Anchored Accumulation Calculus: Mellin Diagonalization and Operator-Theoretic Classification of Scale-Covariant Memory

Abstract

We formalize Anchored Accumulation Calculus (AAC), an operator-theoretic framework for scale-covariant memory on the multiplicative group R+. By defining memory as causal accumulation along exponential scale trajectories, we establish that AAC operators on a dense core are unitarily equivalent to translation-invariant Fourier multipliers. This construction allows us to define the maximal closed extension of these operators, avoiding ad hoc integration-by-parts arguments. We subsequently classify their generators: completely monotone memory kernels are rigorously shown to generate scale-invariant L´evy subordinators via Bernstein’s theorem, while heavy-tailed kernels are unitarily equivalent to Mellinbased Hadamard fractional derivatives. Finally, to demonstrate that AAC is not merely a reformulation of Markovian semigroups, we analyze oscillatory memory kernels, proving they generate dispersive, non-dissipative unitary groups on the scale domain.

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