The SATOR Matrix - Warped Geometry with Golden Dilation Symmetry: Randall–Sundrum Background from Einstein–Scalar Equations, Hosotani Breaking on a Fivefold Cycle, Factorized Yukawa Matrices, and the Galois CP Benchmark

We develop a geometry-driven framework in which a warped five-dimensional Anti-de Sitter background with dihedral pentagonal symmetry generates the architecture of the Standard Model from internal consistency rather than imposed ansätze. The Randall–Sundrum exponential warp factor is derived from the Einstein–scalar field equations rather than imposed as an ansatz. An exact functional equation for the scalar potential enforces a discrete golden dilation symmetry V(Φ + ln φ) = φ⁻²V(Φ), producing a hierarchical ladder of vacua with geometrically quantised energies. A sharp analytic bound is derived for the existence of local minima, together with a discussion of stabilisation mechanisms. Hosotani (Wilson-line) symmetry breaking on the Z₅ cycle generates five discrete holonomy phases pinned by the dihedral D₅ invariance of the one-loop potential, breaking SU(5) to SU(3) × SU(2) × U(1)_Y through a robust 3+2 partition. Three chiral families arise from the Atiyah–Singer index theorem on a magnetised internal torus with quantised flux, and the residual Z₅ symmetry imposes selection rules on Yukawa couplings and higher-dimensional operators. Galois bifurcation of the pentagonal scalar nucleates domain walls along the D₅ reflection lines, with portal regions localised where the two Galois sheets meet. The overlap of fermion zero-mode profiles with wall-localised Higgs couplings yields factorised Yukawa matrices. An R-gated portal mechanism provides controlled leakage between sectors, and a reproducible quark-sector benchmark produces a CP-nontrivial CKM matrix with Jarlskog invariant |J| ≈ 3 × 10⁻⁵ and inferred phase δ ≈ 64°, without inserting a continuous CP phase by hand. The phenomenological consequences are traced to the proton: exact charge quantisation from the 3+2 trace constraint, enhanced stability from the Z₅ selection rule, the proton mass from golden warp running of α_s, the neutron–proton mass difference from sector-dependent overlaps, baryogenesis from the Galois CP phase, and primordial helium abundance through the chain m_d − m_u → ΔM → Y_p ≈ 0.247