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

A single non-preferential axiom — no point, direction, or configuration of the pre-geometric vacuum is preferred over any other — uniquely determines the topology of the universe as real projective four-space RP4 = S4/Z2 and derives its principal geometric consequences without free parameters. This paper provides the foundational derivation of the IHC framework, establishing the chain from axiom to topology to field content. The non-preferential condition is shown to force constant sectional curvature (Lemma A), uniquely selecting S4 by the Killing-Hopf theorem (Lemma B), with the antipodal identification being the unique free Z2 action consistent with the axiom (Lemma E). Euclidean signature is derived from the non-preferential condition via the time-orientation argument (Lemma D). Existence of the geometry is guaranteed by the spinodal instability of the trivial vacuum Psi=0 under the conformally-coupled quartic action, seeded by Bunch-Davies quantum fluctuations (Lemma C, Companion Proofs). The measurement operator M-hat is defined formally as the unique bounded Hermitian involution on the field configuration space with empty fixed-point set. The KAM stability theorem independently selects N=33 nested Clifford tori from the same Fibonacci convergence criterion as the Hurwitz spectral condition. All quantitative predictions — Omega_Lambda = 0.6889, r_s = 153.2 Mpc, the lepton mass spectrum, and the gauge angles — are derived in the companion papers; the present paper establishes the necessary geometric foundations.