This paper develops a mathematically rigorous, philosophically grounded framework for evaluating artificial memory systems, rooted in the metaphysical structure of Leibniz's Monadology. Building on a previously formalized metric, the Artificial Age Score (AAS), the study maps twenty core propositions from the Monadology to an information-theoretic architecture. In this design, each monad functions as a modular unit defined by a truth score, a redundancy parameter, and a weighted contribution to a global memory penalty function. Smooth logarithmic transformations operationalize these quantities and yield interpretable, bounded metrics for memory aging, representational stability, and salience. Classical metaphysical notions of perception, apperception, and appetition are reformulated as entropy, gradient dynamics, and internal representation fidelity. Logical principles, including the laws of non-contradiction and sufficient reason, are encoded as regularization constraints guiding memory evolution. A central contribution is a set of first principles proofs establishing refinement invariance, structural decomposability, and monotonicity under scale transformation, aligned with the metaphysical structure of monads. The framework's formal organization is structured into six thematic bundles derived from Monadology, aligning each mathematical proof with its corresponding philosophical domain. Beyond evaluation, the framework offers a principled blueprint for building Al memory architectures that are modular, interpretable, and provably sound.