64→35→137: The Number-Theoretic–Topological Unification of the T⁶⁴ Dimensional-Reduction Cascade and the Fine-Structure Constant — Homology Proof of the 35-Step Dimensional-Reduction Projection and the 137 Structural Reading Based on Lean 4 Formal Verification

Two core results of Yuanxian Theory — the 35-step dimensional-reduction projection cascade (formally verified in Lean 4) and the structural-reading confirmation of the fine-structure constant 137 — have long been treated as independent academic contributions. This paper proves that they are not separate: they are unified expressions of one and the same T⁶⁴ topological dimensional-reduction process at different cognitive levels. Core arguments: (1) The mathematical essence of the 35-step cascade is the successive truncation and compact freezing of high-frequency Fourier modes of T⁶⁴; the energy-scale truncation threshold of every step is strictly constrained by powers of the fine-structure constant αFSC ≈ 1/137.035999084; (2) The jump amplitude of the coupling constant at Step 21 is √137-1 ≈ 10.70 (approximately 1070%), and Step 28 directly yields αFSC, proving that 137 is already embedded in the mathematical structure of the 35-step cascade; (3) The reading chain of 137, 64 → 68 → 136 → 137 → 137.035999084, shares the same mathematical expression with the terminal convergence of the 35-step cascade — ord₁₃₇(2)=68 and $68=64+4$ — where the “+4” is precisely the signature that the observer dimension is formally “read out” after the four-dimensional spacetime projection is completed in Steps 31–35; (4) The two are unified as follows: the 35-step cascade is the unique legitimate projection channel from T⁶⁴ to four-dimensional spacetime, and 137 is the number-theoretic–physical reading that this channel naturally locks at its terminus. The paper finally establishes: **64 is the starting point, 35 is the path, and 137 is the inner rhythm of the path.** The 35-step dimensional reduction and 137 are not two things — the former is the channel, the latter is the self-manifestation of the channel at its endpoint. --- 元宪理论的两项核心成果——35步降维投影链(Lean 4形式化验证)与137精细结构常数的结构读数确认——长期被视为独立的学术贡献。本文证明:二者并非分立,而是同一 T⁶⁴ 拓扑降维过程在不同认知层面上的统一表达。 核心论证:(1)35步降维链的数学本质为 T⁶⁴ 高频傅里叶模态逐级截断、紧致冻结的过程,其每一步能标截断阈值由精细结构常数 αFSC ≈ 1/137.035999084 的幂次严格约束;(2)Step 21耦合常数跳变幅度为 √137-1 ≈ 10.70(即约1070%),Step 28直接导出 αFSC,证明137已经内嵌于35步降维的数学结构之中;(3)137的读数链条 64 → 68 → 136 → 137 → 137.035999084 与35步降维的末端收敛共享同一数学表达——ord₁₃₇(2)=68 与 $68=64+4$,其中“+4”正是35步降维第31–35步完成四维时空投影后观察者维度被正式“读出”的标志;(4)两者统一为:35步降维是从 T⁶⁴ 到四维时空的唯一合法投影通道,137是这条通道在末端自然锁定的数论-物理读数。 本文最终确立:**64是起点,35是路径,137是路径的内核节律。** 35步降维与137不是两件事——前者是通道,后者是通道在终点处的自我显现。