On the Current Space of Sequence Evolution Operators

We develop a current-space theory for sequence evolution operators. Inherited weights become nonnegative transport current on a directed network, with classification allocating current and history aggregation preserving its provenance. Finite modules admit exact exit, residence, feedback, and elimination formulas. Green transport and finite cuts separate absorption, recurrent retention, and spatial escape, while end measures and invariant observables distinguish geometric channels from persistent modes. Return cycles characterize stationary normalization, and capacities and finite-energy flows supply recurrence tests in reversible networks. Directed return matrices and periodic correctors provide alternatives without detailed balance. Two-current couplings yield merging criteria and sharp waiting-time classifications on a one-ended model. Information loss identifies when an autonomous quotient also preserves source provenance; approximate quotient and path-law estimates quantify weaker comparisons. An occupation-weighted relative-entropy budget controls some perturbations beyond the absolute-discrepancy criterion. Boundary integration represents all bounded harmonic observables on an unmerged history tree, but the current-proportional electrical shadow is null recurrent and its trace energy is noncoercive. State-dependent clocks separate update paths from physical residence and finite-time escape. Directed entropy identities and symmetric energy bounds control finite-space relaxation, while commutators quantify random-activation splitting. Empirical Kirchhoff identities, Poisson corrections, and bounded martingale estimates recover occupation and pair currents from finite records. Fluctuation energy has an exact cycle obstruction, while hidden residence mixtures limit identification after state aggregation. Physical-time observations recover visible generators and distinguish event frequencies from residence fractions. Explicit examples separate topology, directional drift, residence, harmonic information, mixing, and irreversible circulation. The basic matrix, probabilistic, and information principles are classical; we give their proofs or precise use within this sequence-operator organization and state the hypotheses needed for each conclusion. Keywords sequence evolution operator; inherited current; finite module; Kirchhoff flow; exit matrix; feedback; boundary transport; entropy; electrical potential.