A child who learns to add small numbers can add numbers of any length. By contrast, LSTMs and Transformers trained only on short sequences fail when presented with longer ones. The root cause lies in what the learner reads when it makes a decision.\n\nAs the input grows, variables such as position, count or time keep taking new values—they are coordinates. A learner that reads a single coordinate encounters values in every longer input that were never covered by its experience. Deciding whether two items are the same has the same two answers at any scale: coincidence or not. A learner that reads only coincidences finds nothing in a long input that short inputs lacked, so finite experience fixes a rule that applies to every size.\n\nMemory carries the size of the input while the rule stays finite, which determines the kind of memory a generalizing brain can have: memory that grows with the input but what the learner reads of it stays finite and looks the same from every place. Hence it must be composed of cycles, and the past is accessed by asking what happened when those cycles last stood where they stand now. The cycles are the grid modules, the cells that detect their return are place cells, the records of the cycles form the hippocampal index; a prefrontal controller moves the cycles without ever seeing them.\n\nReading only coincidences costs no extra power: what finite experience fixes in such a learner is exactly what a Turing machine can compute step‑by‑step with its input. This explains why a circuit that evolved to find the way home can also learn arithmetic and grammar.\n\nIn a network of neurons this machine of cycles appears as dynamics revealed through its symmetry, and symbols are its invariants. Trained on strings up to length 40, the machine achieves at least 99% accuracy on average over lengths 41–500 on nine benchmark tasks, five of which are unsolvable by any LSTM or Transformer. When supplied with new cycles after training, it reverses strings of length 15,015 without error and fails beyond that exactly as the theory predicts; its controllers run unchanged on real neurons.\n\nReview: The paper offers a testable framework linking spatial memory cycles to universal computation, shedding light on how finite brains can generalize without bound.