Large language models can synthesize a game's rules as executable code - a Code World Model (CWM) - which a classical planner then searches over. Such models are typically accepted when they reach high transition accuracy on sampled trajectories. We argue this is the wrong notion of adequacy for planning. We show four things. (1) An LLM-synthesized CWM can pass a sampling gate at 100% transition accuracy and be $\geq 98\%$ state-accurate on the planner's own search distribution, yet lose systematically at play, because the $<1\%$ it gets wrong is exactly the pivotal dynamics; the play cost of the omitted rule is $0.091$ (seed-clustered 95% CI $[0.065,0.117]$, $n=4800$). We call this the verified-vs-correct gap, and confirm it end-to-end through the synthesis pipeline. (2) The harm follows a quantitative law, $\mathrm{danger}=\mathrm{play\_cost}\times(1-\mathrm{rarity})^N$, whose $(1-\mathrm{rarity})^N$ gate-miss factor is proven exact and whose play cost is empirically bounded. (3) The failure is not repaired by more data: LLM synthesis behaves as rule translation, not rule inference, and did not infer the omitted rule across models (GPT-5.x) and data regimes (including DAgger and targeted examples). (4) The same mechanism recurs on the belief-inference function of imperfect-information CWMs: we prove a coverage bound (a size-$N$ gate is identifying when $N\gtrsim b^{d_{\max}}$), explaining why shallow games such as Kuhn poker show no gap, and hand-construct Beacon, a verified-but-wrong inference function that passes the gate yet loses every game. These results suggest adequacy for planning-oriented world models should be measured on the search distribution or by play directly, not by prediction accuracy on sampled transitions.
A code world model accepted by a sampling gate can be exactly right on everything the gate can see and arbitrarily wrong beyond it. We characterize what a certified model can know, and what its errors can cost, when the omission is an annular freeze mode enclosing an unreachable interior. The gate quotient makes the question precise: acceptance-with-certainty determines the model exactly on the reachable query set; beyond reach is gauge. On a minimal ring instrument we prove the extreme case (a wrong-topology filled-disc artifact unfalsifiable by any sampling gate and bitwise harmless at play) and measure, with LLM synthesis across three model families, how one knob (a channel of width gamma) walks the same artifact through three regimes: unfalsifiable-and-harmless, falsifiable-and-costly, and instantly falsified. Three principles organize the empirics. First, danger is topology relative to reach: a channel the planner can use collapses the blind model's exploitation (play cost 1.09 to ~0 over a knee at gamma ~ 0.1), while a hidden channel with the same first Betti number keeps it at full strength (1.12). Second, repair is parameter-bound and sensor-bound: no family recovers the region from outside evidence; from inside, models pose the right topology but cannot pin its parameters, and the posed topology tracks the guiding persistent-homology summary's wrong beta_1 (a sensor with a measured geometric resolution limit), not the truth. Third, mitigation must match the error's dimension and direction: point fences fail against the one-dimensional boundary, a dimension-matched persisted fence collapses exploitation to a two-lesson transient (0.999 to 0.058), and the dual freedom certificate collapses the invented-mode failure symmetrically (1.769 to 0.029). In n dimensions the shell makes misidentification near-certain while the danger stays fully exploitable: the two axes are independent.
An LLM synthesizes an executable world model that a classical planner searches, and the model is accepted when it reproduces sampled transitions, and it is asked what that acceptance certifies in continuous control.
Javier Aguilar Martín· 0 citations
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