Bridging Mathematics and Physics: Modified Kuramoto Model and Phase Formulation of Topological Quantum Entanglement for Global Dynamics in Starling Murmurations and Mycorrhizal Networks / 數學與物理的架橋：修改版庫拉莫托模型與拓撲量子糾纏的相位公式化——為萬鳥同步與菌根網路全域動態提供之機制

This paper introduces a rigorous mathematical and physical bridging framework, designated as Paper III_004, to resolve the spatio-temporal coherence anomalies observed in macroscopic biological networks, specifically starling murmurations and mycorrhizal systems. Classical synchronization paradigms fail to explain how decentralized entities achieve instantaneous, long-range phase-locking across extensive spatial scales without dissipative transmission delays. By elevating the classical Kuramoto model onto a complex topological manifold, we introduce a non-local topological gauge potential A_ij explicitly bound to the First Chern Number C via the path integral of the Berry connection. The derived exact non-local phase difference formula demonstrates that localized natural frequency variations are instantaneously absorbed by geometric phase compensations under topological protection. This mathematical architecture provides a deterministic, quantum-like quantization mechanism for ecosphere global dynamics, establishing an epistemological convergence with the Yogacara Buddhist doctrine of the Alayavijnana, wherein spatial distance is revealed as an illusory projection of the "manifested division" (Xiang-Fen) coordinated seamlessly by the unified "perceiving division" (Jian-Fen). 本論文提出了一個嚴謹的數學與物理架橋框架（編號 III_004），用以解決在巨觀生物網路（特別是萬鳥同步與菌根系統）中觀察到的時空相干性異常。古典同步範式無法解釋去中心化實體如何在廣闊的空間尺度上實現瞬時、長程的相位鎖定，而免於耗散性傳輸延遲。藉由將古典庫拉莫托模型提升至複數拓撲流形上，我們引入了一個非局域拓撲規範勢 A_ij，該規範勢透過貝里聯絡的路徑積分，與第一陳數 C 進行顯式綁定。導出的精確非局域相位差公式證明了，局域的內稟頻率變異在拓撲保護下，會瞬間被幾何相位補償所吸收。這一數學架構為生態圈全域動態提供了一種決定性的、類量子的量化機制，並建立了與佛教唯識學阿賴耶識教義的認識論會通——其中空間距離被揭示為「相分」的虛幻投影，由統一的「見分」進行無縫協調。