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

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 | (x, t) |² is a "topological intersection density. " By analyzing a single ultra-relativistic worldline X^ () that intersects the constant-time hypersurface ₜ at N () 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 (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 (): Instead of a fundamental constant of nature, 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 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 ₂ₑ₈ₓ, providing a dynamical solution to the measurement problem. Unified Regularization: The paper establishes the "Universal Topological Cancellation, " showing that the same scaling mechanism N () ^- (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