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拓扑相位信号：体几何的动态作用机制 1. 基础范式转向：从静态全息学到依赖于量子态的全息学 标准全息对偶将静态边界态与静态体几何（或稳态时空）关联起来。本研究对这一范式提出挑战，提出拓扑相位信号定理（Topological Phase Signalling Theorem，TPST），该定理指出：若全局幺正算符$U( ho)$为全局量子态的泛函，则可通过依赖于量子态的相位$phi[ ho]$绕过标准无信号定理。我们证明，这并非仅局限于有限维系统的数学趣闻，而是当将Ryu-Takayanagi（RT）曲面视为动态生成元$hat{G} = hat{mathcal{A}}(gamma_B)/(4G_N)$时自然涌现的基础性质。 2. 几何反作用的作用机制 我们给出闭式推导，阐明边界区域$A$上的局域操作$V_A$如何诱导体时空内RT曲面$gamma_B$发生非局域重组。与标准微扰引力中“度规变化由外部物质场作为源”的设定不同，本研究模型表明，几何反作用的源正是“拓扑信号”本身——即相位泛函$phi[ ho]$的变化。我们证明，RT面积的一阶变分$delta langle mathcal{A}(gamma_B) angle$受线性化爱因斯坦方程约束，证实该信号机制可维持全息体-边界一致性。 3. 数学严谨性与操作可实现性 针对依赖于量子态的动力学的常见质疑是其可能存在非么正性或能量不守恒问题。我们通过以下方式解决这一质疑： - 定义积分核：推导积分核$mathcal{K}_{gamma_B}^{ab}(x')$，该核可将边界能量密度涨落$delta langle T_{ab} angle$定量映射为体时空度规微扰$h_{mu u}$。 - 辅助系统建模：通过构造带辅助系统的玩具哈密顿量（toy Hamiltonian），我们明确证明，该信号传输协议可通过一系列局域操作、幺正耦合以及弱测量/反馈读出在物理上实现。全局能量守恒$Delta langle H_{ ext{tot}} angle = 0$得以保持，有效将拓扑相位信号定理（TPST）置于量子场论的标准公理框架之内，同时拓展了其应用能力。 4. 纠缠发散的正则化处理 本研究最重要的理论贡献之一是对纠缠熵中紫外（UV）发散的处理。我们证明，依赖于量子态的相位$phi[ ho]$可充当“动态正则化因子”：当能量密度涨落趋近紫外极限时，全局幺正算符$U( ho)$会引发几何变换，有效“拉伸”纠缠楔（entanglement wedge），从而提供自洽的物理截断$epsilon_{ m eff}(phi[ ho])$，避免非物理奇点的出现。这将纠缠熵的正则化从人为的经验性设定转变为依赖于量子态的几何本身的涌现性质。 5. 延伸影响：纠缠楔的重塑机制 通过将拓扑相位信号定理（TPST）应用于施瓦西-反德西特（Schwarzschild-AdS）背景，我们给出了一种具体机制，可在不违背因果律的前提下实现信息在体时空的重新分布。我们的变换具有依赖于量子态的特性，这表明纠缠楔重构并非静态映射，而是一个动态过程。这为探索局域边界操作与引力内部结构之间的相互作用提供了全新的严谨途径，也为研究信息如何编码于体几何之中提供了潜在的理论框架。 本稿件目前处于正式同行评审阶段，非最终版本。 版权所有©2026 亚历克斯·德·朱塞佩（Alex De Giuseppe），保留所有权利。 本作品受版权保护。任何形式的抄袭、未经授权的复制，或未经适当引用即盗用本研究的思想、数学结果或文本，均违反学术规范、知识产权标准及普通法。 未经作者明确书面许可，不得进行商业使用、改编或创作衍生作品。 如需联系、引用、咨询合作或反馈，请致邮：[e-mail] 用于确认作品所有权的哈希文件已生成。