Anchored Accumulation Calculus: Grammar-Relative Memory Representations and Residual Constraint Closure across Differential and Functional Operators

Abstract

Anchored Accumulation Calculus (AAC) provides an exact discrete-time realization of a class of non-Markovian memory operators by augmenting the dynamics with anchored exponential modes. This paper defines that realization precisely and develops a grammar-relative extension in which the central question is whether an accumulated output is already an admissible observable of a given recursive state. The corresponding Residual Constraint Closure (RCC) test is a cohomological equation in a declared observable algebra. For linear differential states, RCC reduces to a rational row equation; in a minimal scalar differential module, this yields an exact dichotomy between preservation of differential order and an increase by one for a primitive. We construct finite projective states for logarithms of solutions of finite-order linear differential equations and prove a differential-algebraicity equivalence for polynomial-logarithm primitives. For polynomial logarithmic sources, we identify the scalar differential order with the dimension of the apolar derivative space under a full-rank monodromy hypothesis. Our principal algorithmic theorem gives a total-degree bound of one more than the source degree and a finite exact residue-based decision procedure for polynomial-logarithmic RCC under an independent-divisor hypothesis. Exact Gamma and q-Pochhammer examples illustrate how shift and dilation grammars can close accumulations that are not closed in a smaller differential grammar. A Bessel canonical-product example connects the framework with classical Rayleigh sums. The paper distinguishes these specific results from the established theories on which they rely; it does not claim that differential algebra, realization theory, apolarity, or rational summability is new.

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