The Curvature Route Runs Through a Wall: Open Geometry Reaches the Ladder's Rate and Fails the Acoustic Distances

Spatial curvature is the oldest degeneracy in the microwave-background analysis: at fixed physical matter density and fixed acoustic scale, opening the geometry raises the Hubble constant the microwave background infers, while closing it lowers the value — which is why the Planck primary data alone, left free to choose curvature, drift to a closed geometry with an implausibly low expansion rate. This paper computes the full degeneracy track, solving for the Hubble constant that preserves the acoustic scale at each curvature, and then grades each point on the track against the DESI acoustic distances. The track runs from H0 = 53.52 km/s/Mpc at curvature −0.044 (the Planck-alone preference) through 67.36 at flatness to 72.96 at +0.01 and 80.70 at +0.02: an open geometry of +0.0094 or so would indeed put the microwave inference on the SH0ES value. But the acoustic distances exclude the entire useful stretch of the track: the four-point BAO chi-squared rises from 9.74 at flatness to 12.53 at +0.005, 33.87 at +0.01, and 161.41 at +0.02, while the closed direction collapses immediately (20.74 at −0.005, 352.38 at −0.044). The shift parameter, notably, is exactly constant along the whole track at 1.7492 — curvature is the direction R cannot see, which is the analytic reason the degeneracy exists. Curvature is therefore not a resolution of the Hubble tension but a relocation of it into the best-measured distances in cosmology, and the joint constraint (Planck plus BAO curvature of −0.0007 ± 0.0019) already sits two orders of magnitude inside the opening the route would need. This study is computed from published values; it is not a measurement.