Inverted Hypersphere Cosmology: First Principles from a Single Axiom

We take as sole axiom the logical instability of non-existence — the postulate that a state of nothing is not self-consistent, so existence is necessary — and trace its physical consequences. The axiom is metaphysical and pre-temporal; what follows from it is a chain of mathematical necessities. Because the totality of existence admits no external reference frame, its ground state must be self-consistent under its own description; together with the CPT theorem of quantum field theory on curved spacetime, this forces the ground state to be the CPT-invariant de Sitter vacuum dS₄/CPT = RP⁴. From this real projective four-space we derive, as a connected sequence, the spacetime dimension n = 4, the unique Cohesion Field action, the Clifford tori and the golden ratio φ as the self-similar stable modes, the Z₃ ternary class structure, and the shell number N = 33 = 3M with M = 11. These fix the geometric origin of the IHC predictions developed in the companion papers: the fine-structure constant α⁻¹ = 137.036, the weak mixing angle sin²θ_W = 3φ⁻¹/8, the coherence parameter β_coh = 6 cos(π/23), the dark-energy fraction Ω_Λ = 0.6882, and the charged-lepton mass scaling m ∝ φᵏ. We distinguish throughout what is deductive from what is not: the geometric and structural content follows from the axiom by standard mathematics, while the quantitative normalisations connecting this structure to measured magnitudes — a single overall mass scale, and the unit coefficient of the self-observation correction e⁻ᵅ to the electron mass — are identified as the points where the metaphysical axiom must be bridged to physical numbers, and are flagged explicitly. For the e⁻ᵅ correction we derive the exponential form (by two independent routes), the electromagnetic coupling, and the selection of a timeless topological count over a dynamical magnitude, isolating the single residual identification to one sharply stated point. We further show that the free photon is not an independent postulate but the fold-even transverse conformal harmonic selected by the same antipodal structure, with its masslessness and the vanishing of the electromagnetic θ-term following from the non-orientability of RP⁴.