How Long Is a Recurrent Circuit Faithful? Retained Memory and Fading Error

A recurrent circuit must carry its own state, not receive a corrected state before each short test. Certifying persistent execution is difficult when retained units hold information indefinitely: the usual contraction test on the model–circuit comparison can fail even when their output discrepancy is small. We separate retained circuit memory from fading error. An exact finite memory transducer, combined with protocol-constrained error contraction, gives convergent all-time risk bounds and failure histories that can be replayed without requiring the retained memory to contract. We quantify the error that remains when memory summaries are coarse and give a complementary reset–stutter certificate for token-conditioned affine recurrences. The method builds on invariant-set and differential-verification principles rather than replacing them. A coupled write/hold example exercises the new premise. Six fully trained parity recurrences produce smaller circuits certified at tolerance $0.1$ under a reset-bounded token grammar; five of six refreshed selections at that tolerance fail exact persistent evaluation. Separate lower-calibrated scaling tests compare adaptive suffix refinement against block-aware invariant bounds, including difficult unresolved cases. Finally, tolerance reanalysis of fixed dense nonlinear networks show that local success followed by later failure depends strongly on the chosen metric and operating point.