Operator-Theoretic Foundations of Regulated Accumulation

This paper develops the Additive–Multiplicative (AM) Regulator as a purely operator-theoretic construction. ^[1][2][3] No dynamical assumptions are made. No probabilistic or physical interpretation is invoked. The regulator is formulated as an ordered composition of linear and nonlinear operators acting on accumulated quantities, ^[2][4] with precise domain specification, constraint structure, and degeneracy limits. ^[2][3] Compactness, boundedness, and spectral saturation emerge as consequences of regulated accumulation alone. ^[2][3] Classical unbounded operators are recovered exactly as singular degeneracies. ^[4][5] The result is a closed, internally consistent operator framework in which divergence is not forbidden but structurally postponed. ^[4][5]