优先轴Z代数:公理体系、结构理论与内禀微分几何的统一框架

本文给出优先轴Z代数的完整公理化基础与内禀微分几何。在七条公理(Z1–Z7)定义的实16维半单结合代数框架内,Wedderburn定理给出𝒵≅M₂(ℂ)⊕M₂(ℂ),自举定理表明优先轴是代数自洽性的强制输出;作为裸代数𝒵≅ℂl(3)(复Clifford代数),本文贡献在于优先轴表示及其结构理论与几何构造。结果表明:公理相互独立,{Z1,Z2,Z4,Z5}极小;中心、Peirce手征扇区、冻活二分完全确定;Der(𝒵)≅𝔰𝔩(2,ℂ)ℝ⊕𝔰𝔩(2,ℂ)ℝ关闭开放问题O.23;纤维丛不可能性与Cl(1,3)不可嵌入性划定外部边界;内禀导子Dτ=adτ满足谱刚性Dτ²|Im=4·id,积分与传播子有闭式;无挠条件唯一锁定代数联络∇WY=½[W,Y],曲率具纯代数显式,物理切空间为K≡−1的三维Einstein空间(Ric=−2η、R=−6)。验证限于标准表示,物理诠释属后续任务;112项命题(77+35)经精确符号验证独立复验通过(残差严格为零),Z代数由此获得从公理到几何的完整自洽基础。 This paper presents the complete axiomatic foundations and the intrinsic differential geometry of the priority-axis Z-algebra. Within the framework of a real 16-dimensional semisimple associative algebra defined by seven axioms (Z1–Z7), the Wedderburn theorem yields Z ≅ M₂(C) ⊕ M₂(C), and a bootstrap theorem shows that the priority axis is a forced output of algebraic self-consistency; as a bare real associative algebra, Z ≅ Cl(3) (the complex Clifford algebra), and the contribution of this work lies in the priority-axis presentation together with its structure theory and geometric construction. The results show that the seven axioms are mutually independent with {Z1, Z2, Z4, Z5} minimal; the centre, chiral Peirce sectors and the frozen-active dichotomy are completely determined; Der(Z) ≅ sl(2, C)R ⊕ sl(2, C)R closes open problem O.23; fibre-bundle impossibility and non-embeddability into Cl(1, 3) delimit the external boundary; the intrinsic derivation Dτ = adτ obeys the spectral-rigidity constraint Dτ²|Im = 4 · id, with closed-form integration and propagator; torsion-freeness uniquely fixes the algebraic connection ∇WY = ½[W, Y], the curvature admits the purely algebraic expression R(X, Y)Z = −¼[[X, Y], Z], and the physical tangent space is a three-dimensional Einstein space of constant curvature K ≡ −1 (Ric = −2η, R = −6). Verification is confined to the standard representation, and physical interpretation is deferred to subsequent volumes; all 112 propositions (77 in the algebraic part and 35 in the differential-geometric part) pass independent re-verification by exact symbolic computation (strictly zero residual, with no floating-point tolerance and no random seed), so the Z-algebra obtains a complete and self-consistent foundation from axioms to geometry.