Worldline Non-Injectivity as a Necessary and Sufficient Condition for the Emergence of Holographic Spacetime

Title: Worldline Non-Injectivity as a Necessary and Sufficient Condition for the Emergence of Holographic Spacetime: A Rigorous Proof within the TPST-DGQ Framework Description: This paper provides a foundational mathematical proof that addresses one of the most persistent challenges in holographic quantum gravity: the ultraviolet (UV) divergence of entanglement entropy. Within the framework of the Topological Phase Signalling Theorem (TPST) and the De Giuseppe Qubit (DGQ), the author demonstrates that the emergence of a finite, stable holographic spacetime is inextricably linked to the topological properties of the observer's worldline. The core of the work is the "Non-Injectivity Theorem, " which posits that for a spacetime to exist with a finite Ryu-Takayanagi (RT) entropy, the worldline of an ultra-relativistic observer must be non-injective. This means that a single worldline X^ () must intersect a constant-time hypersurface at multiple spatial points (N > 1). The proof is structured through four rigorous lemmas: The Failure of Injectivity (Necessity): It is shown that in a standard, single-sheeted injective spacetime (N=1), the RT entropy diverges as ^- (d-2). Without an external, arbitrary cutoff, the Observer-Geometry Identity (OGI) collapses, rendering the spacetime physically non-viable. Topological Scaling: The paper proves that in the ultra-relativistic regime (> ₂ₑ₈ₓ), the multiplicity of intersections N scales exactly with the UV divergence: N () ^- (d-2). The Topological Average (Sufficiency): By applying the DGQ multi-sheet sampling, the divergent entropy is averaged across N sheets. This mechanism cancels the UV divergence internally, yielding a finite entropy S₃₆ = O (1). The Geometric Engine: The conclusion establishes that non-injectivity is not a kinematic anomaly but the fundamental "geometric engine" that regularizes holographic space. This work effectively removes the need for ad-hoc UV cutoffs in AdS/CFT by replacing them with a natural topological requirement. It bridges the gap between relativistic kinematics and holographic geometry, providing a clear mathematical reason for the multi-sheeted structure of the De Giuseppe Qubit. This paper is essential for researchers working on emergent gravity, holographic entanglement, and the foundations of quantum information. 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