Self-Stirring Sequences: Exact Dynamics and Statistical Rigidity

A self-stirring sequence appends the base-b expansion of a current digit count, followed by that digit, and immediately updates all counts. We develop its exact dynamics in every base and determine how its statistical constraints change when the readings are perturbed. Every exact policy produces a normal sequence, with uniform quantitative bounds for growing words. For the cyclic policy we obtain an explicit logarithmically periodic length expansion, natural query moments, a digital-path invariance principle, and a sharp prefix-resolution threshold. The Brownian limit in digit depth coexists with freezing over subexponential lags in generation time. Carries force macroscopic local accelerations and an infinite sequence of logarithmic-power discrepancy lower bounds. In base two, an exact quotient exhibits arbitrarily large merging families near balance, while the cumulative drift reduces to an explicit Takagi term and a precisely defined low-digit remainder. For readings with error K L^α, 0 ≤ α < 1, every empirical limit has a forced uniform Bernoulli component of mass 1 − α; each invariant residual measure η is attainable and its fiber has Hausdorff dimension αh(η)/log b. No such output is strongly normal, whereas every positive error L^1+o(1) permits a full-dimensional strongly normal set from each seed. Finally, an exact series test determines the Lebesgue size of realizable outputs for every eventually nondecreasing error function. All statements are proved for the count-dependent updates themselves; no independence assumption is imposed on the feedback orbit.