SU(32) Stepwise Breaking as a Yuanxian Dynamical Closed Loop: The Ontological Dimensional-Reduction Path from T⁶⁴ Topological Root Filling to the Standard-Model Gauge Group — Parameter-Free Symmetry Breaking Based on the 64-35-137 Core Number-Theoret

Within the $\mathcal{T}^{64}$ topological ontological framework of Yuanxian Theory, this paper systematically analyses the stepwise symmetry-breaking path of gauge groups and proves that the path is fully embedded in the 35-step standardised iteration of the Yuanxian dimensional-reduction dynamical closed loop, introducing no free parameters external to the system at any stage. The following core conclusions are established: (1) the rank 31 of $\mathrm{SU}(32)$ precisely matches the antecedent symmetry structure after compactification of the 60-dimensional kernel space, realises a half-filling of the 64 root nodes of $\mathcal{T}^{64}$ topology, and completes the initial arrangement of 32 roots; (2) $\mathrm{SU}(32)$ breaks stepwise via maximal-subgroup embeddings, every step satisfying the constraints of rank conservation and representation-dimension conservation; (3) the 24 generators of the flavour symmetry group $\mathrm{SU}(5)_{\mathrm{flavor}}$ are entirely absorbed by the gauge system during dimensional reduction; the residual cokernel structure left in the fermion representation space — the tensor product of the two-dimensional irreducible representation of the permutation group $S_{3}$ — naturally yields three generations of fermions; (4) the three stages of breaking correspond respectively to step 28, step 32 and the final three steps of the 35-step dimensional-reduction iteration, each corresponding to a phase-calibration node of the 68-step discrete self-inspection closed loop. --- 本文在元宪 $\mathcal{T}^{64}$ 拓扑本体论框架下,系统解析规范群逐级对称性破缺路径,证明该路径完全嵌入元宪降维动力学闭环的35步标准化迭代中,全程不引入任何体系外自由参数。 本文确立以下核心结论:(1)$\mathrm{SU}(32)$ 的秩31恰好适配60维核空间紧致化后的前序对称结构,与 $\mathcal{T}^{64}$ 拓扑的64个根系节点实现半充填,完成32个根系的初始排布;(2)$\mathrm{SU}(32)$ 经由极大子群嵌入逐级破缺,每一步均满足秩守恒与表示维数守恒约束;(3)$\mathrm{SU}(5)_{\mathrm{flavor}}$ 味对称群的24个生成元在降维过程中全部被规范体系吸收,其在费米子表示空间留下的余核结构——置换群 $S_{3}$ 的二维不可约表示的张量积——自然产生3代费米子;(4)三级破缺分别对应35步降维迭代的第28步、第32步和最后3步,每一步都对应68步离散自检闭环的一个相位校准节点。