Structural Principle of Entanglement (PSE)

Abstract

Version 6.3 introduces the regularization operator for the emergent domain and provides its full structural formalization, together with the rigorous definition of complexity thresholds that ensure the stability and projectability of the emergent structure. Version 6.2 introduces quantum maps and places quantum theory within the PSE framework through a complete mathematical formalization. The conceptual structure has been clarified, and the release includes a concise analysis of observational data potentially consistent with the emergent structure of the PSE. Version 6.1.1 introduces the explicit treatment of emergent rigidity and the definition of the Φrig map, refining the geometric projection framework and strengthening the internal coherence of the PSE formalization.Version 6.1 of the PSE introduces, relative to version 6.0, a more rigorous definition of the emergent domain X and a complete bibliography in the conceptual framework; in the mathematical formalization it adds the functional spaces of the PSE, the isomorphisms of the Fundamental Domain, the definition of the compressive map, and the diagram of the ontological chain.The 6.0 release of the PSE — Structural Principle of Entanglement represents the stabilized formulation of the theory, following the complete consolidation of its fundamental ontological chain: 𝑺 ⟶ 𝑫𝑺(𝒇) ⟶ 𝑪𝒇 ⟶ 𝑫s(𝒆) ⟶ 𝑪𝒆 ⟶𝒟physical. The transitions between the levels of the ontological chain are carried out by the operators 𝔇, Δ, Fλ, Gλ, and Φ, each with a specific structural role defined in the formalization of the PSE. In this structure, the chain is no longer referenced to the metric as its final object, but to the projection map Φ, which formalizes the transition from the pregeometric emergent domain to the physical domain. All physical quantities — metric, fields, geometric invariants, energetic and dynamical quantities — arise as compressed images of the emergent structure. Φ is not an ontological operator: it expresses the level of emergent structural complexity at which physical quantities become definable. Version 6.0 introduces a significantly expanded conceptual framework, now including: a precise theoretical positioning of the PSE within contemporary ontological approaches to quantum theory, a clarified physical interpretation of emergent quantities and their non‑fundamental status, a more comprehensive general introduction, outlining motivations, structure, and implications of the theory, a refined distinction between emergent structural objects and physical quantities, ensuring that the physical domain is understood as a deterministic projection of the emergent complexity of S, not as an ontological layer. The release definitively consolidates: the general conceptual framework of the PSE and its emergent chain, the rigorous mathematical formalization of the emergent domain, the definition of the fundamental operators 𝔇, Δ, Fλ, Gλ, and Φ, the proof of the functional uniqueness of the projection Φ, the deterministic derivation of the metric and emergent fields as components of Φ, the physical corollaries concerning curvature, derived tensor fields, and emergent stability. The release includes four documents: Conceptual framework (IT) — complete exposition of the theoretical structure, ontological assumptions, emergent chain, theoretical positioning, and physical interpretation. Mathematical formalization (IT) — definitions, operators, Theorems, Corollaries, stability and functional determination. Conceptual framework (EN) — rigorous English version intended for international dissemination. Mathematical formalization (EN) — rigorous English version intended for international dissemination. Version 6.0 is declared conceptually final: the theoretical structure is complete, coherent, and ready for future integration with applied physical models, while remaining open to future development. This work was developed with the support of artificial intelligence. Author: Andrea Toricelliindependent researcherEmail: a.toricelli@hotmail.it Although I am not a physicist by training, I have done my best to present this thesis as clearly and rigorously as possible; my sole intention is to offer a useful contribution to the scientific community.