Geometric Origin of Quantum Entanglement from Worldline Non-Injectivity: Area Law, Decoherence, and Spacetime Connectivity

What is the geometric origin of quantum entanglement? In standard quantum mechanics, entanglement is a postulate: composite systems occupy tensor product Hilbert spaces, and states need not factorize. Its emergence from deeper physical principles has remained an open question. This paper provides a rigorous answer: entanglement is a geometric consequence of worldline non-injectivity. The framework rests on a single kinematic fact established in the companion paper "Worldline Non-Injectivity as a Necessary and Sufficient Condition for the Emergence of Holographic Spacetime": when a massive body travels at ultra-relativistic velocities ₂ₑ₈ₓ, its worldline X^ () intersects the simultaneity foliation ₜ in N > 1 distinct spatial points. These N intersections are not N distinct particles, but N temporal appearances of the same physical entity. The central achievement of this paper is the derivation of the **Ontological Identity Principle (OIP) ** — the statement that all N appearances are manifestations of one entity, and that the multi-sheet state is a coherent superposition with uniform coefficients |cₙ| = 1/N — not as a postulate, but as a **theorem**. The proof rests on three independent lemmas, each grounded in results already established within the TPST–DGQ framework: 1. **Unitarity of proper-time transport (Lemma 1): ** The continuity of the worldline X^ () C^ (R) and the Hamiltonian structure of the theory imply that the evolution operator between proper times is unitary, forcing the total state to be a coherent superposition over all sheets. 2. **Topological superselection rule (Lemma 2): ** No local physical observable can distinguish the n-th sheet from the m-th sheet without breaking the C^ regularity of X^ (). Since no topological defect separates the sheets, there is no conserved quantum number capable of differentiating them, forbidding classical mixtures and enforcing quantum coherence. 3. **Permutation symmetry (Lemma 3): ** The sheets are indistinguishable manifestations of the same entity, so the physics is invariant under any permutation SN of the sheet labels. The unique pure state invariant under the full permutation group SN is the uniform superposition with coefficients |cₙ| = 1/N. From this derived OIP, the paper obtains five major results: - **Area law for entanglement entropy: ** The entanglement entropy between the physical sector and the sheet sector is S₄₍ₓ = N. Using the UV scaling N () ^- (d-2) (Lemma 2 of the companion paper on non-injectivity), this becomes S₄₍ₓ = (d-2) (1/), reproducing the Srednicki area law for entanglement entropy in quantum field theory. - **Isomorphism with holographic entanglement: ** We construct an explicit linear isomorphism between the sheet Hilbert space Hₒ₇₄₄ₓₒ = ² (ZN) and the space of Ryu–Takayanagi minimal surfaces anchored at the boundary of region A. Under this isomorphism, the entanglement entropy S₄₍ₓ = N coincides with the holographic entanglement entropy SA^RT after the identification N Area (A) / (4GN). This makes precise the connection between the geometric entanglement derived here and the holographic entanglement entropy of AdS/CFT. - **Lorentz invariance of entanglement entropy: ** Although the sheet number N depends on the observer's foliation, the entanglement entropy S₄₍ₓ = N is a Lorentz scalar. The topological projection operator P₀ = 1N/N commutes with all Lorentz generators, and S₄₍ₓ is invariant under any change of inertial frame. This resolves the apparent tension between the observer-dependence of N and the physical requirement of covariance. - **Geometric decoherence: ** Quantum decoherence is the partial trace over the sheet degrees of freedom. When inter-sheet perturbations (derived from the Maxwell Topological Emergence Identity in the companion paper on electromagnetic fields) induce distinct states |ₙ on different sheets, the off-diagonal elements of the physical density matrix decay as 1/N, producing classicality without any external environment. - **Van Raamsdonk conjecture as a theorem: ** The equivalence between spacetime connectivity and quantum entanglement — conjectured by Van Raamsdonk (2010) and formalized as ER = EPR by Maldacena and Susskind (2013) — is proved as a theorem within the TPST–DGQ framework. Non-injectivity (N > 1), entanglement (S₄₍ₓ > 0), and geometric connectivity of the N spatial intersections are three equivalent statements. Removing entanglement (by measuring the sheet index) forces N 1 and disconnects the geometry, exactly as Van Raamsdonk conjectured. The paper further introduces a **Fock space structure** for variable sheet number, describing quantum fluctuations that create or destroy fold points. The associated creation and annihilation operators satisfy the bosonic algebra, identifying the fold points as quanta of a topological field. For N = 2, the De Giuseppe Qubit (DGQ) is the maximally entangled Bell state of the framework. The O (1) gate complexity and 1/N decoherence suppression proved in the companion paper on the DGQ follow directly from the entanglement structure derived here. The **universal cancellation identity** N () ^d-2 = O (1) — alrKeywordseady shown in companion papers to regularize holographic entropy, Coulomb self-energy, the cosmological constant, and the intersection density ||² — now operates at nine levels of physical theory, confirming worldline non-injectivity as the geometric engine of entanglement. 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. 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