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

Topological Phase Signalling as a Dynamic Mechanism for Bulk Geometry 1. The Foundational Departure: From Static to State-Dependent Holography Standard holographic duality relates static boundary states to static bulk geometries (or stationary spacetimes). Our work challenges this paradigm by introducing the Topological Phase Signalling Theorem (TPST), which posits that if global unitaries $U(\rho)$ are functional of the global state, the standard no-signalling theorems are bypassed through a state-dependent phase $\phi[\rho]$. We demonstrate that this is not merely a mathematical curiosity of finite-dimensional systems, but a fundamental property that emerges naturally when the Ryu–Takayanagi (RT) surface is treated as a dynamic generator $\hat{G} = \hat{\mathcal{A}}(\gamma_B)/(4G_N)$. 2. The Mechanism of Geometric Backreaction We provide a closed-form derivation showing how a local operation $V_A$ on the boundary region $A$ induces a non-local reorganization of the RT surface $\gamma_B$ in the bulk. Unlike standard perturbative gravity, where metric changes are sourced by external matter fields, our model shows that the geometric backreaction is sourced by the "topological signalling" itself—the change in the phase functional $\phi[\rho]$. We prove that the first-order variation of the RT area $\delta \langle \mathcal{A}(\gamma_B) \rangle$ is constrained by the linearized Einstein equations, confirming that this signalling mechanism preserves the holographic bulk-boundary consistency. 3. Mathematical Rigor and Operational Realizability A common critique of state-dependent dynamics is the potential for non-unitarity or energy violation. We address this by: Defining the Kernel: We derive the integral kernel $\mathcal{K}_{\gamma_B}^{ab}(x')$ that quantitatively maps the boundary energy density variation $\delta \langle T_{ab} \rangle$ to the bulk metric perturbation $h_{\mu\nu}$. Ancillary Modeling: By constructing a toy Hamiltonian with an ancillary system, we explicitly show that the signalling protocol is physically realizable via a sequence of local operations, unitary coupling, and weak-measurement/feedback readout. The global energy conservation $\Delta \langle H_{\text{tot}} \rangle = 0$ is preserved, effectively placing the TPST within the standard axioms of quantum field theory while extending its capabilities. 4. Regulating Entanglement Divergences One of the most significant theoretical contributions of this work is the treatment of UV divergences in entanglement entropy. We show that the state-dependent phase $\phi[\rho]$ acts as a "dynamical regulator." In regions where energy density fluctuations approach the UV limit, the global unitary $U(\rho)$ induces a geometric transformation that effectively "stretches" the entanglement wedge, providing a self-consistent physical cutoff $\epsilon_{\rm eff}(\phi[\rho])$ that prevents non-physical singularities. This moves the regularization of entanglement entropy from a manual, heuristic choice to an emergent property of the state-dependent geometry itself. 5. Broader Impact: Reshaping Entanglement Wedges By applying the TPST to the Schwarzschild-AdS background, we provide a concrete mechanism for how information is redistributed across the bulk without violating causality. The state-dependent nature of our transformation suggests that the entanglement wedge reconstruction is not a static map but a dynamical process. This provides a new, rigorous avenue for exploring the interplay between local boundary operations and the interior structure of gravity, offering a potential framework for addressing how information is encoded in the geometry of the bulk 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