We present the Inverted Hypersphere Cosmology (IHC) framework, in which information-theoretic constraints imposed by the RP4 antipodal identification — a topological self-measurement operator that couples UV and IR vacuum modes — determine the cosmological constant and baryon acoustic oscillation (BAO) scale without parameters fitted to data. Specifically, IHC predicts the dark energy density parameter Ω_Λ = 0.6889 ± 0.0006 and the BAO sound horizon r_s^IHC = 153.2 Mpc from real projective 4-space (RP4) topology; neither quantity is fitted to BAO or CMB data. A second independent derivation via the RP4 UV–IR Casimir seesaw gives Ω_Λ = √(1262/(270π²)) = 0.6882 with no free parameters; the 0.10% agreement between the two derivations constitutes a non-trivial internal consistency check. The universe is modelled as RP4 containing N = 33 nested toroidal structures scaling by the golden ratio φ = (1 + √5)/2, generating a geometric suppression factor β = 1345 ± 50 with coherence amplitude β_coh = 6cos(π/23) derived from the Dirac spectrum on RP4. The ratio ξ = r_s^IHC / r_s^CAMB = 1.0367 is a topological invariant that cancels exactly in all dimensionless CMB and BAO observables, but is observable only through the H(z) step at z₁ = 0.754, where the amplitude ξ−1 enters D_H additively rather than as a ratio, breaking the ratio degeneracy. Against seven independent BAO surveys (33 measurements, z = 0.106–2.33), IHC achieves χ²/n = 0.916 versus ΛCDM's 1.196 (Δχ² = +9.22) with zero parameters fitted to BAO data. DESI DR2 (13 observables) gives χ²/dof = 0.98, matching ΛCDM with two fewer fitted parameters. Exact Bayesian evidence computed via dynesty nested sampling gives ln B(IHC/ΛCDM) = +4.76 (moderate evidence on the Jeffreys scale). A joint four-parameter MCMC places the IHC zero-parameter prediction at Mahalanobis distance 0.70σ from the posterior mean, within the joint 68% credible region. Survey consistency tests show all six pairwise tensions below 1.1σ; a posterior predictive check yields p-value = 0.61, confirming model adequacy.