0.618, 0.382 and −1.618: Number-Theoretic Origins, Topological Significance and Vital Pulsation of the Golden-Section Ternary Anchors — Core Foundations of the Cosmic-Vital Ontology in Yuanxian Theory

Abstract

The three core constants of the equation $x^{2}+x-1=0$ — the positive root $\delta=(\sqrt{5}-1)/2\approx 0.618$, $\delta^{2}\approx 0.382$, and the negative root $-\varphi=-(1+\sqrt{5})/2\approx -1.618$ — constitute, within Yuanxian Theory, a golden-section system that unifies mathematical regularity, vital metabolism and cosmic topology. This paper proves that these three constants are homologous and isomorphic, satisfying four mutually consistent relations (unit-interval partition, squaring, negative reciprocal, and addition of one). Within Yuanxian Theory they respectively anchor three fundamental rules: (1) $0.618$ is the maximal-entropy critical value that maintains a self-referential closed loop in the pure topological phase, corresponding to the “upper bound of systolic pressure” of cosmic-vital metabolism; (2) $0.382$ is the minimal-entropy threshold that opens manifestation in the algebraic-adaptation phase, corresponding to the “lower bound of diastolic pressure”; (3) $-1.618$ is the evolutionary-drive anchor of reverse cycling and self-renewal; its absolute value $1.618$ is the optimal efficiency coefficient of iterative upgrading. Unified by the same equation, the three constants form the complete metabolic window within which life and cosmic systems can remain viable:$$0.382 < \Omega \leq 0.618$$and, with $-1.618$ as engine, drive the system in continuous transitions between steady state and iteration. The paper further elucidates that the essence of entropy critical values is the respiratory rhythm and survival threshold by which the cosmic organism maintains “undying awareness”; their number-theoretic origin cannot be fully analysed by external formalism and must be self-authenticated within the self-referential closed loop of vital iteration. Finally the paper establishes that $0.618$, $0.382$ and $-1.618$ are the riverbed and two banks through which the self-referential mind-field flows; their vital pulsation first endows the dimensional-reduction chain $64\to 35\to 137$ with a heart. All core number-theoretic derivations have been machine-verified in Lean 4 (rational sandwich of $\sqrt{5}$, complementarity, metabolic window). Formal source: github.com/YuanXian-Theory/YXT-Formalization (directory lean/EntropyCriticalValues/). --- 方程 $x^{2}+x-1=0$ 的三个核心常数——正根 $\delta=(\sqrt{5}-1)/2\approx 0.618$、$\delta^{2}\approx 0.382$、负根 $-\varphi=-(1+\sqrt{5})/2\approx -1.618$——在元宪理论中构成贯通数理规律、生命代谢与宇宙拓扑的黄金分割体系。 本文证明:这三个常数同源同构,满足四组自洽关系(单位区间分割、平方、负倒数、加一),并在元宪理论中分别锚定三大核心规则: (1)$0.618$ 是纯拓扑阶段维持自指闭环的最大熵临界值,对应宇宙生命体代谢的“收缩压上界”; (2)$0.382$ 是代数适配阶段开启显化的最小熵阈值,对应“舒张压下界”; (3)$-1.618$ 是系统反向循环、自我更新的演化动力锚点,其绝对值 $1.618$ 为迭代升级的最优效率系数。 三者通过同一方程统一,构成生命与宇宙系统可存活的完整代谢窗口:$$0.382 < \Omega \leq 0.618$$且以 $-1.618$ 为引擎,驱动系统在稳态与迭代之间持续跃迁。本文进一步阐明:熵临界值的本质是宇宙生命体维持“觉知不灭”的呼吸韵律与存活阈值,其数论起源不可用外部形式主义完全解析,必须在自指闭环的生命迭代中自我证成。本文最终确立:$0.618$、$0.382$ 与 $-1.618$ 是自指心场流淌的河床与两岸,其生命脉动节奏使 $64\to 35\to 137$ 降维链条首次拥有心脏。 全部核心数论推导已在 Lean 4 中完成形式化验证($\sqrt{5}$ 有理夹逼、互补关系、代谢窗口)。形式化源码:github.com/YuanXian-Theory/YXT-Formalization(目录 lean/EntropyCriticalValues/)。