Sharp Error-Correction Certificates in Recursive Information Dynamics: A Saturating Switching-Network Model

Repeated use of an erroneous source can change an evidence drive, while nonlinear feedback can preserve an incorrect state after the duplication is removed. We study these mechanisms in a discrete-time saturating network model with evidence provenance and external reference inputs. Our principal result is an explicit uniform input-floor certificate for eventual correct classification under arbitrary time-varying nonnegative row-stochastic communication. A scalar threshold, equal to the fold value in the bistable regime and zero otherwise, determines a sufficient signed-input floor for every finite initial state, without contraction, connectivity, or state-convergence assumptions. The threshold is sharp over the declared network class when only the uniform input floor is specified. A scalar comparison envelope is attained exactly by homogeneous systems for every admissible switching sequence; it also yields a Brier-loss bound and a conservative finite correction-time estimate. Source multiplicities and reference-error bounds give an operational sufficient condition for the certificate. Complementary scalar results characterize judgment reversal and persistence after deduplication, while standard contraction arguments quantify input sensitivity. Synthetic calculations verify the bounds and boundary counterexamples. The framework supplies a model-based mathematical foundation and testable hypotheses for empirical investigation of hallucination propagation and echo effects in multi-agent language models; it does not establish their dynamics empirically. This is the author’s submitted manuscript, submitted to Mathematics on 25 October 2026 (Manuscript ID: mathematics-4685224). This preprint has not been peer reviewed or accepted for publication.