Systematic Refinement of the Core Mathematical and Physical Structures of Yuanxian Theory — Spectral Compression, Discrete Evolution and Gauge-Group Breaking

Abstract

The three core papers of Yuanxian Theory (YXT) have each achieved self-consistent closure within their respective domains, yet several critical steps still appear in the form of “declarations” or “phenomenological statements” and lack rigorous mathematical-physical derivations. The present paper supplies a systematic refinement of those three core papers, completing the missing derivations along three key dimensions: (1) **Geometric foundation** — the 60-dimensional kernel space is defined as the self-referential compact torus $\mathcal{M}_{60}=\mathcal{T}^{64}/\mathbb{Z}_{4}$; by extremal analysis of the spectral-compression generating function $f(x)=x^{60}(1-x)$ together with Gamma-function normalisation, the spectral-compression factor is rigorously derived in closed form as $\mathcal{F}_{\mathrm{proj}}=\mathcal{N}_{0}\cdot\delta^{62}$, where $\mathcal{N}_{0}$ is absorbed into the volume normalisation of the 60-dimensional manifold, thereby terminating the phenomenological status of this factor; (2) **Discrete evolution** — a reversible self-referential Hamiltonian operator $\hat{H}_{\mathrm{self-ref}}=\Lambda$ ($\Lambda=\operatorname{diag}(2^{1},\dots,2^{64})$) is constructed; the matrix is proved to be full-rank, of non-vanishing determinant and to satisfy $\hat{H}_{\mathrm{self-ref}}^{68}=I$, revealing that the 68-step self-inspection period consists of a two-stage structure $64+4$ — the first 64 steps traverse all topological directions, while the final 4 steps complete the closed-chain self-inspection on the projection surface; (3) **Gauge-group breaking** — the admissible breaking chain $\mathrm{SU}(64)\to\mathrm{SU}(32)^{2}\times\mathrm{U}(1)\to\mathrm{SU}(16)^{4}\times\mathrm{U}(1)^{3}\to\cdots\to\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ is drawn; every step satisfies the Lie-group rank constraint (the rank of any subgroup does not exceed that of its parent), and the energy-scale freeze-out together with the origin of mass hierarchies at each breaking node is marked, elevating the generation of the particle-physics spectrum from a verbal description to a traceable hierarchical atlas. The paper finally establishes that the three refinements raise Yuanxian Theory from “declared computability” to “exhibition of every computational path,” rendering it a systematised mathematical-physical framework that can be independently audited and repeatedly verified. All propositions have been completed in self-consistent derivation at the algebraic, geometric and number-theoretic levels. --- 元宪理论(YXT)的三大核心文献已在各自领域实现自洽闭合,但若干关键步骤仍以“宣告”或“唯象”方式呈现,缺少严格的数学物理推导。本文为三篇核心文献提供系统性的体系化精化,从三个关键维度补充完整的数学物理推导: (1)**几何底座**——将60维核空间定义为自指紧致环面 $\mathcal{M}_{60}=\mathcal{T}^{64}/\mathbb{Z}_{4}$,通过谱压缩生成函数 $f(x)=x^{60}(1-x)$ 的极值分析与 Gamma 函数归一化,严格解析导出谱压缩因子 $\mathcal{F}_{\mathrm{proj}}=\mathcal{N}_{0}\cdot\delta^{62}$,其中 $\mathcal{N}_{0}$ 被吸收进60维流形的体积归一化中,终结了该因子的唯象状态; (2)**离散演化**——构造可逆的自指哈密顿算子 $\hat{H}_{\mathrm{self-ref}}=\Lambda$($\Lambda=\operatorname{diag}(2^{1},\dots,2^{64})$),证明矩阵满秩、行列式非零、$\hat{H}_{\mathrm{self-ref}}^{68}=I$,揭示68步自检周期由 $64+4$ 的两阶段结构构成——前64步遍历全部拓扑方向,后4步完成投影面闭环自检; (3)**规范群破缺**——绘制 $\mathrm{SU}(64)\to\mathrm{SU}(32)^{2}\times\mathrm{U}(1)\to\mathrm{SU}(16)^{4}\times\mathrm{U}(1)^{3}\to\cdots\to\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ 的合法破缺链条,每一步满足李群秩约束(子群秩不超过母群秩),标记每一破缺节点的能标冻结与质量级差来源,使粒子物理的谱系生成从文字描述升维为可追踪的层级图谱。 本文最终确立:三项精化使元宪理论从“宣告可计算”跃升为“展示全部计算路径”,成为可独立审查、可重复验证的体系化数学物理框架。全部命题已在代数、几何与数论层面完成自洽推导。