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| Format: | Recurso digital |
| Idioma: | anglès |
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2026
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| Accés en línia: | https://doi.org/10.5281/zenodo.18948614 |
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- <div dir="ltr"> <p> </p> <h3><strong>Title: Holographic Extension of the Topological Phase Signalling Theorem: Entanglement-Induced Bulk Geometry Dynamics</strong></h3> <p> </p> <p><strong>Author:</strong> Alex De Giuseppe</p> <p> <strong>Abstract / Executive Summary</strong></p> <p>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 $\gamma_{B}$, $\hat{G} = \frac{\hat{\mathcal{A}}(\gamma_{B})}{4G_{N}}$.</p> <div> </div> <p> </p> <p>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.</p> <div> </div> <p> </p> <h3><strong>Core Theoretical Novelties</strong></h3> <ul> <li> <p> </p> <p><strong>State-Dependent Unitary on the Code Subspace:</strong> 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.</p> <div> </div> <p> </p> </li> <li> <p> </p> <p><strong>Causal Amplification and RT Phase Transitions:</strong> 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.</p> <div> </div> <p> </p> </li> <li> <p> </p> <p><strong>The Observer-Geometry Identity (OGI):</strong> 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.</p> <div> </div> <p> </p> </li> </ul> <h3><strong>The Three Fundamental Equations</strong></h3> <p>The culmination of the TPST holographic extension is captured in three explicit mathematical results that bridge entanglement, geometry, and gravitational dynamics.</p> <p><strong>1. The Entropic-Geometric Response Formula</strong> 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:</p> <div> </div> <p> </p> <div> <div>$$\delta S_{B} = \frac{8\pi R_{B}^{2}}{L_{A}} \left[ \frac{a R_{B}}{R_{B}^{2} - a^{2}} + \frac{1}{2R_{B}^{3}} \arctan\left(\frac{a}{R_{B}}\right) \right] (\delta E)^{2}$$</div> </div> <p>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.</p> <div> </div> <p> </p> <p><strong>2. The Observer-State Gravitational Equation</strong> 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:</p> <div> </div> <p> </p> <div> <div>$$G_{\mu\nu} + \Lambda[\rho^{*}]g_{\mu\nu} = 8\pi G_{N}T_{\mu\nu}$$</div> </div> <p>Where the emergent cosmological constant is defined as:</p> <div> <div>$$\Lambda[\rho^{*}] = 4\pi G_{N}\lambda^{2}\langle T_{00}\rangle_{A}[\rho^{*}]$$</div> </div> <p>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.</p> <div> </div> <p> </p> <p><strong>3. The TPST Master Equation</strong> 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.</p> <div> </div> <p> </p> <div> <div>$$G_{\mu\nu} + 4\pi G_{N}\lambda^{2}\langle T_{00}\rangle_{A}[\rho^{*}]g_{\mu\nu} = 8\pi G_{N}T_{\mu\nu} + \frac{8\pi R_{B}^{2}}{L_{A}}\mathcal{K}(a, R_{B})\frac{(\delta E)^{2}}{c_{d}}h_{\mu\nu}|_{\gamma_{B}}$$</div> </div> <p>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.</p> <div> </div> <div> <p dir="ltr">This manuscript is current in Official Peer Review.</p> <p dir="ltr">Not final version.<br>Copyright©2026 Alex De Giuseppe.<br>All rights reserved.</p> <p dir="ltr">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.</p> <p dir="ltr">No commercial use, adaptation, or derivative works are permitted without explicit written permission from the author.</p> <p dir="ltr">For correspondence, citations, collaboration inquiries, or feedback please contact:<br>degiuseppealex@gmail.com</p> <p dir="ltr">The hash files that determine ownership have been created</p> </div> </div>