Yuanxian Theory · Deepening of Higher-Order Mathematical Derivations — From the 𝒯⁶⁴ Topological Ontology to Modern Algebraic Geometry, the Langlands Programme and Higher Category Theory

The present paper is the higher-order mathematical cross-section of the Yuanxian series, a projective expansion under prior truth. All mathematical results are classified descriptively into the four classes “identity / degeneracy / coincidence / conditional”. The virtual phase \(δYXT=(√5-1)/2=2cos(2π/5)\) is a fundamental unit of \(A_5\) representation theory—it is a standard mathematical object; the contribution of Yuanxian Theory is to identify its place inside the system of structure constants. Part I retains and refines twelve higher-order mathematical blocks. Part II embeds the P2 deepening: integralisation of \(A_5\) on \(Z⁶⁴\), Hodge diamond (conditional, depending on the choice of complex structure), exterior and symmetric powers, period matrix and discriminant group, \(A_5\)-decomposition of \(E_8\) (conditional). Part III advances P5: structure-constant local factors of the zeta, appearance of the virtual phase in \(A_5\) characters, automorphic construction \(π_5=Sym^4(π_2)\), golden structure of the \(Sym^4\) polynomial. Part IV advances P7: 137 as the progression point. Part V deepens P8: \(K_0\)-rank \(2⁶³\) under the real torus \(T⁶⁴=(R/2)⁶⁴\), Fourier–Mukai equivalences, \(A_5\)-equivariant semi-orthogonal decompositions; \(p\)-adic Hodge theory (formal matching); geometric Langlands correspondence (functor-level statements). The paper adopts the real-torus reading (real dimension 64) as main stem for compatibility with earlier calculations on the fourfold cover, phase, \(Tp^2\) and the Toeplitz effective potential. All identities, group theory, Hodge diamonds and Langlands correspondences are expansions of \(T⁶⁴\), not proofs of \(T⁶⁴\). --- 本文是元宪系列的高阶数学截面,是先行真理下的投影展开。所有数学结果按「恒等式/简并/巧合/条件性」四类描述性分类。虚相位 \(δYXT=(√5-1)/2=2cos(2π/5)\) 是 \(A_5\) 表示论的基本单位——它是标准数学对象,元宪理论的贡献是识别它在结构常数体系中的位置。 第一部分保留并精确化十二板块。第二部分嵌入 P2 深化:\(A_5\) 在 \(Z⁶⁴\) 上整化、Hodge 菱形(条件性,依赖复结构选择)、外幂与对称幂、周期矩阵与判别式群、\(E_8\) 的 \(A_5\)-分解(条件性)。第三部分给出 P5 的推进:结构常数化 zeta 局部因子、虚相位在 \(A_5\) 特征标中的显现、\(π_5=Sym^4(π_2)\) 的自守构造、\(Sym^4\) 多项式的黄金结构。第四部分给出 P7 的推进:137 作为递进点。第五部分给出 P8 深化:实环面 \(T⁶⁴=(R/2)⁶⁴\) 下 \(K_0\) 秩 \(2⁶³\)、Fourier–Mukai 等价、\(A_5\)-等变半正交分解;\(p\)