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Born Rule as a Dynamical Necessity: Phase-Density Formalism and Topological Quantization

Souhaib zayed Awad alkarim alnur صهيب زائد عوض الكريم النور

Published Feb 10, 2026• 0 citations
Publisher

Abstract

This paper presents a derivation of the Born rule from first principles using a phase-density formalism. We show that the probability density is necessarily the square of the wavefunction amplitude due to linearity, local conservation, and independence from absolute phase. The phase is naturally defined on a compact space S^1, leading to topological quantization and explaining the appearance of the quantum of action ħ. This approach unifies the origin of quantum probability and phase quantization from a deterministic, geometric perspective.

DOI
10.5281/zenodo.18598533

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Quantum Mechanics as Topological Intersection Theory: The Born Rule, Wavefunction Collapse, and Planck's Constant from Worldline Non-Injectivity

Alex De Giuseppe

New edited version Title: Quantum Mechanics as Topological Intersection Theory: The Born Rule, Wavefunction Collapse, and Planck’s Constant from Worldline Non-Injectivity Abstract: This work presents a comprehensive derivation of the foundational principles of quantum mechanics by shifting the paradigm from stochastic postulates to a deterministic geometric framework. Utilizing the TPST-DGQ (Topological Phase Signalling Theorem - Discrete Geometric Quantization) framework, the paper demonstrates that the core axioms of quantum theory emerge as necessary theorems from the geometry of non-injective worldlines. Key Mathematical and Physical Derivations: Geometric Origin of the Born Rule: The paper proves that the probability density $|\psi(x,t)|^2$ is a "topological intersection density." By analyzing a single ultra-relativistic worldline $X^\mu(\tau)$ that intersects the constant-time hypersurface $\Sigma_t$ at $N(\epsilon)$ distinct points, the Born rule is derived as the normalized frequency of these intersections. This replaces the epistemic interpretation of probability with a frequentist geometric count. The Wavefunction as a Transport Representation: The wavefunction $\psi(x,t)$ is defined as the complex square root of the intersection density. Its phase is shown to be the result of proper-time transport along the continuous segments of the worldline connecting different intersection points. Emergence of the Schrödinger Equation: The manuscript derives the Schrödinger equation as the fundamental transport law for the intersection density. This derivation is shown to be valid under the condition of "inter-sheet phase coherence," a state forced by the Ontological Identity Principle and the finiteness of the action. Fundamental Nature of Planck’s Constant ($\hbar$): Instead of a fundamental constant of nature, $\hbar$ is derived as the minimum action per radian of phase accumulated between stable folds of the worldline. It is functionally linked to the UV cutoff $\epsilon$ and the rest mass $m$, providing a parameter-free basis for the quantum of action. Deterministic Wavefunction Collapse: Collapse is redefined as a topological transition from a non-injective state ($N > 1$) to an injective state ($N = 1$). This occurs through a physical interaction (measurement) that reduces the Lorentz factor below a critical threshold $\gamma_{crit}$, providing a dynamical solution to the measurement problem. Unified Regularization: The paper establishes the "Universal Topological Cancellation," showing that the same scaling mechanism $N(\epsilon) \sim \epsilon^{-(d-2)}$ that regularizes the Ryu-Takayanagi entropy in holography and the Coulomb self-energy in classical electrodynamics also ensures the UV-finiteness of the quantum intersection density. This research unifies quantum mechanics, classical electrodynamics, and holographic gravity under a single topological principle: worldline non-injectivity as a necessary and sufficient condition for finite physics at every level. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created

•Jan 16, 2026•0 citations

The Wave Function as an Energy-Quantum Density Amplitude

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The mathematical and physical foundations of the Born rule remain open questions, largely because the wave function lacks a mathematical identity and a physical origin. Here we uncover a mathematical structure of the wave function by treating the phase as a circular variable. This structure reveals that the wave function is the pullback of the phase probability-density amplitude to position space via the phase field. Because the phase probability distribution forms an equal-area contour in the complex plane, its probability-area element yields the modulus squared as the position probability density, revealing the Born rule as a geometric relation rooted in the circular topology of the phase variable. The same structure governs the phase-energy distribution of electromagnetic waves. By quantizing phase energy into energy quanta, in the single-quantum limit the energy-quantum density reduces to the probability density, and its amplitude becomes the probability-density amplitude. This therefore identifies the energy-quantum density amplitude in the single-quantum limit as the physical origin of the wave function. To verify this, we derive the time-evolution equation for the energy-quantum density amplitude from five physical principles. This yields the Gross-Pitaevskii equation for many-body systems and, in the single-quantum limit, the Schrödinger equation. The framework also recovers Maxwell's equations in the classical limit. Overall, this study introduces the energy-quantum density amplitude from the classical phase-energy distribution, and derives the three core postulates of quantum mechanics: the wave function, the Born rule, and the Schrödinger equation, thereby identifying the position probability density as the energy-quantum density in the single-quantum limit.

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AbstractThe Born rule—assigning quantum probabilities as —remains a postulate in standard quantummechanics. Despite numerous attempts (Gleason's theorem, decoherence, envariance, decision theory), noderivation has eliminated the need for an additional probabilistic axiom. This paper presents a geometric derivationof the Born rule from three minimal principles: (1) physical states are complex linear fields, (2) reality includes anunobservable scale dimension , and (3) observers are restricted to fixed-scale hypersurfaces. Under theseconditions, the unique operational density consistent with linearity, conservation, and phase invariance is thequadratic form . Marginalizing over unobservable scale degrees of freedom yields . Under scale-localization or scale-decoherence, this reduces to , reproducing the Born rulewithout assumption. If the ultraviolet limit of the manifold is a compact genus- surface, topological constraintsproduce structured statistical signatures: for , a 138-mode periodic correction emerges, providing afalsifiable experimental prediction. Measurement is reinterpreted as dimensional projection, not wavefunctioncollapse. Quantum probabilities arise as geometric shadows of deterministic higher-dimensional dynamics.Keywords: Born rule, quantum probability, dimensional flow, marginalization, scale space, genus, quantumfoundations, measurement problem Full Theory Portfolio at: https://drive.google.com/drive/folders/1fdKdo3edGqXVx95IntIumXlzKq22s-yw?usp=drive_link

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The Geometric Mandate of Local Continuity: Deriving the Deterministic Foundation of Quantum Fields

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The probabilistic interpretation of quantum mechanics rests fundamentally on the Born rule, which asserts that the physical density of a state is given by the squared magnitude of its wavefunction (ρ=|ψ|^2). Historically, this exponent of (α=2) has been treated as an unprovable, axiomatic postulate justified solely by empirical alignment. This paper presents a rigorous, parameter-agnostic derivation that turns this narrative on its head. By introducing ( α) as a completely free geometric parameter ( ρ=|ψ|^α) and demanding strict adherence to the universal principle of Local Continuity: ∂ρ/∂t+∇⋅J ⃗=0 I show that the algebra itself systematically eliminates all alternative mappings. If ( α≠2), the local density variation inherently picks up non-local dependencies on the absolute background field amplitude, violating localized field interactions. The mathematics strictly demands ( α=2) as a unique geometric threshold required to isolate unobservable phase variables and preserve local conservation laws. Consequently, I demonstrate that the foundational framework of quantum mechanics is not natively probabilistic or random; rather, it is a strictly deterministic, hydrodynamic field theory whose fluid-like conservation mandate uniquely dictates the geometry of its measurement space.

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In order to make the quantum mechanics a closed theory one has to derive the Born rule from the first principles, like the Schroedinger equation (SEq), rather than postulate it. The Born rule was in certain sense derived in several articles, e.g. in [1] and [2]. In this work some arguments of previous authors are simplified and made more “physical”. It is shown how to derive the Born rule using the conservation of quantum state norm (Ψ|Ψ) that is the unitary evolution property determined by the SEq. It is this property that makes the probability equal to the square of the amplitude modulus. Possible modification of the Born rule is briefly discussed in the case when the evolution of the system is described by a nonlinear Schroedinger equation. We also present arguments in the spirit of the Many-World Interpretation to explain the origin of probabilistic behavior. Simply speaking, the randomness appears as a result of representing the wave function by using a detector of discrete nature that is found only in one state at a time, out of two or more possible states.

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SDRIS: The Geometric Origin of Probability - Deriving the Born Rule and Wave-Particle Duality from the Dimensional Reduction of Recursive Amplitudes

Jan Patrick Maier-Lutz

Paper 102 of the SDRIS Research Series Following the geometric derivation of forces (Paper 100) and non-locality (Paper 101), this work addresses the origin of quantum probability. Abstract: The Born Rule, which states that the probability of a quantum measurement is proportional to the square of the wavefunction's amplitude ($P = |\psi|^2$), is one of the most fundamental yet least understood axioms of quantum mechanics. Standard theory treats it as an empirical postulate without a rigorous geometric derivation. In this paper, we derive the Born Rule as a necessary consequence of the Static-Dynamic Recursive Information Space (SDRIS) framework. We demonstrate that the "amplitude" $|\psi|$ corresponds to the linear Euclidean length of the information vector in the 5-dimensional bulk, while the "probability" $P$ corresponds to the interaction flux through the 3-dimensional projection interface. Since the information flux through a holographic boundary scales with the surface area ($A \propto r^2$) rather than the linear radius, the quadratic relationship is geometrically inevitable. Furthermore, we show that Wave-Particle Duality is resolved as an observational artifact: The "Wave" is the continuous rotation of the vector in the recursive time dimension ($\tau$), while the "Particle" is the discrete intersection of this vector with the static metric slice ($t$). Thus, quantum randomness is identified not as intrinsic indeterminism, but as the geometric cross-section of a higher-dimensional determinism. Includes Computational Verification: This publication includes a Python simulation (SDRIS_Born_Rule_Simulation_Code_v1.0.ipynb) that numerically validates the flux-projection mechanism, confirming that the measurement of a deterministic 5D-vector yields a statistical distribution strictly following the Born Rule.

•Jan 1, 2026•1 citation
This paper presents a derivation of the Born rule from first principles using a phase-density formalism. We show that the probability density is necessarily the square of the wavefunction amplitude due to linearity, local conservation, and independence from absolute phase. The phase is naturally defined on a compact space S^1, leading to topological quantization and explaining the appearance of the quantum of action ħ. This approach unifies the origin of quantum probability and phase quantization from a deterministic, geometric perspective.
New edited version Title: Quantum Mechanics as Topological Intersection Theory: The Born Rule, Wavefunction Collapse, and Planck’s Constant from Worldline Non-Injectivity Abstract: This work presents a comprehensive derivation of the foundational principles of quantum mechanics by shifting the paradigm from stochastic postulates to a deterministic geometric framework. Utilizing the TPST-DGQ (Topological Phase Signalling Theorem - Discrete Geometric Quantization) framework, the paper demonstrates that the core axioms of quantum theory emerge as necessary theorems from the geometry of non-injective worldlines. Key Mathematical and Physical Derivations: Geometric Origin of the Born Rule: The paper proves that the probability density ∣ψ(x,t)∣2|\psi(x,t)|^2∣ψ(x,t)∣2 is a "topological intersection density." By analyzing a single ultra-relativistic worldline Xμ(τ)X^\mu(\tau)Xμ(τ) that intersects the constant-time hypersurface Σt\Sigma_tΣt​ at N(ϵ)N(\epsilon)N(ϵ) distinct points, the Born rule is derived as the normalized frequency of these intersections. This replaces the epistemic interpretation of probability with a frequentist geometric count. The Wavefunction as a Transport Representation: The wavefunction ψ(x,t)\psi(x,t)ψ(x,t) is defined as the complex square root of the intersection density. Its phase is shown to be the result of proper-time transport along the continuous segments of the worldline connecting different intersection points. Emergence of the Schrödinger Equation: The manuscript derives the Schrödinger equation as the fundamental transport law for the intersection density. This derivation is shown to be valid under the condition of "inter-sheet phase coherence," a state forced by the Ontological Identity Principle and the finiteness of the action. Fundamental Nature of Planck’s Constant (ℏ\hbarℏ): Instead of a fundamental constant of nature, ℏ\hbarℏ is derived as the minimum action per radian of phase accumulated between stable folds of the worldline. It is functionally linked to the UV cutoff ϵ\epsilonϵ and the rest mass mmm, providing a parameter-free basis for the quantum of action. Deterministic Wavefunction Collapse: Collapse is redefined as a topological transition from a non-injective state (N>1N > 1N>1) to an injective state (N=1N = 1N=1). This occurs through a physical interaction (measurement) that reduces the Lorentz factor below a critical threshold γcrit\gamma_{crit}γcrit​, providing a dynamical solution to the measurement problem. Unified Regularization: The paper establishes the "Universal Topological Cancellation," showing that the same scaling mechanism N(ϵ)∼ϵ−(d−2)N(\epsilon) \sim \epsilon^{-(d-2)}N(ϵ)∼ϵ−(d−2) that regularizes the Ryu-Takayanagi entropy in holography and the Coulomb self-energy in classical electrodynamics also ensures the UV-finiteness of the quantum intersection density. This research unifies quantum mechanics, classical electrodynamics, and holographic gravity under a single topological principle: worldline non-injectivity as a necessary and sufficient condition for finite physics at every level. This manuscript is current in Official Peer Review. Not final version.Copyright©2026 Alex De Giuseppe.All rights reserved. This work is protected by copyright. Any form of plagiarism, unauthorized reproduction, or misappropriation of ideas, mathematically results, or text without proper citation constitutes a violation of academic and intellectual property standards and common laws. No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author. For correspondence, citations, collaboration inquiries, or feedback please contact:degiuseppealex@gmail.com The hash files that determine ownership have been created
The mathematical and physical foundations of the Born rule remain open questions, largely because the wave function lacks a mathematical identity and a physical origin. Here we uncover a mathematical structure of the wave function by treating the phase as a circular variable. This structure reveals that the wave function is the pullback of the phase probability-density amplitude to position space via the phase field. Because the phase probability distribution forms an equal-area contour in the complex plane, its probability-area element yields the modulus squared as the position probability density, revealing the Born rule as a geometric relation rooted in the circular topology of the phase variable. The same structure governs the phase-energy distribution of electromagnetic waves. By quantizing phase energy into energy quanta, in the single-quantum limit the energy-quantum density reduces to the probability density, and its amplitude becomes the probability-density amplitude. This therefore identifies the energy-quantum density amplitude in the single-quantum limit as the physical origin of the wave function. To verify this, we derive the time-evolution equation for the energy-quantum density amplitude from five physical principles. This yields the Gross-Pitaevskii equation for many-body systems and, in the single-quantum limit, the Schrödinger equation. The framework also recovers Maxwell's equations in the classical limit. Overall, this study introduces the energy-quantum density amplitude from the classical phase-energy distribution, and derives the three core postulates of quantum mechanics: the wave function, the Born rule, and the Schrödinger equation, thereby identifying the position probability density as the energy-quantum density in the single-quantum limit.
AbstractThe Born rule—assigning quantum probabilities as —remains a postulate in standard quantummechanics. Despite numerous attempts (Gleason's theorem, decoherence, envariance, decision theory), noderivation has eliminated the need for an additional probabilistic axiom. This paper presents a geometric derivationof the Born rule from three minimal principles: (1) physical states are complex linear fields, (2) reality includes anunobservable scale dimension , and (3) observers are restricted to fixed-scale hypersurfaces. Under theseconditions, the unique operational density consistent with linearity, conservation, and phase invariance is thequadratic form . Marginalizing over unobservable scale degrees of freedom yields . Under scale-localization or scale-decoherence, this reduces to , reproducing the Born rulewithout assumption. If the ultraviolet limit of the manifold is a compact genus- surface, topological constraintsproduce structured statistical signatures: for , a 138-mode periodic correction emerges, providing afalsifiable experimental prediction. Measurement is reinterpreted as dimensional projection, not wavefunctioncollapse. Quantum probabilities arise as geometric shadows of deterministic higher-dimensional dynamics.Keywords: Born rule, quantum probability, dimensional flow, marginalization, scale space, genus, quantumfoundations, measurement problem Full Theory Portfolio at: https://drive.google.com/drive/folders/1fdKdo3edGqXVx95IntIumXlzKq22s-yw?usp=drive_link
The probabilistic interpretation of quantum mechanics rests fundamentally on the Born rule, which asserts that the physical density of a state is given by the squared magnitude of its wavefunction (ρ=|ψ|^2). Historically, this exponent of (α=2) has been treated as an unprovable, axiomatic postulate justified solely by empirical alignment. This paper presents a rigorous, parameter-agnostic derivation that turns this narrative on its head. By introducing ( α) as a completely free geometric parameter ( ρ=|ψ|^α) and demanding strict adherence to the universal principle of Local Continuity: ∂ρ/∂t+∇⋅J ⃗=0 I show that the algebra itself systematically eliminates all alternative mappings. If ( α≠2), the local density variation inherently picks up non-local dependencies on the absolute background field amplitude, violating localized field interactions. The mathematics strictly demands ( α=2) as a unique geometric threshold required to isolate unobservable phase variables and preserve local conservation laws. Consequently, I demonstrate that the foundational framework of quantum mechanics is not natively probabilistic or random; rather, it is a strictly deterministic, hydrodynamic field theory whose fluid-like conservation mandate uniquely dictates the geometry of its measurement space.
In order to make the quantum mechanics a closed theory one has to derive the Born rule from the first principles, like the Schroedinger equation (SEq), rather than postulate it. The Born rule was in certain sense derived in several articles, e.g. in [1] and [2]. In this work some arguments of previous authors are simplified and made more “physical”. It is shown how to derive the Born rule using the conservation of quantum state norm (Ψ|Ψ) that is the unitary evolution property determined by the SEq. It is this property that makes the probability equal to the square of the amplitude modulus. Possible modification of the Born rule is briefly discussed in the case when the evolution of the system is described by a nonlinear Schroedinger equation. We also present arguments in the spirit of the Many-World Interpretation to explain the origin of probabilistic behavior. Simply speaking, the randomness appears as a result of representing the wave function by using a detector of discrete nature that is found only in one state at a time, out of two or more possible states.
Paper 102 of the SDRIS Research Series Following the geometric derivation of forces (Paper 100) and non-locality (Paper 101), this work addresses the origin of quantum probability. Abstract: The Born Rule, which states that the probability of a quantum measurement is proportional to the square of the wavefunction's amplitude (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2), is one of the most fundamental yet least understood axioms of quantum mechanics. Standard theory treats it as an empirical postulate without a rigorous geometric derivation. In this paper, we derive the Born Rule as a necessary consequence of the Static-Dynamic Recursive Information Space (SDRIS) framework. We demonstrate that the "amplitude" ∣ψ∣|\psi|∣ψ∣ corresponds to the linear Euclidean length of the information vector in the 5-dimensional bulk, while the "probability" PPP corresponds to the interaction flux through the 3-dimensional projection interface. Since the information flux through a holographic boundary scales with the surface area (A∝r2A \propto r^2A∝r2) rather than the linear radius, the quadratic relationship is geometrically inevitable. Furthermore, we show that Wave-Particle Duality is resolved as an observational artifact: The "Wave" is the continuous rotation of the vector in the recursive time dimension (τ\tauτ), while the "Particle" is the discrete intersection of this vector with the static metric slice (ttt). Thus, quantum randomness is identified not as intrinsic indeterminism, but as the geometric cross-section of a higher-dimensional determinism. Includes Computational Verification: This publication includes a Python simulation (SDRIS_Born_Rule_Simulation_Code_v1.0.ipynb) that numerically validates the flux-projection mechanism, confirming that the measurement of a deterministic 5D-vector yields a statistical distribution strictly following the Born Rule.