The cosmological constant problem, the fermion mass hierarchy, and the strong-CP problem are three of the deepest unsolved puzzles in theoretical physics. We present a framework that addresses all three from one axiom: the pre-geometric state is a non-preferential void — no direction, scale, or configuration is preferred. This is the Hartle–Hawking no-boundary state. Four forced steps follow: the non-preferential condition selects S⁴ (Killing–Hopf theorem); the unique involution central in Isom(S⁴)=O(5) is the antipodal map; the physical manifold is therefore RP⁴ = S⁴/Z₂. The CPT theorem of QFT on curved spacetime confirms CPT as a consequence of RP⁴, not a premise. The 55 spectrally stable modes at harmonic degree l=4 distribute equally across the 5 embedding directions of R⁵ — the same non-preferential principle that forces RP⁴ — giving M=55/5=11, N=3M=33 nested toroidal shells. This fixes the complete IHC structure with no free parameters. The main results are: The UV–IR Casimir seesaw gives Ω_Λ = 0.6882, agreeing with Planck 2018 at 0.48σ. A second independent route gives Ω_Λ = 0.6889; the 0.10% agreement between two structurally independent derivations is a non-trivial internal consistency check. Against 33 BAO measurements from seven surveys (z = 0.106–2.33), the framework achieves χ²/n = 0.916 versus ΛCDM's 1.196, with zero parameters fitted and Bayesian evidence ln B = +4.76. The Weinberg angle sin²θ_W = 3φ⁻¹/8 = 0.23176 follows from the 24-cell structure. All six quark masses and three charged lepton masses are predicted with RMS deviation 0.24% from PDG. The proton-to-electron mass ratio m_p/m_e = 1836 (0.008%) and neutron–proton mass difference 1.289 MeV (−0.34%) follow from the chain spectrum. The RP⁴ topology forces θ_QCD = 0 exactly, resolving the strong-CP problem without an axion. The Ψ-field action on RP⁴, uniquely fixed by conformal invariance (ξ=1/6) and the Z₂ projection, gives w_Λ = −1 exactly and Ω_K = 0, ruling out dynamical dark energy independently of the DESI signal. The SO(10) GUT group is derived from RP⁴ geometry. The intermediate scale is k_PS = M(N_co+1) = 253, giving E_PS ≈ 1.1×10¹¹ GeV; broken generators equal N=33; Weinberg angle running satisfies sin²θ_W(M_Z) = sin²θ_W(E_GUT) × φ⁻¹ exactly. The electron mass is derived: m_e/m_P = φ⁻⁷⁸ × 33⁻⁴ × e⁻ᵅ (0.001%), where α is itself determined geometrically by the k=8 shell. The only external input is the Planck mass.