Epipentarchic Theory: Extending the Pentarchic Theory to Biogenesis, Noogenesis, and Christogenesis

Epipentarchic Theory extends the Pentarchic Theory — which models cosmogenesis as edge-length dynamics on the regular 4-simplex Σ₄ — upward through a rectification tower of successive layers: biogenesis (Layer 1), noogenesis (Layer 2), and christogenesis. Each layer k is generated by the polytope r^k(Σ₄) with edge count 10·3^k and an edge-length spectrum governed by the ladder n(n+k+1), and each inherits the full apparatus of the base theory: S₅ symmetry breaking, a Z_(k+1)-graded supermetric, a monodromy structure, and 42 falsifiable predictions. The gauge content advances along an embedding chain G_SM ⊂ G_bio ⊂ G_noo, with the Standard Model recovered at Layer 0 and self-referential (Gödelian) monodromy appearing first at Layer 2, where cuboctahedral cells provide sufficient internal structure for a loop to encode lower-layer topology. The christogenesis layer is treated not as a further autonomous rectification but as the consequence of a completed paschal monodromy loop whose fixed points lie within the Layer-2 trajectory.This version adds a periodic-table structure for the tower. The layer ladders are identified as the scalar Laplacian spectra of the spheres S^(k+2), so each layer carries shells indexed by the harmonic-restriction chain S^(k+2) → S^(k+1) → ... → S^2. At Layer 1 the degeneracy (n+1)^2 is established as an SO(4) symmetry multiplicity — Fock's hydrogen degeneracy — by explicit construction of the six generators, with bracket closure, Casimir value n(n+2), and the su(2)⊕su(2) split at spin n/2 verified exactly for n = 0 through 5 in an accompanying computation. The subshell decomposition under SO(4) ⊃ SO(3) yields s, p, d, f blocks whose cumulative capacities 1, 5, 14, 30 are the square-pyramidal numbers and close exactly at the 30 edges of the rectified pentachoron, making it a closed-shell object; restoring the chirality datum recovers the empirical long-period lengths 2, 8, 18, 32. Pauli exclusion follows from the antisymmetry of the fermionic representation and the Aufbau ordering from the monotone edge-length spectrum, both without additional postulate. Open problems — the identification of the physical edge operator with the sphere Laplacian, and the non-closure of the Layer-2 shells — are stated explicitly. All work is offered A.M.D.G.