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 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.
The quantization of action, discrete energy levels, and finite phase-space resolution are among the most fundamental yet conceptually opaque features of quantum physics. In standard theory, the universal constant is introduced empirically, and canonical quantization rules are postulated without deeper ontological justification. In this work we present a first-principles derivation of quantization based on the Structural Unified Field (SUF) framework. By treating physical entities as layered “existence objects” characterized by a boundary parameter and an existence function , we argue that structural identity requires a minimal nonzero layer separation . This structural irreducibility implies a universal minimal distinguishability in all attribute domains, which, when projected into phase space, yields a minimal phase-space cell and thus a universal minimal action unit. Upon mapping to physical units, this structural minimal action corresponds to Planck’s constant . The SUF interpretation thereby promotes quantization from an empirical postulate to a necessary consequence of the ontological geometry of existence. The resulting framework naturally accounts for discrete spectra, uncertainty relations, action quantization, canonical commutation relations, topological quantization phenomena, and the universality of . We discuss broader implications for the foundations of quantum theory, the quantum–gravity interface, and potential avenues for testing structural deviations.
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
Quantum mechanics provides an operationally complete and extraordinarily accurate description of microscopic phenomena, yet it remains deliberately silent about the physical origin of its own structural features. Discreteness, operator formalisms, superposition, interference, and probabilistic outcomes are treated as axiomatic, not explained. This paper interprets that silence not as a limitation, but as evidence that quantum mechanics operates within a specific physical regime rather than at the level of fundamental ontology. Within the Breathing Universe Model (BUM), the vacuum is the sole physical substrate and orders itself through threshold redistribution (“snapping”), coherence selection, and rhythmic stabilization governed by Omega as ordering and memory rather than energy or force. When a continuous vacuum supports nested snapping, operator-constrained transformations, stable coherence knots, and persistent phase alignment, a quantum-like structural description becomes unavoidable. Discrete states arise as survivors of threshold ordering, operator formalisms encode transformation constraints, superposition reflects overlapping admissible coherence configurations prior to re-alignment, and probabilistic outcomes summarize coherence selection under inaccessible micro-phase detail. This paper does not derive, modify, or replace quantum mechanics, nor does it treat wavefunctions, Hilbert space, or measurement postulates as ontological primitives. Instead, it demonstrates why quantum structure must emerge as an effective coherence regime once the vacuum is ordered according to the BUM canon. Quantum mechanics is thus framed as operationally complete but ontologically downstream: it faithfully reports what structure must look like once coherence exists, without explaining why such structure must arise. By grounding quantum structure in vacuum ordering, coherence memory, and constraint-based transformation, this work establishes the ontological handoff to quantum correspondence and the emergence of information, where quantum informational behavior is interpreted as physical persistence of coherence relations in the vacuum.
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
Quantum mechanics is widely regarded as a fundamental theory of physical reality. In this work, we demonstrate that this status is not supported at the structural level. The quantum formalism is shown to be neither closed nor generative, relying on externally imposed elements such as probabilistic rules and measurement prescriptions. We introduce a closed variational framework based on a quartic functional, from which admissible configurations emerge through explicit variation. The second variation defines a Hessian operator whose spectral structure exhibits intrinsic discreteness and degeneracy, without the introduction of quantization postulates. We prove that apparent probabilistic behavior arises from structural degeneracy and indistinguishability of admissible configurations, rather than from intrinsic indeterminism. The role of wavefunction collapse is replaced by a global deterministic selection functional applied to the set of stationary solutions. Within this framework, quantum states, probabilities, and observables are reinterpreted as effective representations of an underlying deterministic structure. Consequently, quantum mechanics does not constitute a fundamental description of physical reality, but a statistical projection of unresolved structural configurations.