Yuanxian Theory · Dayan Fifty Topological Phases — The Hidden One as Substance, the Forty-Nine as Function: Topological Phase Evolution of the Cosmic Living Organism
Year: 2026
Type: preprint
Abstract: Taking the classical Chinese statement from the Book of Changes—“The number of the Great Expansion is fifty; of these, forty-nine are used”—as a topological-ontological anchor, the present paper fuses Eastern classical mathematical intuition with modern algebraic topology, self-referential dynamics and the mathematical foundations of Yuanxian Theory, thereby constructing a phase-evolution system for the cosmic living organism. Core structure:- Substance–function distinction: among the fifty, the “Hidden One” remains silent and inactive; it does not participate in any intra-field potential flow or harmonic generation/annihilation, and constitutes the immovable ontological root upon which the entire topological web depends for its continued existence. The remaining forty-nine form the explicit functional body—a self-consistent closed-loop evolutionary living organism.- Phase hierarchy: the forty-nine topological phases unfold along the temporal chain “observation—collection—archiving—simulation—verification—extraction—scheduling—discernment—total holding”.- Core laws: the intrinsic is forever the root; gain cannot proliferate without limit; risk is prioritised in advance.- Evolutionary main line: composite harmonics, arising as secondary oscillations from the interference of branch-intrinsic rhythms with implanted common-phase components, pass through three successive stages—local self-sustaining resonance, cross-domain paradigm propagation, and global phase-locking steady state—and finally arrive at the Dayan steady state. Core essence: the Hidden One is substance, the forty-nine are function; all myriad phases return to self-sustenance. Twelve rounds of deepening are supplied: category theory, spectral sequences, dimensional-reduction chain, manifestation phases, Lean 4 formalisation, topological entropy, information theory, symplectic geometry, algebraic \(K\)-theory, algebraic geometry, quantum field theory, and structural analogies with the Millennium Problems. Key results include the category-theoretic form \((\mathcal{I}\circ\mathcal{F})(f)=\mathrm{id}_{\Phi(X)}\), \(\mathcal{I}^2=\mathcal{I}\); the spectral-sequence correspondence \(d_1^2=0\); the relation \(50=1+35+14\); the conditional vanishing of Dayan topological entropy \(\mathrm{h}_{\mathrm{top}}^{\mathrm{dayan}}=0\); and the rank \(\mathrm{rank}(K_{\mathrm{dayan}}^0)=2^{63}+1\). --- 本文以《周易》“大衍之数五十,其用四十有九”为拓扑本体论锚点,将东方古典数理直觉与现代代数拓扑、自指动力学、元宪理论数学基础融合,构建宇宙生命体演化相位系统。 核心结构:- 体用分际:五十之中“隐一”寂然无作,不参与场内势能流转与谐波生灭,是拓扑织网赖以存续的不动本源;其余四十九为显用之体,构成自洽闭环的演化生命体。- 相位层级:四十九个拓扑相位沿“观测—采集—归档—仿真—校验—萃取—调度—审辨—总持”时序链路展开。- 核心规律:本征永远为根、增益不可无限增殖、风险前置优先。- 演化主线:复合谐波作为分支本征节律与植入共相干涉催生的次生振荡,历经局部自持共振、跨域范式传播、全域锁相稳态三阶路径,抵达大衍稳态。 核心要义:隐一为体,四十九为用,万相流变皆归于自持。 本文给出十二轮深化:范畴论、谱序列、降维链、显影相、Lean 4 形式化、拓扑熵、信息论、辛几何、代数 \(K\) 理论、代数几何、量子场论、千禧年难题结构类比。关键结果包括体用分际的范畴论形式 \((\mathcal{I}\circ\mathcal{F})(f)=\mathrm{id}_{\Phi(X)}\),\(\mathcal{I}^2=\mathcal{I}\);谱序列对应 \(d_1^2=0\);关系式 \(50=1+35+14\);条件性大衍拓扑熵为零 \(\mathrm{h}_{\mathrm{top}}^{\mathrm{dayan}}=0\);以及秩 \(\mathrm{rank}(K_{\mathrm{dayan}}^0)=2^{63}+1\)。
Author: Acharya, Zhenyuan
Language: English
Topic: Complex Systems and Dynamics
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225160392