Holographic Extension as a Dynamic Mechanics for Bulk Geometry CTC with Topological Phase Signalling Theorem

Project Description: Topological Phase Signalling & Holographic Spacetime Dynamics Overview: A New Paradigm for Quantum Gravity This research introduces a fundamental departure from the static interpretations of the AdS/CFT correspondence. By formalizing the Topological Phase Signalling Theorem (TPST)—a novel theoretical framework developed to govern state-dependent unitary evolutions—we demonstrate that the bulk geometry is not a static background, but a dynamical, programmable entity responsive to quantum information at the boundary. Core Contributions The TPST Framework: We move beyond finite-dimensional qubit models to establish a rigorous mapping between continuous quantum fields on the CFT boundary and the bulk metric. The introduction of a state-dependent phase functional $\phi[\rho]$ allows for a non-linear, controlled feedback loop between boundary observables and the geometry of the Ryu–Takayanagi (RT) surface. The "Holographic Lever" (Causal Amplification): Our work unveils a critical regime in which infinitesimal perturbations at the boundary trigger macroscopic geometric transitions in the bulk. This "leverage" effect provides the first quantitative measure of how entanglement acts as the primary engine for spacetime structure, offering an operational path to "sculpting" geometry. Solving the Non-Linearity Paradox: We address the long-standing challenge of Gisin-Polchinski-type non-linearities. Our derivation proves that these dynamics, which would otherwise threaten the causality of quantum mechanics, are naturally regularized and mediated by the dynamical bulk fields. This ensures that the protocol remains strictly within the bounds of standard causality while enabling sophisticated geometric control. Dynamical Entanglement Regularization: We redefine UV divergence management. By demonstrating that the phase functional $\phi[\rho]$ induces an effective cutoff $\epsilon_{\rm eff}$ through geometric backreaction, we transition from heuristic UV-cutoff choices to a self-consistent, emergent property of the state-dependent geometry. Operational Realizability: Moving from abstract theory to protocol, we provide the kernel $\mathcal{K}_{\gamma_B}^{ab}(x')$ that maps boundary energy-stress variations to bulk metric perturbations, establishing a bridge for future experimental verification in quantum circuit simulations and tensor networks. Potential Impact This manuscript redefines the entanglement wedge reconstruction not as a fixed mapping, but as a dynamical process. By establishing the TPST as a foundational pillar, we offer a new framework for addressing the Black Hole Information Paradox and the emergence of gravity from quantum information (the "It from Qubit" program). This work transforms the bulk-boundary duality into an active, programmable interface. 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:[e-mail] The hash files that determine ownership have been created