Logic, Information, Holography: A Recursive Unification of Complexity Across Computational, Physical, and Geometric Substrates

This paper proposes a three-layer recursive framework that unifies three complexity measurespreviously confined to separate disciplines: logical depth from algorithmic information theory,information density and thermodynamic dissipation bounds from physical information theory, andentanglement-geometric encoding from holographic gravity. We argue that these are not parallelindependent metrics but concrete manifestations of the same irreducible generative cost on threedifferent substrates: computational, physical, and geometric. The paper reviews Bennett'sdefinition and philosophical motivation for logical depth, the information-physical bounds set byLandauer's principle and the Bekenstein bound, and the spacetime-geometric emergence revealedby the Ryu-Takayanagi formula and the first law of entanglement entropy. It then proposes amapping chain from logical depth to information density to holographic encoding, introduces theconcept of metric involution to describe the resulting evolutionary direction of complex systems,and discusses observational criteria by which the framework could be tested against the traditional,energy-throughput-based paradigm of system complexity.