The DESI Expansion Anomaly as a Topological Shell Crossing: RP4 Geometry Predicts z = 0.754

DESI DR2 reports a preference for dynamical dark energy over the standard cosmological constant model at between 2.8 and 4.2 standard deviations. The collaboration characterises this as a phantom crossing — dark energy whose equation of state passes through w = −1 — somewhere in the redshift range between 0.4 and 0.8.

We show that this signal is the expected observational signature of a zero-parameter prediction made by Inverted Hypersphere Cosmology (IHC): a discrete localised step in the Hubble expansion rate at redshift z₁ = 0.754, derived entirely from the geometry of the k=1 co-rotating shell at radius R₁ = R_H φ⁻¹. This prediction was in place before DESI DR2 was released.

The phantom crossing is not real. On RP4, the dark energy equation of state is w_Λ = −1 exactly — it cannot evolve. What DESI is seeing is the artefact of fitting a smooth parametric curve to a discrete topological feature. A smooth fitter applied to a sudden step in the expansion rate will always produce an apparent phantom crossing as its best approximation to something it was not designed to model.

Comparing the IHC expansion history against all 13 DESI DR2 measurements gives a goodness-of-fit of 0.983 per degree of freedom, against 1.438 under the flat-topology model, with zero parameters adjusted to fit the data. The two most discrepant measurements in the DESI dataset — the Hubble distance measurements at redshifts 0.51 and 0.71, which sit on either side of the predicted shell crossing — are brought from tensions of −1.80 and −2.14 standard deviations down to −0.31 and −0.91. When the step location is freed and recovered independently from the data, the posterior peaks at z₁ = 0.708 ± 0.188, placing the zero-parameter IHC prediction within 0.25 standard deviations of the data-preferred value. DESI five-year observations will provide a decisive test at approximately fifty standard deviations of separation between the IHC expansion history and the flat-topology cosmological constant model.