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 field of potential — a state of total superposition encompassing all possible configurations. This superposition is not a quantum mechanical postulate: it is the logical origin of quantum mechanics, which emerges as a consequence of the framework rather than a prerequisite for it. A Recursive Nothing Paradox shows that pure potentiality is itself a form of nothingness at the level of actuality, forcing collapse. 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 phi=(1+sqrt(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 . phi^(-78) . 33^(-4) . exp(-alpha) . phi^k . (1 +/- phi^(-1)/(n x 8)).

Four 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 inside 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+sqrt(5+d). Second, the electron-to-Planck mass ratio m_e/m_P = phi^(-78) x 33^(-4) x exp(-alpha) is derived completely from the action principle in four steps: (A) phi^(-2) per shell from the spectral zeta function zeta_k(s)=(R_k^2/2)^s Z_2(s); (B) phi^(-2N) from N independent shell transitions; (C) phi^(-12) from the six Goldstone modes of SO(5) to SO(2)xSO(2); and (D) N^(-4) from the globally trivial normal bundle (nabla^nu = 0, proved directly). The factor exp(-alpha) is the all-orders topological correction from the conformally coupled field on curved S4 (0.001% error). Third, the sign rule g(n) = 1 + (-1)^n sin^2(theta_W)/(n x 3) is derived from the GL lepton coupling wave Psi_lep(k) = A sin(pi(k-6)/6) with period T=12=base-24/2, whose exact half-period identity Psi(k+6) = -Psi(k) gives the sign alternation for all generations. Fourth, the cosmological constant coherence factor beta_coh = 6cos(pi/23) is derived from the eigenvalue structure of the 22 co-rotating shells. The sole external input throughout is m_P.