Memorization isn't failure. It's reduction to a known problem. The failure starts where reduction ends.
The standard framing: LLMs either “reason” (genuine inference) or “memorize” (brittle pattern matching). But this dichotomy misses a third mode (arXiv:2603.21350): reduction — mapping a novel instance onto a familiar problem and applying the known solution. Reduction isn't reasoning (it doesn't derive the answer from first principles) and it isn't memorization (it handles novel instances). It's the space between.
The reduction ladder: a sequence of modifications that progressively move puzzle instances away from canonical forms, making it harder for models to reduce while preserving the underlying logical structure. At the bottom of the ladder, the puzzle is recognizable — a standard hat puzzle, a standard Knights and Knaves variant. At the top, the same logical structure is embedded in unfamiliar framing, unusual constraints, or inverted roles.
The finding: some large models succeed through reduction, mapping novel-seeming problems onto familiar templates. Others fail earlier. But all models struggle once epistemic reasoning is required — the deeper inference needed to determine world-states from incomplete information about what other agents know. The ladder's top rung, where reduction is impossible and the model must actually track nested knowledge states, is universally difficult.
The structural insight: the important boundary is not between memorization and reasoning. It's between problems that can be reduced to known patterns and problems that can't. Reduction is computationally cheap and handles most instances. Genuine epistemic reasoning — what you know about what I know about what you know — requires maintaining nested mental models, and no current model reliably does this. The capability gap is at the reduction boundary, not the memorization boundary.