Emergent Modular Dynamics in Stable Informational Partitions: An Algebraic Reconstruction within TSI

Paper 3 of the Theory of Informational Selection (TSI) program. This work investigates whether Stable Informational Partitions (SIPs), previously introduced as emergent structures arising from informational selection, admit a rigorous mathematical reconstruction. Starting from an explicit correlational representation of stable informational organization, the paper develops a systematic sequence leading from primitive informational transformations to informational *-algebras, informational states, Hilbert-space representations, von Neumann algebras, and ultimately to Tomita-Takesaki modular structures. The reconstruction establishes the chain A central result is the emergence of modular dynamics without the prior introduction of spacetime, Hamiltonian evolution, or external temporal parameters. Within the TSI framework, modular flow constitutes the first mathematically rigorous candidate for an intrinsic notion of evolution arising entirely from informational organization. While the Thermal Time Hypothesis of Connes and Rovelli reconstructs a notion of time from a pre-existing operator-algebraic framework, the present work investigates a deeper question: how the operator-algebraic structures themselves, together with their modular dynamics, may emerge from a more fundamental informational ontology governed by correlational stability and informational selection. The work does not attempt to derive geometry or gravitation. Instead, it establishes the algebraic and modular foundations required for subsequent stages of the TSI program, providing a mathematical bridge between informational ontology and future informational-geometric reconstruction.