Title: Holographic Extension of the Topological Phase Signalling Theorem: Entanglement-Induced Bulk Geometry Dynamics Author: Alex De Giuseppe Abstract / Executive Summary This paper presents a rigorous derivation of an $AdS/CFT$ implementation of the Topological Phase Signalling Theorem (TPST). Transitioning from a finite-dimensional qubit construction to continuous quantum fields on a boundary Conformal Field Theory (CFT), the framework introduces a state-dependent global unitary evolution $U(\rho) = \exp(-i\phi[\rho]\hat{G})$. The generator $\hat{G}$ is mapped to the bulk via identification with the area operator of the Ryu-Takayanagi (RT) surface $\gamma_{B}$, $\hat{G} = \frac{\hat{\mathcal{A}}(\gamma_{B})}{4G_{N}}$. This work establishes a self-consistent, observer-inclusive paradigm for holographic quantum gravity where the bulk causal structure is dynamically derived from the quantum state. Furthermore, the framework operates with zero free parameters: the phase-geometry coupling constant is uniquely fixed to $\lambda = \frac{2}{\sqrt{L}}$ via the Brown-Henneaux relation. Core Theoretical Novelties State-Dependent Unitary on the Code Subspace: The unitary $U(\rho)$ is rigorously proven to be well-posed and bounded on the semiclassical code subspace $\mathcal{H}_{code}$ using a regularised area operator, avoiding the unbounded domain pathologies of the full Hilbert space. Causal Amplification and RT Phase Transitions: Near the critical manifold where the RT surface is tangent to the bulk null cone of the perturbed region, the geometric sensitivity diverges. An infinitesimally small boundary phase perturbation triggers a discontinuous, macroscopic $O(N^{2})$ jump of the RT surface without violating bulk causality. The Observer-Geometry Identity (OGI): The standard external causal constraint collapses under observer inclusion. The system reaches a fixed point characterised by the identity $\rho^{*} = \mathcal{G}[\rho^{*}] = \mathcal{O}[\rho^{*}]$, demonstrating that the boundary state, its generated bulk geometry, and the included observer are three representations of the same fixed point. This generalises the ER=EPR correspondence to fully self-referential regimes. The Three Fundamental Equations The culmination of the TPST holographic extension is captured in three explicit mathematical results that bridge entanglement, geometry, and gravitational dynamics. 1. The Entropic-Geometric Response Formula This is the first fully explicit, parameter-free formula in the holographic literature mapping a local boundary energy perturbation $\delta E$ (in region $A$) to a measurable quadratic variation of entanglement entropy $\delta S_{B}$ (in region $B$) via the RT surface. In $AdS_{3}/CFT_{2}$, it reads: $$\delta S_{B} = \frac{8\pi R_{B}^{2}}{L_{A}} \left[ \frac{a R_{B}}{R_{B}^{2} - a^{2}} + \frac{1}{2R_{B}^{3}} \arctan\left(\frac{a}{R_{B}}\right) \right] (\delta E)^{2}$$ This formula predicts a universal quadratic law $\delta S_{B} \propto (\delta E)^{2}$ and mathematically captures the logarithmic divergence at the causal amplification threshold ($a \rightarrow R_{B}^{-}$), offering direct testability for MERA tensor-network simulations. 2. The Observer-State Gravitational Equation At the self-consistent fixed point $\rho^{*}$, the standard Einstein field equations are modified. The cosmological constant $\Lambda$ ceases to be a fundamental free parameter and emerges dynamically as a functional of the observer's quantum state: $$G_{\mu\nu} + \Lambda[\rho^{*}]g_{\mu\nu} = 8\pi G_{N}T_{\mu\nu}$$ Where the emergent cosmological constant is defined as: $$\Lambda[\rho^{*}] = 4\pi G_{N}\lambda^{2}\langle T_{00}\rangle_{A}[\rho^{*}]$$ This equation establishes that spacetime curvature is determined by the energy the observer assigns to their own local region, providing a phase-topological mechanism for vacuum selection via discrete winding sectors. 3. The TPST Master Equation Unifying the local perturbative response with the global non-perturbative fixed point, the TPST Master Equation encodes the fully self-referential coupling between state, geometry, and measurement. $$G_{\mu\nu} + 4\pi G_{N}\lambda^{2}\langle T_{00}\rangle_{A}[\rho^{*}]g_{\mu\nu} = 8\pi G_{N}T_{\mu\nu} + \frac{8\pi R_{B}^{2}}{L_{A}}\mathcal{K}(a, R_{B})\frac{(\delta E)^{2}}{c_{d}}h_{\mu\nu}|_{\gamma_{B}}$$ This single tensorial equation subsumes standard general relativity in the classical limit ($\delta E \rightarrow 0$), the Entropic-Geometric Response in the perturbative limit, and introduces a completely novel regime where the bulk metric is simultaneously sourced by matter stress-energy and the quadratic entanglement response of the RT surface. 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