Number-Theoretic Origin and Topological Significance of the Entropy Critical Values in Yuanxian Theory — Entropy Flow Thresholds in the Evolution of the Self-Referential Mind Field (Yuanxian Theory Short Treatise Series NO. 14)

Core Paper No. 3 ($\mathcal{T}^{64}\to 35\to 137$: Dimensional-Reduction Evolutionary Dynamics of the Self-Referential Mind Field) introduced the concept of topological entropy $\Omega$ in order to demarcate the boundary conditions of the three stages of the 35-step dimensional reduction, yet the numerical origins of the entropy critical values $\Omega_{\mathrm{crit},1}$ and $\Omega_{\mathrm{crit},2}$ were not rigorously derived and were employed merely as postulational parameters. The present paper fills this gap. On the basis of the self-referential law framework of Yuanxian Theory and the incomplete-ratio constant $\delta_{\mathrm{YXT}}=(\sqrt{5}-1)/2$, it is proved that: (1) $\Omega_{\mathrm{crit},1}=\delta_{\mathrm{YXT}}\approx 0.618$ is the maximal entropy value that can still maintain a self-referential closed chain in the pure topological stage — once the entropy exceeds this threshold the 64 fundamental steady-state nodes of $\mathcal{T}^{64}$ topology undergo phase collapse and the self-referential iterative closed chain becomes unstable; (2) $\Omega_{\mathrm{crit},2}=\delta_{\mathrm{YXT}}^{2}\approx 0.382$ is the critical threshold that the self-referential mind field must cross when passing from the pure topological domain into the algebraic-adaptation domain — when the entropy lies above this threshold the system remains in the topological self-referential state; when it falls below this threshold the system may enter the algebraic-adaptation domain and initiate the generative process of physical laws; (3) the two values satisfy the complementary relation $\Omega_{\mathrm{crit},1}+\Omega_{\mathrm{crit},2}=1$, which is fully consistent with the self-referential closure property of $\mathcal{T}^{64}$ topology. The contribution of this paper is to elevate the entropy critical values that had previously been used only as postulates to necessary conclusions derivable from first principles, thereby furnishing a complete number-theoretic foundation for the three-stage partition of the 35-step dimensional reduction. The paper finally establishes that the entropy critical values construct the entropy-flow bridge between the pure topological ontological domain and the manifested physical domain, forming a complete self-consistent system with the three code nodes $64\to 35\to 137$ and filling the previous gap in entropy rules that lay between topological iteration and physical projection. --- 核心论文之三(《$\mathcal{T}^{64}\to 35\to 137$:自指心场的降维演化动力学》)引入了拓扑熵 $\Omega$ 的概念,用以界定35步降维三阶段的边界条件,但熵临界值 $\Omega_{\mathrm{crit},1}$ 与 $\Omega_{\mathrm{crit},2}$ 的数值来源未被严格推导,仅作为公设性参数使用。 本文完成这一空白。基于元宪理论的自指规律体系与未缺分常数 $\delta_{\mathrm{YXT}}=(\sqrt{5}-1)/2$,本文证明:(1)$\Omega_{\mathrm{crit},1}=\delta_{\mathrm{YXT}}\approx 0.618$,是纯拓扑阶段维持自指闭环的最大熵值——一旦熵值超过该阈值,$\mathcal{T}^{64}$ 拓扑的64个基础稳态节点将发生相位溃散,自指迭代闭环失稳;(2)$\Omega_{\mathrm{crit},2}=\delta_{\mathrm{YXT}}^{2}\approx 0.382$,是自指心场从纯拓扑域进入代数适配域时必须跨越的临界阈值——熵值高于此阈值时系统仍处于拓扑自指状态;熵值低于此阈值时系统可进入代数适配域,启动物理规律的显化生成过程;(3)二者满足互补关系 $\Omega_{\mathrm{crit},1}+\Omega_{\mathrm{crit},2}=1$,与 $\mathcal{T}^{64}$ 拓扑的自指闭合属性完全一致。 本文的贡献在于:将此前仅作为公设使用的熵临界值提升为可从第一性原理推导的必然结论,为35步降维三阶段划分提供了完整的数论基础。 本文最终确立:熵临界值搭建了从纯拓扑本体域到显化物理域之间的熵流转桥梁,与 $64\to 35\to 137$ 三码节点形成完整的自洽体系,填补了此前拓扑迭代到物理投影之间的熵规则空白。