From Informational Distinguishability to Emergent Geometry: A Geometric Reconstruction within TSI

This article develops the fourth stage of the Theory of Informational Selection (TSI) and investigates the emergence of geometry from stable informational organization. Building upon the algebraic-modular framework established in previous works, the paper introduces a reconstruction program in which informational distinguishability gives rise to a geometry on informational state space through the Bures metric and the associated Quantum Fisher Information. A realization map then transfers this geometry to stable informational modes, allowing the reconstruction of an emergent metric geometry on the physically distinguishable modal organization. The work shows how locality, continuum behavior, effective dimensionality, and large-scale geometric organization can emerge as collective properties of stable informational structures without assuming a pre-existing spacetime background, manifold structure, or predefined dimensionality. A central result is the Fundamental Informational-to-Geometric Reconstruction Theorem, which establishes the mathematical bridge between informational distinguishability and emergent geometric organization. The paper further introduces the notions of Regular Geometric Phase, Geometric Closure, and Post-Geometric Dimensional Selection, providing a systematic framework for understanding how stable geometric regimes may arise from informational foundations. The article completes the geometric stage of the TSI program and lays the foundations for the next phase of the theory, devoted to the reconstruction of geometric dynamics and emergent gravitational behavior.