Group-Representation Embedding of the Combinatorial Mirror–Permutation Group and Fixed-Point Counting: Algebraic Manifestation of the Symmetry Self-Inspection Section of the Cosmic Living Organism — Ontological Re-reading and Mathematical-Rigour Calibration of the Group-Representation Embedding ρ: G_mirror → End(Π₃₅) within the Yuanxian-Theory T⁶⁴ (Five-Layer Structure: Law – Axiom – Reading – Hypothesis – Interpretation)

**[Manifestation layer]** The cosmos is a living organism; \(T⁶⁴\) is the form of the cosmic living organism. Once the ontological positioning of Yuanxian Theory—“the cosmos is a living organism”—has been established, all mathematical conclusions inside the \(T⁶⁴\) ontology are re-manifested as geometric manifestations of the topological self-inspection of the cosmic living organism. Building on the group-representation embedding of the combinatorial mirror–permutation group, the present paper completes the cognitive ascent from “mathematical construction” to “algebraic manifestation of the symmetry self-inspection section of the cosmic living organism”. **Five-layer structure**- **Law layer** (irreplaceable): Burnside’s lemma; fixed-point count \(|Fix(Gₘᵢᵣᵣₒᵣ)|=2²⁹\).- **Axiom layer** (replaceable human entrances): definition of \(Gₘᵢᵣᵣₒᵣ\) (A1), fixed-point-free condition (A2), group-representation embedding \(ρ:Gₘᵢᵣᵣₒᵣ(Π₃₅)\) (A3), 35-step stratified group reduction (A4).- **Reading layer**: \(48=36+12\) (R1), \(2²⁹\) (R2).- **Hypothesis layer**: \(O(3)\) correspondence locked at the step-35 section (H1), origin of the 12 inversion fixed-point equivalence classes (H2).- **Interpretation layer**: ontological re-reading of \(Gₘᵢᵣᵣₒᵣ\), \(ρ\), \(ρ(R)=I\) and \(48=36+12\) as self-inspection symmetry readings of the cosmic living organism. **Core results** Under the fixed-point-free condition and 35-step stratified reduction, successive halving yields \(|Fix(Gₘᵢᵣᵣₒᵣ)|=2⁶⁴/2³⁵=2²⁹\). The count \(48=36+12\) records the complete closed-chain class total on the step-35 fixed-point section (36 combinatorial + 12 inversion classes). Burnside’s lemma is retained only for contrast: it counts orbits, not the global fixed-point set. **Criterion**: Laws are the terrain; axioms are the entrance. The ontological positioning does not enter an external-confirmation framework; its criterion is “read out / not read out”. Manifestation precedes; confirmation follows. --- **【显影层】** 宇宙是生命体;\(T⁶⁴\) 是宇宙生命体的形态。 元宪理论的本体论定位「宇宙是生命体」确立后,\(T⁶⁴\) 本体论中的全部数理结论都应被重新显影为宇宙生命体拓扑自检的几何显影。本文以组合镜像—置换群的群表示嵌入为基础,完成从“数理构造”到“宇宙生命体对称自检截面代数显影”的认知升维。 **五层结构**- **规律层**(不可替换):Burnside 引理;冻结点计数 \(|Fix(Gₘᵢᵣᵣₒᵣ)|=2²⁹\)。- **公理层**(可替换的人的表述入口):\(Gₘᵢᵣᵣₒᵣ\) 定义(A1)、无固定点条件(A2)、群表示嵌入 \(ρ:Gₘᵢᵣᵣₒᵣ(Π₃₅)\)(A3)、35 步分层群约化(A4)。- **读数层**:\(48=36+12\)(R1)、\(2²⁹\)(R2)。- **假设层**:第 35 步截面锁定的 \(O(3)\) 对应(H1)、12 个反演不动点等价类的来源(H2)。- **解释层**:将 \(Gₘᵢᵣᵣₒᵣ\)、\(ρ\)、\(ρ(R)=I\) 与 \(48=36+12\) 重读为宇宙生命体的自检对称性读数。 **核心结果** 在无固定点条件与 35 步分层约化下,逐次减半得 \(|Fix(Gₘᵢᵣᵣₒᵣ)|=2⁶⁴/2³⁵=2²⁹\)。计数 \(48=36+12\) 记录第 35 步冻结点截面上的完整闭链类总数(36 个组合类 + 12 个反演类)。Burnside 引理仅作对照:它计算轨道数,而非全局冻结点集。 **判据**:规律是地形,公理是入口。本体定位不进入外部确认框架,其判据是「读出/未读出」。显影先行,确认在后。