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 kGUT = M x 24 + 8 = 11 x 24 + 8 = 272, giving EGUT = 1. 005 x 10¹5 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 kPS = M (Nco+1) = 11x23 = 253, giving an intermediate scale EPS ~ 1. 1 x 10¹1 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 sin² (thetaW) (EW) = sin² (thetaW) (GUT) x phi^-1, an exact identity following from phi² = 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 mₙuₜau = mₜ² / EGUT = 0. 030 eV, consistent with atmospheric neutrino oscillations.
Peacock et al. (Fri,) studied this question.