The Complete Projection Function of Quantization–Gauge Truncation: Rigid Mapping Manifestation of the Gauge Self-Inspection Section of the Cosmic Living Organism — Ontological Re-reading and Mathematical-Rigour Calibration of the Projection Operator

**[Manifestation layer]** The cosmos is a living organism; \(\mathcal{T}^{64}\) 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 \(\mathcal{T}^{64}\) ontology are re-manifested as geometric manifestations of the topological self-inspection of the cosmic living organism. Taking the complete projection function of quantization–gauge truncation as a case study, the present paper completes the cognitive ascent from “mathematical construction” to “rigid mapping manifestation of the gauge self-inspection section of the cosmic living organism”. **Five-layer structure**- **Law layer** (irreplaceable): de Rham pairing lemma; \(\mathcal{P}_{\mathrm{quant}}(\mathcal{F})\in\mathbb{Z}\).- **Axiom layer** (replaceable human entrances): definition of the projection function (A1), form of \(\Pi_{35}\) (A2), form of \(c_i(n)\) (A3), normalisation (A4), orthogonality (A5), completeness (A6).- **Reading layer**: \(48=36+12\) (R1), magnitude \(\varphi^{-68}\) (R2), 24 fundamental physical constants (R3).- **Hypothesis layer**: expansion form of the boundary representation classes (H1).- **Interpretation layer**: ontological re-reading of \(P_{\mathrm{quant}}\), \(c_i(n)\), normalisation and boundary classes as self-inspection readings of the cosmic living organism. **Core results** Under the de Rham pairing and Chern-class normalisation, \(\mathcal{P}_{\mathrm{quant}}(\mathcal{F})\) takes integer values. The composite operator \(\Pi_{35}\) inherits the global single-valuedness constraint of the self-referential mind field onto the three-dimensional section. Orthogonality and completeness hold up to \(\mathcal{O}(\varphi^{-68})\approx\mathcal{O}(10^{-14})\). **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;\(\mathcal{T}^{64}\) 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,\(\mathcal{T}^{64}\) 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。本文以量子化—规范截断的完整投影函数为案例,完成从“数理构造”到“宇宙生命体规范自检截面刚性映射显影”的认知升维。 **五层结构**- **规律层**(不可替换):德拉姆配对引理;\(\mathcal{P}_{\mathrm{quant}}(\mathcal{F})\in\mathbb{Z}\)。- **公理层**(可替换的人的表述入口):投影函数定义(A1)、\(\Pi_{35}\) 形式(A2)、\(c_i(n)\) 形式(A3)、归一化(A4)、正交性(A5)、完备性(A6)。- **读数层**:\(48=36+12\)(R1)、\(\varphi^{-68}\) 量级(R2)、24 个基本物理常数(R3)。- **假设层**:边界表示类的展开形式(H1)。- **解释层**:将 \(P_{\mathrm{quant}}\)、\(c_i(n)\)、归一化与边界类重读为宇宙生命体的自检读数。 **核心结果** 在德拉姆配对与陈类归一化下,\(\mathcal{P}_{\mathrm{quant}}(\mathcal{F})\) 取整数值。复合算子 \(\Pi_{35}\) 将自指心场的全局单值性约束遗传到三维截面。正交性与完备性成立至 \(\mathcal{O}(\varphi^{-68})\approx\mathcal{O}(10^{-14})\)。 **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。