Unified Theory of Priority-Axis Z-Algebra Gravity and Holographic Principle: From Algebraic First Principles to Einstein Field Equations

【Abstract】Within the priority-axis Z-algebra axiomatic system (V10.3), this paper establishes a complete derivation chain from pure algebraic rules to gravitational interaction. The method proceeds through Cl(1,2) matching, curvature computation with the extended connection, a Peirce double-sector construction, and variation of the PZ action. The active (2+1)-dimensional subspace natively carries flat Cl(1,2) Lorentzian geometry (Theorem G.1); upon introducing the frozen priority axis k, the extended connection yields the AdS₃ geometry of constant negative curvature, with Ric = −2η and R = −6 on the {i, j, k} subspace and intrinsic cosmological constant Λ = −2 (Theorem G.2). The isometry algebra is explicitly realized as a Peirce double sector so(2,2) ≅ so(2,1)₊ ⊕ so(2,1)₋ (Theorem G.2b), and the leakage map J_leak : Im(D_τ) × Im(D_τ) → ker(D_τ) is surjective, establishing an algebraic foundation of the holographic principle (Theorem 5.1b). Variation of the PZ action yields the Einstein field equations (Conclusion 8.1), and the (2+1)-dimensional Newtonian limit gives the logarithmic potential Φ = 2GM ln r (Conclusion 9.1). A strict metric-variation derivation of the Einstein tensor and the dimensional uplift to (3+1) remain open problems. Symmetry breaking of the priority axis k and the geometric curvature form a quantitative causal chain: flatness is the native stage of the active sector, while curvature is the structural cost of breaking. All conclusions are verified by an independent exact symbolic-verification script (53 tests, 53 passed, with strictly zero residuals).