Grand Unification from RP⁴ Topology: SO(10) as the Geometric GUT Group of the Inverted Hypersphere Cosmology

We derive Grand Unified Theory (GUT) gauge group SO(10) as a mathematical consequence of the RP4 = S4/Z2 topology of the Inverted Hypersphere Cosmology (IHC) framework, with no additional assumptions. The derivation proceeds through five steps: (i) the golden ratio phi = (1+sqrt(5))/2 that governs the IHC shell hierarchy encodes the 5-dimensional ambient geometry of S4 in its very definition, via sqrt(5) = phi + phi^-1; (ii) the isometry group of S4 is SO(5); (iii) the Z2 antipodal identification x ~ -x of RP4 maps one SO(5) orbit to its antipodal image, generating a second independent SO(5) factor; (iv) SO(5)xSO(5) is a maximal subgroup of SO(10); (v) SO(8) triality selects the spinor embedding 16 = 8s + 8c that accommodates one complete Standard Model generation per SO(10) representation. The GUT scale emerges from the formula k_GUT = M x 24 + 8 = 11 x 24 + 8 = 272, giving E_GUT = 1.005 x 10^15 GeV, consistent with the standard SO(10)/SU(5) GUT scale. The complete SO(10) to Standard Model symmetry breaking chain is derived geometrically: SO(10) at k=272 breaks to SO(5)xSO(5) at k_PS = M(N_co+1) = 11x23 = 253, giving an intermediate scale E_PS ~ 1.1 x 10^11 GeV consistent with the Pati-Salam range. The Z3 shell classes of IHC are identified with the three SO(8) triality representations, assigning gauge bosons, left-handed and right-handed fermions without additional input. The total number of broken generators is 45 - 12 = 33 = N, the IHC shell count. The Weinberg angle running from GUT to electroweak scale satisfies sin^2(theta_W)(EW) = sin^2(theta_W)(GUT) x phi^-1, an exact identity following from phi^2 = phi + 1. Three additional results follow: the 24-cell provides exactly three SM generations as three Z8 cycles of 8 fermion states; a geometric mechanism for the hierarchy problem without supersymmetry; and the type-I seesaw gives m_nu_tau = m_t^2 / E_GUT = 0.030 eV, consistent with atmospheric neutrino oscillations.