fixed-action term and a reallocation term; and it requires the control that replacement methods omit, the reduced network trained from scratch on the task. Under the contract, fidelity is neither necessary nor sufficient, and three settings show both directions. On GPT-2, a substitute trained through the continuation with no fidelity target beats a published block-replacement method’s fidelity optimum at all five boundaries tested in distribution; under distribution shift the fidelity optimum keeps the lower loss. On a pretrained recursive model, the continuation-trained substitute is 0.36–0.37 nats below the un-adapted model in distribution at 24% fewer recursion MACs, with up to thirty times the fidelity optimum’s boundary error. On an eight-expert prototype over six synthetic data families, a from-scratch control beats every arm that inherits the trained weights on all 60 family, model, and split combinations, and anchoring expert occupancy to the original path cuts the distance from 0.78 to 0.26 without consistently lowering error. A replacement claim thereby becomes a question its continuation can answer: what it still achieves, at what cost, against which control.