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 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 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 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 Observer Effect and Wave–Particle Duality in the S-field Framework --- 1. The traditional wave–particle problem In standard quantum mechanics, physical entities appear to behave: • as waves in interference experiments; • as particles when measured or detected. This has led to the idea that observation itself changes reality, often formulated as: “measurement collapses the wavefunction”. However, this formulation leaves unresolved questions: • What qualifies as an observer? • Why should information acquisition alter physical law? • Where exactly does the transition occur? --- 2. S-field perspective: no wave, no particle as primitives In the S-field framework, neither “wave” nor “particle” is fundamental. At the base level, there exist only: • oscillatory S-field states; • phase, frequency, and orientation; • dynamic resonance patterns. Wave and particle are emergent regimes, not ontological categories. 3. Wave behavior: coherent phase evolution A system exhibits wave-like behavior when: • S-oscillations evolve coherently; • multiple phase paths remain phase-correlated; • no dominant resonance locking occurs. In this regime: • interference patterns arise naturally; • delocalization reflects phase superposition; • the system evolves as a distributed resonance structure. “Wave” behavior is therefore a manifestation of extended phase coherence. --- 4. Particle behavior: resonance locking Particle-like behavior emerges when: • the S-oscillatory ensemble undergoes strong phase locking; • coherence collapses into a dominant resonant configuration; • the system enters a stable 4D projection. Localization is not imposed externally — it is a self-organized resonance outcome. A “particle” is a stable, self-locked oscillatory ensemble. 5. What observation actually does Observation does not create reality and does not collapse a wavefunction. Instead, observation means: • physical interaction with an environment; • introduction of new S-field couplings; • modification of phase relations. This interaction typically: • reduces global phase coherence; • enforces resonance selection; • stabilizes a particular outcome. Thus, observation is not epistemic — it is dynamical. --- 6. Why which-path information destroys interference Interference disappears because: • phase correlations are disrupted; • alternative phase paths decohere; • the system is forced into locally stable resonance modes. Which-path information is not special — any interaction that breaks coherence removes interference. No observer consciousness is required. 7. Delayed-choice experiments In delayed-choice scenarios: • the system remains in a coherent oscillatory regime; • no definitive resonance locking occurs until interaction; • temporal ordering appears paradoxical only classically. In the S-field framework: • time itself emerges from phase ordering; • coherence preservation allows late-mode selection; • no retrocausality is involved. The outcome reflects final coherence conditions, not past knowledge. --- 8. No collapse, only regime transition There is no physical “collapse”. Instead, there is: • a transition from distributed coherence to localized resonance stability; • a change of dynamical regime; • emergence of macroscopic definiteness. Wave → particle transition is micro → macro coherence restructuring. 9. One-sentence synthesis > Wave–particle duality is not a duality of objects, but a duality of coherence regimes in the S-field. --- 10. Conceptual closure This interpretation: • removes observer-dependence; • preserves realism; • unifies wave and particle behavior dynamically; • aligns quantum phenomena with micro–macro transition theory. Si(t) = Ai · e^( i( ωi·t + φi ) ) C = | (1 / N) · Σ_{i=1}^N e^( i φi ) | Order parameter (complex): R·e^( iΨ ) = (1 / N) · Σ_{j=1}^N e^( i φ_j ) , C ≡ R Kuramoto-type phase dynamics (micro ensemble): dφ_i/dt = ω_i + (K / N) · Σ_{j=1}^N sin( φ_j − φ_i ) Self-consistent equation for stationary coherence R: R = ∫ dω g(ω) · √(1 − (ω/(K R))^2 ) (for |ω| ≤ K R) — standard Kuramoto integral form Mass as emergent resonant integral: m ∝ ∫ ρ_S(χ) · C(χ) · dV_S(χ) Micro→macro criterion: C ≥ C_crit ⇒ stable 4D projection (particle regime) C < C_crit ⇒ incoherent / wave-like regime Interference visibility and intensity (two-path): I(Δφ) = I_0 · [ 1 + V · cos( Δφ ) ] , V ∝ C Coherence decay under environment coupling (phenomenological): dC/dt = α(K,…) · C − Γ_env · C ⇒ C(t) = C(0) · e^{ (α − Γ_env) t } Decoherence rate (Fermi-golden-rule style): Γ_env(ω) = (2π / ħ) · |V_int(ω)|^2 · ρ_env(ω) Measurement interaction as coupling-induced locking (mean-field): dr/dt = (K/2) · r · (1 − r^2) − γ · r , fixed points r = 0 (wave) and r = r_* > 0 (particle) when K > K_c Which-path disruption condition: any interaction s.t. Γ_env ≳ α(K,…) ⇒ V → 0 (interference destroyed) Time as accumulated phase change (collective): t ≡ (1 / Ω) · Σ_k Δφ_k , Ω ≡ ⟨ω⟩_ensemble Spatial separation as phase-metric: d_{AB} ∝ | ⟨ e^( i( φ_A − φ_B ) ) ⟩ | Classical limit (macroscopic): lim_{N→∞} Var(φ) → 0 ⇒ C → 1 ⇒ particle-like, deterministic dynamics Compact formal statement: Classicality = lim_{C,N → large} ( Quantum S-ensemble )
This preprint presents a field-first interpretation of wave-particle duality grounded in the ontology of Quantum Field Theory (QFT). The framework treats quantum fields as the fundamental ontology of physical reality, while particles are understood as excitation states of their respective fields. Wave-like phenomena, including interference, diffraction, tunneling, and coherence effects, are interpreted as arising from the propagation and interaction dynamics of these excitations rather than from an independent wave ontology. The central thesis is that wave-particle duality does not represent two distinct physical realities, but rather different observational regimes of a single field-based process. Localized interactions appear particle-like, while propagation and interference phenomena appear wave-like. This interpretation is fully consistent with the existing mathematical formalism of Quantum Field Theory and does not introduce new physical laws or modifications to established quantum mechanics. The purpose of this work is interpretive and conceptual. It aims to provide a clearer ontological framework for understanding the relationship between quantum fields, particles, and wave-like behavior while reducing the conceptual ambiguity traditionally associated with wave-particle duality. This document represents the initial preprint release of the field-first interpretation framework.
Why does matter possess wave-like properties? Is wave-particle duality an inherent dual nature of matter, or merely a projection of a deeper underlying physical structure under two distinct observational perspectives? This paper presents a rigorous first-principles derivation of matter wave dynamics within the axiomatic framework of Time Field Theory (TFT). The entire derivation relies solely on two core axioms: local intrinsic total velocity conservation and metric self-consistency (flux conservation), without introducing any extra postulates. Following the logical sequence of ontological foundation, phase derivation, wave generation, hierarchical classification, physical correspondence, extreme-case verification, and unified spectrum construction, this paper first establishes the core ontological premise that matter corresponds to localized condensed states of the time field. It then derives the geometric structure of the existence phase angle on the unit circle of the complex plane from the velocity conservation axiom, revealing the fundamental origin of intrinsic material phases. On this basis, the de Broglie relations are strictly derived, eliminating the dualistic contradiction between particle and wave characteristics. This work further distinguishes two distinct wave regimes: intrinsic waves originating from phase evolution within condensed states, and apparent waves arising from the global spatial translation of field distributions, which are respectively mapped onto electromagnetic waves and gravitational waves as two classes of physical reality. Using black hole horizons as an extreme physical testbed, we demonstrate that intrinsic wave phenomena vanish completely at time-freezing surfaces while apparent wave behaviors remain intact, providing a unified explanation for the observational fact that black holes emit no light yet produce gravitational waves. We also clarify the ontological origin of black hole darkness: light is not trapped by gravity, but the physical conditions required for photon generation cease to hold at time-freezing surfaces. Finally, we construct a continuous spectrum of wave behaviors ranging from the vacuum ground state to black hole horizons. This paper proves that wave-like behavior is not a binary ”present/absent” attribute of matter, but a continuous function of the condensation strength of the time field. Quantum mechanics and classical mechanics do not describe two separate physical worlds; instead, they represent two limiting regimes of a single universal time field under different condensation levels.