Engram: Episodic Composition of Certified Computation. Abstention, Observability and Auditability in Looping Transformer Decoders

Long algorithmic computations tolerate wrong guesses, but not wrong commitments: an error that is kept breaks everything built on it. This paper reports Engram, a single-layer looping transformer decoder, and the pedagogy it demonstrates: training step by step on targets an oracle (an exact solver) has verified, prioritising abstention, observability and auditability; it holds a promise that next-token approximation does not make by construction. Training is for near-perfect precision, which makes recall a function of compute. Abstention is recoverable whereas commitment is not, and composition makes that asymmetry exponential: precision fixed at zero is what turns compute into recall instead of kept errors. The model's latent iterative window is short and never grows: the episode is bound and long problems are solved by composing episodes. The problem is held in its written partial solution, which every episode reads and extends; the window never needs to hold more than one episode's work. Trained on 8-bit parity episodes, the model reaches 256-bit instances by composition; trained on compositions of two permutations in the alternating and symmetric groups of degree five, it composes ten thousand permutations: zero errors in both, on inputs far longer than any the window ever held. On Sudoku-Extreme, served inside a certified search that opens hypotheses on its behalf, it solves exactly or abstains: 1,635 of 2,000 unseen puzzles, zero wrong, with coverage rising from 27.8% at 800 compute episodes per puzzle to 81.8% at 51,200. Across every experiment reported here, over fifty-two million episodes, it has never kept an error. Every step is emitted as ordinary tokens, checkable against the rules of the task the moment it appears. What these results support is a claim about training rather than about architecture: supervising every step against an oracle, and training abstention wherever no verified step exists, is what makes computation safe to compose, and it is this regime, more than the model, that produced the record above.