The Free-Recipe Limit: Every Recipe Effect Measures Which Premise of an Idealised Learner Broke

来源摘要

arXiv:2609.26160v1 Announce Type: new Abstract: Fix a corpus and send recipe search to infinity: try every order of the skills, every arrangement from blocked to interleaved, every composition, and keep the best. Two quantities decide what that search was worth: the diameter of the reachable set it explores, and the resolution at which anyone can tell two endpoints apart. Where the diameter falls below the resolution, no amount of search converts into a decision, and the signature is not an absence of winners but winners that do not survive re-running. We measure this recipe-search wall with 761 fine-tuning runs on 12 base models (0.5B-14B, three pretraining families) over competition-mathematics skills: base checkpoints, supervised fine-tuning under AdamW, exact-match scoring at k=4. Within one coherent domain at fixed volume the three classical freedoms average 0.010-0.021 against a 0.019 floor, and the largest contrast, 0.0619, clears a three-seed resolution and then reads +0.010 and -0.015 on two reruns. The departure with a systematic answer is coherence: halving one pooled corpus and letting the halves write answers under incompatible but equally correct conventions moves arrangement from capability to allocation between conventions, by two orders of magnitude over a same-convention control, and writing the convention into the input switches the phenomenon off. The switch replicates on a second pretraining family and survives an independent re-execution of its own protocol, with a re-execution spread (0.087) smaller than the resolution a search-selected order cell carries (0.144). Order itself is a transient whose sign crosses zero three times inside a single run. Volume, the one lever nobody calls a recipe, is the one that reliably pays. A public scorecard grades