Yuanxian Theory · Deepening of Higher-Order Mathematical Derivations — From the 𝒯⁶⁴ Topological Ontology to Modern Algebraic Geometry, the Langlands Programme and Higher Category Theory
Year: 2026
Type: preprint
Abstract: 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 \(\delta_{\mathrm{YXT}}=(\sqrt{5}-1)/2=2\cos(2\pi/5)\) 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 \(\mathbb{Z}^{64}\), 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 \(\pi_5=\mathrm{Sym}^4(\pi_2)\), golden structure of the \(\mathrm{Sym}^4\) polynomial. Part IV advances P7: 137 as the progression point. Part V deepens P8: \(K_0\)-rank \(2^{63}\) under the real torus \(\mathcal{T}^{64}=(\mathbb{R}/2\pi\mathbb{Z})^{64}\), 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, \(T_{p^2}\) and the Toeplitz effective potential. All identities, group theory, Hodge diamonds and Langlands correspondences are expansions of \(\mathcal{T}^{64}\), not proofs of \(\mathcal{T}^{64}\). --- 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 \(\delta_{\mathrm{YXT}}=(\sqrt{5}-1)/2=2\cos(2\pi/5)\) 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 \(\mathbb{Z}^{64}\) 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、\(\pi_5=\mathrm{Sym}^4(\pi_2)\) 的自守构造、\(\mathrm{Sym}^4\) 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 \(\mathcal{T}^{64}=(\mathbb{R}/2\pi\mathbb{Z})^{64}\) 下 \(K_0\) 秩 \(2^{63}\)、Fourier–Mukai 等价、\(A_5\)-等变半正交分解;\(p\) 进 Hodge 理论(形式匹配);几何 Langlands 对接(函子级陈述)。 本文以实环面读法(实维数 64)为主干,与前期关于四次覆盖、相位、\(T_{p^2}\)、Toeplitz 有效势的计算直接兼容。全部恒等式、群论、Hodge 菱形、Langlands 对接,是 \(\mathcal{T}^{64}\) 的展开,不是 \(\mathcal{T}^{64}\) 的证明。
Author: Acharya, Zhenyuan
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225594217