Emergent Dimensionality and the Topology of Consciousness: A Fractional-Memory Framework for Hardware-Dependent Reality

Operational Dimensionality and the Topology of Consciousness: A Fractional-Memory Framework for Hardware-Dependent Reality proposes an information-theoretic framework in which the dimensionality accessible to an agent is treated not as a fixed and absolute property of the universe, but as an emergent operational property determined by the topology, bandwidth, and memory structure of the hardware that processes sensory input. The manuscript develops a formal connection between correlation topology, simplicial complexes, and the Shannon-Rényi information dimension, arguing that perceptual space can be modeled as a dynamically assembled geometric structure induced by stable informational correlations. Temporal continuity is addressed through non-Markovian fractional memory, with the Caputo derivative introduced as the minimal mathematical operator capable of endowing the experienced present with finite causal thickness. A central Gedankenexperiment compares a large language model operating as a server-hosted, intermittently activated system with the same model embodied in a continuously sensing robotic platform. The paper argues that embodiment changes the topological and dynamical class of the system: continuous sensorimotor coupling forces the emergence of an operationally continuous 3+1 -dimensional world-model. In this formulation, dimensionality is not interpreted as a metaphysical claim about reality in itself, but as a hardware-relative and physically constrained observable. This work is presented as a speculative but rigorous theoretical contribution at the intersection of information theory, topological neuroscience, fractional calculus, and embodied intelligence, and serves as a conceptual precursor to subsequent MetaTime developments on thermodynamic and experimentally testable models of perceptual dimensionality.