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Foundational Definitions of the Framework

Andrea Toricelli

Published May 21, 2026• 0 citations
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Abstract

This document is obsolete; its content has been fully integrated into the consolidated PSE manuscript. This document presents the foundational definitions of the Principle of Emerging Structure (PSE), a theoretical framework in which physical reality arises from a prespatial network of correlations governed by two fundamental principles: evolutionary coherence (E) and structural coherence (C).Within this framework, space and time are not primitive entities but emergent properties derived from the structural relations within the network. Particles are defined as stable, localized, and quantized configurations of coherence—stationary modes of the underlying structure—while fields correspond to diffuse, non‑localized modes of the same dynamics.Interactions are described as structural reorganizations that modify the stability of these modes, eliminating the need for fundamental forces. The wave–particle duality is resolved as an artifact of the emergent description: wave‑like and particle‑like behaviors correspond to different regimes of coherence. Measurement is interpreted as a structural reconfiguration induced by interaction, without invoking collapse or observer‑dependent mechanisms.The uncertainty principle emerges naturally from the impossibility of stabilizing incompatible structural modes simultaneously, providing a geometric rather than epistemic interpretation. Additional concepts such as decoherence and emergent gravity are defined in terms of changes in structural coherence and curvature of structural distance.This document establishes the conceptual and ontological basis of the PSE and serves as a reference for the other works in the series. This work complements the previous document published on Zenodo with DOI: https://doi.org/10.5281/zenodo.20218975. Andrea Toricelli - a.toricelli@hotmail.it

DOI
10.5281/zenodo.20331786

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This document is obsolete; its content has been fully integrated into the consolidated PSE manuscript. This document presents the foundational definitions of the Principle of Emerging Structure (PSE), a theoretical framework in which physical reality arises from a prespatial network of correlations governed by two fundamental principles: evolutionary coherence (E) and structural coherence (C).Within this framework, space and time are not primitive entities but emergent properties derived from the structural relations within the network. Particles are defined as stable, localized, and quantized configurations of coherence—stationary modes of the underlying structure—while fields correspond to diffuse, non‑localized modes of the same dynamics.Interactions are described as structural reorganizations that modify the stability of these modes, eliminating the need for fundamental forces. The wave–particle duality is resolved as an artifact of the emergent description: wave‑like and particle‑like behaviors correspond to different regimes of coherence. Measurement is interpreted as a structural reconfiguration induced by interaction, without invoking collapse or observer‑dependent mechanisms.The uncertainty principle emerges naturally from the impossibility of stabilizing incompatible structural modes simultaneously, providing a geometric rather than epistemic interpretation. Additional concepts such as decoherence and emergent gravity are defined in terms of changes in structural coherence and curvature of structural distance.This document establishes the conceptual and ontological basis of the PSE and serves as a reference for the other works in the series. This work complements the previous document published on Zenodo with DOI: https://doi.org/10.5281/zenodo.20218975. Andrea Toricelli - a.toricelli@hotmail.it
The PSE framework develops a fully structural description of physical phenomena, where space‑time, fields, particles and constants arise as geometric projections of a deeper prespatial correlation network. The theory replaces ontological entities with structural invariants defined by a fundamental reorganization operator 𝔇, which selects admissible modes of coherence through discrete internal operators (rigidity, symmetry, internal transformations). In the PSE, space‑time is not fundamental: it emerges from structural distance via the map Φ, compressing and projecting prespatial geometry. Coherence configurations generate both localized modes (particles) and diffuse modes (fields), while interactions correspond to structural variations of internal invariants rather than dynamical forces. Key physical quantities—mass, energy, spin, charge, c, ℏ, probability—are reinterpreted as emergent geometric effects: mass as minimum rigidity, energy as total rigidity, c as the upper bound of admissible structural variation, ℏ as minimum granularity of admissible modes, probability as structural compatibility between system and detector, entanglement as structural co‑immersion, decoherence as loss of compatibility. The PSE provides a unified structural explanation for quantum statistics, uncertainty, interactions, causal structure, and emergent constants. It also clarifies the limits of the emergent description: space‑time cannot represent co‑immersion, internal operators, structural compatibility, or prespatial geometry. These appear only indirectly as amplitudes, probabilities, apparent collapse, non‑locality and dynamical interactions. This document presents the complete A–P formulation of the PSE, detailing the structural foundations, emergent phenomena, and the geometric origin of the laws of physics. Andrea ToricelliWith the help of Artificial Intelligence systemsa.toricelli@Hotmail.it
This work presents the formulation of wave–particle dualism within the framework of the Structural Principle of Entanglement (PSE), a pregeometric emergentist theory in which dualism is not a postulate of Quantum Mechanics, but a structural transition determined by the emergent correlation density ρ(x)=∫XCe(x,y) dμ(y). The ontological chain S ⟶ DS(f) ⟶ Cf ⟶ DS(e) ⟶ Ce ⟶ ρ(x) ⟶ Φ ⟶ physical domain shows how all physical quantities (metric, fields, dynamics, observables) are compressive images of the emergent structure (DS(e),Ce). The document demonstrates the existence of a local critical threshold ρc(x)∈(0,ρmax⁡), which separates two emergent regimes: Wave regime: ρ(x)<ρc(x) → coherence, diffusion, interference, absence of emergent mass. Particle regime: ρ(x)>ρc(x) → geometric compression, localization, emergent mass, decoherence. Wave–particle dualism therefore results from an emergent phase transition—continuous, monotonic, and anisotropic—fully determined bythe fundamental structure F=(S,DS(f),Cf). The PSE also provides testable physical predictions: an upper limit for interference in complex systems, emergent mass as a monotonic function of ρ, local structural decoherence, induced transitions, a correlational interpretation of dark matter, absence of metric singularities, and geometric–quantum unification. The document includes technical appendices with formal proofs, explicit structural examples, and possible experimental tests. Andrea Toricellia.toricelli@hotmail.it
This work presents a structural interpretation of the reduced Planck constant ℏ within the framework of the Structural Principle of Entanglement (PSE). In this theory, ℏ is not a fundamental physical parameter nor a quantum‑mechanical postulate. Instead, it emerges as the compressed image of a discrete action threshold selected by the regularized emergent structure. The PSE establishes a unidirectional ontological hierarchy: a prespatial structure without geometry, a fundamental reorganization governed by a coherence threshold, an emergent correlation that organizes continuity, a regularization process that filters coherent configurations, and a set of complexity thresholds that preserve discrete structural rigidity. Together, these elements produce a minimum action threshold that is structural rather than physical. Through a compressive and non‑invertible projection, this structural rigidity is formalized in the physical domain as the reduced Planck constant ℏ. As a result, the quantization of action, canonical non‑commutativity, precision limits, and the structure of the quantum state space are not primitive features of nature. They are consequences of the compressive projection and of the discrete rigidity selected in the emergent domain. Quantum physics therefore appears not as a set of fundamental axioms, but as an effective representation of a deeper emergent structure. In this view, ℏ is an ontological property of the PSE hierarchy: the physical signature of a discrete threshold rooted in the prespatial domain. Observable physical reality is a compressed representation of this emergent structure, and ℏ is its most fundamental imprint.
This work introduces the Generative Principle of the Universal Correlational Lattice (UCL), a pre-geometric theoretical framework in which physical reality is described as an evolving network of correlational sites, rather than as fields defined on a pre-existing spacetime background. In the UCL framework, spacetime, dynamics, quantum uncertainty, and effective physical observables emerge from structured correlations encoded in a universal kernel. The theory does not postulate spacetime, wavefunctions, or classical fields as fundamental objects. Instead, it treats them as effective constructs arising from coarse-grained correlational dynamics. A central element of the framework is the introduction of internal times associated with correlational sites, whose relational synchronization generates an emergent external time only at the effective level. Temporal evolution is therefore defined intrinsically, without requiring a fundamental global clock or background metric. The Generative Principle specifies: how admissible correlational configurations are formed, how dynamical evolution arises as a one-parameter flow in correlational space, and how stable, observable regimes emerge through functional selection and decoherence-like processes intrinsic to the lattice. Importantly, the framework provides a well-defined notion of observables as emergent, operational quantities derived from correlational averages, spectral properties of the kernel, and effective coarse-graining procedures. These observables are not postulated ad hoc, but arise naturally from the internal structure of the lattice and its generative dynamics. Quantum uncertainty is interpreted as emergent and contextual, reflecting partial access to the full correlational state rather than a fundamental indeterminism. In principle, the framework allows deterministic underlying dynamics while reproducing effective probabilistic behavior at observable scales. The UCL is constructed to be compatible with known physical theories in appropriate limits, while remaining ontologically independent from both classical spacetime and traditional quantum mechanical foundations. It provides a unified conceptual basis for addressing quantum theory, gravity, and cosmology within a single correlational framework. This paper focuses on the foundational and generative aspects of the UCL. Specific phenomenological models and numerical predictions are deferred to subsequent works, where concrete kernels and parameter regimes are explored.
Abstract Despite its empirical success, quantum mechanics continues to provoke foundational debates concerning the nature of probability, the ontological status of the quantum state, and the mechanism of measurement. [1] This work introduces Distributed Presence (DP) as a structural ontological framework that addresses these issues by reinterpreting quantum phenomena in terms of a system’s distributed mode of existence across its state space. Crucially, this distribution is modal rather than spatial: it concerns how a system is present across possible states, not how it is extended in physical space. Within the DP framework, a quantum system is not assumed to occupy a definite state prior to interaction. Instead, its fundamental mode of existence is an ontologically distributed presence, understood as a structural feature of reality rather than a reflection of epistemic uncertainty. From this standpoint, probability is not introduced as a primitive postulate but emerges operationally from the geometry of presence, yielding a structural derivation of the Born rule. Core quantum phenomena (including superposition, wave–particle duality, state reduction, and entanglement) are recast as different manifestations of a single underlying principle: the modal distribution, reconfiguration, and non‑separability of presence. Entanglement corresponds to the non‑factorizability of presence fractions over a composite state space, while non‑local correlations are interpreted as consequences of shared modal structure rather than dynamical signal transmission, fully consistent with relativistic causality. The framework is developed within a Euclidean geometric setting chosen to prioritize ontological clarity, without altering the predictive formalism of standard quantum mechanics or its Hilbert‑space representation. By grounding randomness, probability, and quantum behavior in the notion of modal distributed presence, DP offers a coherent foundational program that bridges the conceptual gap between unitary evolution and definite outcomes, providing an alternative to both epistemic interpretations and wave‑function realism.
This document extends Relational Quantum Theory (RTK) to address a foundational gap in physics: the ontological status of fields. Standard physics describes electromagnetic, gravitational, and gauge fields as fundamental entities pervading spacetime, yet provides no explanation for what fields are or why space possesses the capacity to host them. Central Thesis: Physical fields emerge as propagating coherence patterns in the underlying relational network. They are not substances filling space, nor mysterious influences acting at a distance, but coordinated modulations of the relationships that constitute reality at the deepest level. Key Contributions: Field ontology resolved: Fields are patterns in relations, not separate fundamental entities Finite propagation explained: Speed of light c emerges from network locality—the maximum rate at which coherence patterns can propagate through local relational structure Virtual particles demystified: Virtual particles are transient coherence patterns in extended algebraic space (ℍ, 𝕆) that don't reach stable ℂ configuration Gauge groups derived: U(1), SU(2), SU(3) emerge as automorphism groups of division algebras ℂ, ℍ, 𝕆 Entanglement vs. fields distinguished: Entanglement = fundamental graph edges (ignores emergent distance); Fields = emergent coherence patterns (respects emergent distance) Gravity's special status: Gravity is network geometry itself, not a pattern within the network—explaining its universality and the difficulty of quantization Vacuum structure: Relational sea of unstable relations; vacuum energy cancels between U₊ and U₋ sectors Hierarchy of Emergence: Level Structure Physics Fundamental Relations — Emergent 1 Network Quantum correlations Emergent 2 Coherence patterns Fields Emergent 3 Stable ℂ clusters Particles Emergent 4 Network geometry Spacetime + gravity Epistemological Status: This document presents PROPOSITIONS extending the core RTK axioms. It is compatible with the main RTK framework but addresses questions not directly covered in the main document.