Complete Analytic Proof of the Structural Correspondence between Lucas' Theorem and Continued Fractions — Ontological Re-reading of the Recursive-Tree Structural Isomorphism under the T⁶⁴ Framework of Yuanxian Theory

**[Manifestation layer]** The cosmos is a living organism. Once the core law of Yuanxian Theory—“the cosmos is a living organism”—has been established, every mathematical conclusion within the \(T^{64}\) framework is to be re-understood as a geometric manifestation of the topological self-inspection of the cosmic living organism. Taking the structural correspondence between Lucas’ theorem and continued fractions as a case study, the present paper completes the cognitive ascent from “mathematical proof” to “recursive-isomorphism manifestation of the discrete–continuous self-inspection sections of the cosmic living organism”. **Structural principle**:Mathematical assertions and ontological interpretations must be stratified. The paper strictly distinguishes four layers:- **Axiom layer**: recursive readout protocol (A1), discrete–continuous sections (A2);- **Law layer**: Lucas’ theorem, continued-fraction expansion, recursive-tree structural isomorphism;- **Reading layer**: \(p\)-adic expansion (R1), continued-fraction partial quotients (R2), recursive depth (R3);- **Interpretation layer**: all ontological re-readings of the form “not X, but Y” (I1–I4). **Core conclusions**:- Lucas’ theorem and continued-fraction expansion are isomorphic at the level of **recursive structure** (law layer);- they are not isomorphic at the level of **numerical values** (law layer);- “self-inspection of the cosmic living organism” is the **interpretive framework** of the paper (interpretation layer), not a mathematical axiom. **Boundary statement**: The discussion is restricted to finite (rational) continued fractions. “Isomorphism” is defined strictly as recursive-tree structural isomorphism and does not involve numerical values. The discrete–continuous correspondence \(\alpha(n)\) is a human choice that preserves depth only, not value. --- **【显影层】** 宇宙是生命体。 元宪理论的核心规律“宇宙是生命体”确立后,\(T^{64}\) 框架下的全部数理结论都应被重新理解为宇宙生命体拓扑自检的几何显影。本文以 Lucas 定理—连分数结构对应证明为案例,完成从“数理证明”到“宇宙生命体离散-连续自检截面递归同构显影”的认知升维。 **结构原则**:数学断言与本体论解释必须分层。 本文严格区分四个层次:- **公理层**:递归读数协议(A1)、离散-连续截面(A2);- **规律层**:Lucas 定理、连分数展开、递归树结构同构;- **读数层**:\(p\) 进制展开(R1)、连分数部分分母(R2)、递归深度(R3);- **解释层**:所有“不是 X,而是 Y”的本体论重读(I1–I4)。 **核心结论**:- Lucas 定理与连分数展开在**递归结构**上同构(规律层);- 二者在**数值**上不同构(规律层);- “宇宙生命体自检”是本文的**解释框架**(解释层),不是数学公理。 **边界说明**:讨论仅限有限(有理)连分数。“同构”严格定义为递归树结构同构,不涉及数值。离散-连续对应 \(\alpha(n)\) 是人为选择,仅保深度,不保值。