Inverted Hypersphere Cosmology from a Single Axiom: Self-Observation, ℝP⁴ Topology, and the First-Principles Origin of the Physical Universe

We derive the complete structure of Inverted Hypersphere Cosmology (IHC) from a single logical necessity: nothingness is self-contradictory. The instability of pure nothingness necessitates the existence of a quantum field of potential — a state of total superposition encompassing all possible configurations. Since no external observer exists in this pre-geometric state, the unique self-consistent collapse is self-observation: the field collapses upon itself, producing the IHC geometry. Formalised as the existence of a compact space admitting a fixed-point-free involution, this logical necessity uniquely determines: (i) RP⁴ topology; (ii) n=4 spacetime dimensions, via the quaternionic Hopf fibration and the uniqueness of SO(8) triality; (iii) the Ginzburg–Landau Cohesion Field as the unique dynamically stable self-referential scalar field; (iv) Clifford tori as the unique isotropic stable modes; (v) the golden ratio φ=(1+√5)/2 as the unique self-similar scaling ratio; (vi) Z₃ triality and three fermion generations; (vii) N=33 nested tori from Fibonacci spectral self-termination; and (viii) the Standard Model charged lepton masses from m(k,n) = m_P · φ⁻⁷⁸ · 33⁻⁴ · e^(−α) · φᵏ · (1 ± φ⁻¹/(n×8)).

Three further results are established without free parameters. First, an Arithmetic Self-Consistency Theorem proves that the two independent appearances of 5 in IHC — the ambient dimension of S⁴ ⊂ R⁵ and the discriminant of the self-similarity polynomial x²−x−1 — are the same 5, both equal to dim(H)+1 via the closing identity T=−2+√(5+d), showing T=1 and d=4 are mutually forced. Second, the electron-to-Planck mass ratio m_e/m_P = φ⁻⁷⁸ × 33⁻⁴ × e^(−α) is derived completely from the action principle in four steps: (A) φ⁻² per shell from the spectral zeta function ζ_k(s)=(R_k²/2)^s Z₂(s); (B) φ^(−2N) from N independent shell transitions; (C) φ⁻¹² from the six Goldstone modes of SO(5)→SO(2)×SO(2) breaking; and (D) N⁻⁴ from the globally trivial normal bundle (∇ᵛ≡0, proved directly by symbolic computation). The factor e^(−α) is the all-orders topological correction from the conformally coupled field on curved S⁴, reducing the error from 0.73% to 0.001%. Third, the sign rule g(n) = 1 + (−1)ⁿ sin²θ_W/(n×3) is derived from the GL lepton coupling wave Ψ(k) = A sin(π(k−6)/6) with period T=12=base-24/2, whose exact half-period identity Ψ(k+6) = −Ψ(k) gives the sign alternation for all generations. The cosmological constant coherence factor β_coh = 6cos(π/23) is derived from the eigenvalue structure of the 22 co-rotating shells, closing the derivation of Ω_Λ = 0.6889 from first principles. The sole external input throughout is m_P.