The Pauli Exclusion Principle and the Spin-Statistics Theorem from Worldline Non-Injectivity: Exchange Phase, Rapidity, and Topological Sheet Structure

The Pauli Exclusion Principle and the Spin-Statistics Theorem from Worldline Non-Injectivity: Exchange Phase, Rapidity, and Topological Sheet Structure For nearly a century, the Pauli exclusion principle (PEP) and the spin-statistics theorem have stood as foundational postulates of quantum mechanics and relativistic quantum field theory. The PEP dictates that no two identical fermions can occupy the same quantum state, determining the structure of atoms, the stability of matter, and the properties of neutron stars and white dwarfs. The spin-statistics theorem connects the intrinsic spin of a particle to its statistical behavior: half-integer spin particles (fermions) obey Fermi-Dirac statistics, while integer spin particles (bosons) obey Bose-Einstein statistics. Despite their universal success, these principles have never been derived from a more primitive geometric or topological principle — until now. This paper demonstrates that both the Pauli exclusion principle and the spin-statistics theorem emerge naturally and rigorously from a single geometric fact: the **non-injectivity of a timelike worldline** for Lorentz factors exceeding a critical threshold, \ (> ₂ₑ₈ₓ\). In this regime, a single worldline intersects a constant-time hypersurface \ (ₜ\) in \ (N > 1\) distinct spatial points, generating a multi‑sheet spacetime structure. This phenomenon, known as **worldline non-injectivity**, is the foundation of the TPST–DGQ framework. **Key results and logical flow: ** 1. **Multi‑sheet structure for a single particle: ** For \ (> ₂ₑ₈ₓ\), a particle's worldline produces \ (N () ^- (d-2) \) topological sheets, each carrying the same physical properties (mass, charge, spin) via the Ontological Identity Principle. 2. **Composite worldline for two identical particles: ** By combining two identical particles into a composite worldline in configuration space, we obtain a two‑particle sheet structure. Identical particles share isomorphic sheet structures, reducing the \ (N N\) sheet pairs to \ (N\) independent pairs. 3. **Exchange as a permutation of sheet blocks: ** The exchange operator \ (P₁₂\) swaps the sheet blocks associated with the two particles. This permutation is a topological operation in the two‑particle configuration space. 4. **Exchange phase from winding number: ** The exchange path corresponds to half of a complete winding cycle around the fold structure of the composite worldline. The accumulated topological phase is \ (₄ₗ₂₇₀₍₆₄ = w\), where \ (w\) is the **winding number** of the worldline. Hence the exchange phase factor is \ (= e^i w = (-1) ʷ\). 5. **Spin from winding number: ** The winding number \ (w\) is directly related to the spin \ (s\) by the requirement that a \ (4\) rotation returns the physical state to itself (single‑valuedness). This yields \ (w = 2s\). Therefore \ (= (-1) ^2s\): half‑integer spin gives \ (= -1\) (fermions, antisymmetric), integer spin gives \ (= +1\) (bosons, symmetric). This is the spin‑statistics theorem derived from geometry. 6. **Pauli exclusion principle: ** For two identical fermions (\ (= -1\) ) at spatial coincidence, the two‑particle amplitude satisfies \ (A = \), forcing \ (A = 0\). Thus no two fermions can occupy the same quantum state. The PEP is no longer a postulate; it is a theorem. 7. **Rapidity as inter‑sheet phase measure: ** The rapidity \ (= artanh (v/c) \) is shown to be a monotonic function of the inter‑sheet phase difference \ (ₙ\). The non‑injectivity threshold \ (> ₂ₑ₈ₓ\) corresponds to \ (> ₂ₑ₈ₓ\). The speed of light \ (c\) emerges as the asymptotic limit where the inter‑sheet phase diverges, making \ (v = c\) unreachable for massive particles. 8. **Universal cancellation identity: ** The same algebraic identity \ (N () ^d-2 = O (1) \) that regularizes holographic entropy, Coulomb self‑energy, wavefunction normalization, electromagnetic fields, and the cosmological constant also regularizes the two‑particle exchange amplitude — zero for fermions, finite for bosons. Quantum statistics thus joins the list of phenomena unified under the principle of worldline non‑injectivity. **Implications: ** This work completes the derivation of the foundations of quantum mechanics within the TPST–DGQ framework, following previous derivations of the Born rule, the Schrödinger equation, Planck’s constant, and wavefunction collapse. It also establishes a geometric origin for the spin‑statistics connection, a result that in standard physics relies on the heavy machinery of relativistic quantum field theory (unitarity, causality, positivity of energy). Here, it follows from the topology of worldline folds. 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.