An Omitted Mode Is a Rare Rule: The Sampling-Verification Danger Law in Continuous Code World Models

Abstract

In the Code World Model paradigm an LLM synthesizes an executable world model that a classical planner searches, and the model is accepted when it reproduces sampled transitions. We ask what that acceptance certifies in continuous control. We define the pipeline's danger as an expected risk and isolate its exact factor: the probability that N i.i.d. gate rollouts all miss a critical event of probability r is exactly (1-r)^N; an independent acceptance sample adds its budget to the exponent. On three hybrid instruments the accepted mode-blind model is exploited: the planner is pinned at the mode boundary at a regret of nearly the whole attainable return. We prove a localization budget, valid at boundary points: models with Lipschitz constant at most L differing by eta at a point disagree above tolerance eps on a region of volume at least kappa((eta-eps)/L)^(d+m); the discontinuous reset modes studied pay no such budget. With real LLM synthesis, GPT-5.x repairs an omitted 1D clamp in 105 of 111 mode-containing draws -- every attempt exact on 50 of 56 instrument-stream blocks (95% CI [0.781, 0.960]). On 2D regions no artifact recovers the rule (0/156); eight targeted interventions leave the failure in place, and positive controls locate it: a located rule is not induced, while given form and location the constants follow exactly. A version-space certificate proves identification is class-relative: at the widest dose the declared fit succeeds in 20/20 blocks and every sample-consistent circle is within tolerance in 18/20. We prove a class of entry rules exactly consistent with every sample yet harmless at play, so identifiability is a measurable property of the instrument. Re-scoring all 1034 artifacts on independent samples confirms acceptance certifies sample consistency and no more: where the gate is provably informative it covers about two percent of the exploited planner's queries.