Checkpoint Reload Explains Most of the Measured State Gap in Two State Discovery Reproductions

Saved checkpoints record states along a training trajectory, but generally do not determine the updates at states that would be visited under a different schedule. We study how accurately these checkpoints can reconstruct the endpoint of a sequential reference with prescribed update strengths. Under a common local transition model, two checkpoint-index moment conditions characterize all convex merges that agree with this reference through second order. We then prove an information limit that for nondegenerate profiles, no algorithm using only a fixed-length gradient-descent (GD) history with step size $h$ can achieve $o(h^3)$ endpoint error uniformly over a fixed class of smooth, strongly convex losses. The lower bound follows from two losses with identical GD checkpoint histories but sequential reference endpoints separated by $Ω(h^3)$. Quadratic-Accurate Merging (QAM) achieves a matching uniform $O(h^3)$ endpoint error bound. Its explicit coefficients also define the unique profile-dependent merge that exactly matches the sequential GD reference across all fixed quadratic objectives. Across two public Adam checkpoint trajectories (SmolLM3-3B and OpenEuroLLM-Prelude-9B), three windows and three profiles per model, and 15 tasks, QAM shows mixed results for short windows and broader advantages over Warmup-Stable and Merge (WSM) for longer windows. Matched-moment GSM8K diagnostics further show that local consistency alone does not fully determine downstream scores. These results characterize the reconstruction limits of saved histories, provide a coefficient rule that attains the optimal rate, and assess its practical utility.

Generative world models are increasingly driven as simulators: a planner forks a state, rolls out futures, backtracks, and returns to a visited viewpoint. Recent benchmarks establish that current video world models fail this usage, and attribute it to the model, prescribing new architectures and training objectives. We show this attribution is incomplete, and for an important class of models simply wrong. Snapshotting the state the runtime already holds -- an observation plus RNG state, a memory bank, or a windowed KV context, by architecture -- and restoring it after a genuine excursion reproduces the never-left continuation byte-identically on all three; corrupting only the RNG degrades it. The capability was never missing: request-centric serving discarded it, inheriting from language-model serving the assumption that runtime state is recomputable -- but world-model state carries a non-recomputable kernel. We define Persistent Computational State (PCS), the minimal non-recomputable state that must survive across requests, show it can be discovered by measurement, and build a session-centric runtime over it. Checkpoint and restore cost 0.012 ms against a 1.85 s generation step; resident sessions become host- rather than device-bounded (measured to 1,024); and world memory must be evicted by relevance to the return, not recency -- the inverse of LLM practice.

Language-model checkpoints are commonly selected by pretraining loss or benchmark scores, assuming that the highest-scoring checkpoint will remain the best starting point for subsequent training. We show that this assumption can fail in a full 30B mixture-of-experts training pipeline. The checkpoints that perform better after the full downstream training stack also have higher solution density, i.e., retain downstream performance under local weight perturbations.