Foundations of the AM-Regulator
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This paper develops the foundational mathematical structure underlying the Additive–Multiplicative (AM) Regulator as a constrained operator system acting on accumulated quantities. [1][2] Treating the regulator axioms, operators, and degeneracy limits as fixed, the author investigates the deeper properties required for internal mathematical completeness: topology, convergence, duality, compactness, and categorical structure. [2][3] Classical functional spaces and analytical frameworks arise only as degenerate limits under vanishing constraint. [1][3] An explicit and exact correspondence is established with the Ramanujan-refined spectral–geometric AM-regulator, demonstrating that refined summability and spectral damping are intrinsic consequences of the same constrained accumulation principle. [6][7][8] The result is a foundations-level formalism in which divergence, completeness, and applicability are regulated structurally rather than repaired post hoc. [6][10]