From a Single 11×11 Metric to the Standard Model and Einstein-Maxwell Spacetime: Unipotent Block-Cholesky Geometry of the 11 = 4 + 7 Flag Manifold

Standard Kaluza-Klein and grand unified theories face two persistent structural obstacles. In Kaluza-Klein reductions on a product manifold, off-diagonal gauge connections contaminate the metric determinant with non-minimal radion scalar couplings, whereas monolithic grand unified groups (SU(5), SO(10)) introduce leptoquark gauge bosons that mediate unobserved proton decay. Maintaining strict epistemic transparency between deductively proven mathematical theorems (Tier A) and physical model postulates (Tier B), we resolve both obstacles on a nested 11-dimensional flag manifold M_11 = Z_4 × F_7 composed of five hierarchical stages Ω_0 ⊕ Z_1 ⊕ Z_2 ⊕ Z_3 ⊕ Z_4 of dimensions 1 + 1 + 2 + 3 + 4 = 11, bounded by Ehrenfest orbital stability in d = 3 spatial dimensions. First, we prove via a unipotent Block-Cholesky factorization G_11 = L^T B L (det L ≡ 1) that the T_11 = 66 components of a symmetric 11 × 11 metric split uniquely into 21 block-diagonal stage metrics (B) and 45 off-diagonal Ehresmann connectors (L), factoring the invariant volume measure √|det G_11| = √|det B| without radion determinant mixing. Second, combining the triangular-number identity T_k + T_{k-1} = k^2 with the Jauch-Hill-Schwinger oscillator theorem on compact Euclidean fibers Z_1, Z_2, Z_3, we derive the exact Standard Model gauge algebra u(1)_Y ⊕ su(2)_L ⊕ su(3)_C (1 + 3 + 8 = 12 generators), while Lorentzian signature on Z_4 decouples 4^2 = 16 into T_4 = 10 Einstein metric components and T_3 = 6 Maxwell/Lorentz components. Finally, we prove topologically why parity violation is restricted to Z_2 (det(-I_2) = +1), bypassing Witten's 1981 Riemannian no-go theorem via symplectic Spin^c / Dolbeault phase-space chirality and identify the electroweak Higgs doublet with the 1 × 2 inter-stage connector A^(1,2). ___________________________________________________________________________________________________________ Context within the TPD Research Program:This standalone paper develops and expands the differential-geometric and gauge-algebraic matrix architecture first archived in Chapter 6 of the author's foundational monograph, "Theory of Perspectival Dimensionality (TPD)" (Illustrated English Edition 2.0, DOI: 10.5281/zenodo.23902867; Foundational German Edition DOI: 10.5281/zenodo.23770569), and complements two companion studies on biophysical Kuramoto synchronization in the oncological Warburg transition (DOI: 10.5281/zenodo.23920048) and sub-femtosecond quantum measurement resolution (DOI: 10.5281/zenodo.23925732).