The Priority-Axis Z-Algebra: A Unified Framework of Axioms, Structure Theory, and Intrinsic Differential Geometry

This paper establishes the complete axiomatic foundation and intrinsic differential geometry of the priority-axis Z-algebra, a real 16-dimensional semisimple associative algebra defined by seven axioms Z1–Z7, generated by quaternionic units i, j, k and a hyperbolic time axis τ. The Wedderburn theorem yields Z ≅ M₂(C) ⊕ M₂(C); a bootstrap theorem shows that the priority axis is necessarily forced by algebraic self-consistency. Although Z ≅ Cl(3) as a bare algebra, the novelty lies in the priority-axis presentation and its structure theory. The axioms are mutually independent, with {Z1, Z2, Z4, Z5} a minimal core; the centre, the chiral Peirce sectors and the frozen–active dichotomy are completely determined; the derivation algebra Der(Z) ≅ sl(2,C)ᵣ ⊕ sl(2,C)ᵣ, closing open problem O.23; fibre-bundle impossibility and non-embeddability into Cl(1,3) delimit the external boundary. The intrinsic derivation Dτ = ad_τ obeys the spectral rigidity Dτ²|_Im = 4·id and admits closed-form integration and a propagator; torsion-freeness uniquely fixes the connection ∇_WY = ½[W, Y], and the curvature is purely algebraic, R(X, Y)Z = −¼[[X, Y], Z]; the physical tangent space is a three-dimensional Einstein space of constant curvature K ≡ −1 (Ric = −2η). All 112 propositions (77 algebraic, 35 differential-geometric) pass exact symbolic verification with strictly zero residuals.