Structural Principle of Entanglement (PSE)

Version 7.0 – Advanced Mathematical Formalization – The Structural Principle of Entanglement (PSE) is a theoretical framework that reinterprets the foundations of physics by placing entanglement at the origin of all observable structures. Rather than treating physical geometry, matter, and dynamical laws as primitive ingredients, the PSE reconstructs them as emergent phenomena generated by correlations that exist in a non‑observable pregeometric domain. At its core, the PSE posits that the fundamental level of reality contains no space, distance, objects, or fields. It consists solely of correlation structures—patterns of connection without geometric meaning. These correlations form the basis of a unidirectional and non‑invertible ontological chain through which physical structures gradually emerge. Each level introduces new forms of organization: from raw prespatial correlations, to proto‑configurations, to regular domains that can be interpreted in geometric and dynamical terms. Physical quantities arise not because they are fundamental, but because the underlying correlation structure undergoes processes of compression, projection, and regularization. These processes filter and reorganize complexity, producing stable emergent domains where geometry, matter, probability, and dynamics become meaningful. The PSE therefore draws a sharp conceptual boundary between the non‑observable fundamental domain and the emergent domain where physical interpretation becomes possible. Within this framework: geometry emerges from structured correlations, matter emerges from projection degeneracy, probability emerges from the organization of correlation fibers, quantum behavior arises from the structure of emergent states, dynamical laws appear as collective properties of the emergent domain. The PSE thus provides a unified conceptual foundation for interpreting the origin of physical structures, showing how the familiar features of the physical world can arise from deeper, non‑observable entanglement‑based organization. Version 7.0 marks the first fully formalized mathematical formulation of the PSE. Building on the conceptual architecture established in earlier releases, the framework now incorporates a complete advanced mathematical structure that rigorously defines each level of the emergent ontological chain. Entanglement is treated as the generative principle that organizes fundamental correlations and produces, through mathematically controlled transitions, the observable physical configurations. This release introduces a unified formal system based on discrete pregeometric sets, monotonic families of regularization operators, structural distance and correlation functionals, complexity thresholds, and projection maps. These elements are expressed through a coherent mathematical language that specifies the properties, constraints, and transformation rules governing the passage from the non‑observable fundamental domain to emergent geometric and physical structures. The ontological chain is formulated as a non‑invertible sequence of mathematically defined levels, each endowed with precise structural conditions. The emergent domain acquires distance, regularity, and stability through limit processes, while physical quantities arise from structural compression and projection. Metrics, observables, and dynamical laws are not primitive: they emerge from the mathematical organization of correlations and from the degeneracy of projection fibers. The advanced formalization clarifies the role of emergence as a mathematically constrained ontological process. It rigorously distinguishes fundamental structures, emergent configurations, and physical representations, providing explicit criteria for continuity, stability, complexity, and physical interpretability. The framework now includes the full mathematical logic of regularization, the structure of quantum maps, the emergence of probability, the formation of Hilbert spaces, and the derivation of dynamical evolution. Version 7.0 thus delivers a coherent, self‑contained, and mathematically complete theoretical framework. It provides the foundations required for a fully formal interpretation of physical geometry, matter, gravitation, probability, and quantum dynamics as emergent phenomena generated by entanglement‑based structures. Author: Andrea ToricelliIndependent [email protected]