Generalized Grokers: The Verifier-Rich Substrate Is Autonomously Grokable

The verifier-rich structure that makes trust a computable property of LLM inference can be discovered and maintained autonomously from the corpus itself. An inference run over a trajectory has three reusable artifacts: the strategy that orders its steps, the function computed at each step, and the context each step conditions on. One amortization principle governs all three: keep the reusable artifact off the per-query path and learn it incrementally from actual data, so a model invocation becomes a cheap deterministic read. Probabilistic Language Tries amortize the strategy, LAWS amortizes the function as a certified expert, and groking amortizes the context as an incrementally maintained typed comprehension index. We formalize groking as such an index, give an amortized-comprehension bound stating when it beats per-query re-derivation, and show that its bottom-up order is the operational form of certified composition. An upward-closed uncertainty result confines the index's model-quality residual to an explicit, shrinking flagged set; and, made conservative, the index is a sound over-approximation at every depth for a model of any accuracy—by order-theoretic monotonicity rather than any smoothness assumption—so model quality sets its precision, never its soundness. We show the domains admitting profitable groking are exactly the verifier-rich ones: code, games, formal mathematics, hyperlinked corpora, schemas, and data pipelines.