Geometric obstruction to resolving the Hubble tension: orthogonality of scale and shape in distance measurements

Abstract

We identify a geometric obstruction to resolving the Hubble tension by combining early-time sound-horizon reduction with late-time smooth dark energy. Within $\Lambda$CDM, the BAO--SN matter-density gap $\Delta {\Omega }_m=0.037$ is exactly invariant under the sound-horizon rescaling $\alpha \equiv r_s^{\mathrm{mod}}/r_s^{\Lambda \mathrm{CDM}}$, and late-time $w(z)$ deformations cannot eliminate this gap either: reconciling the two datasets requires \emph{opposite} deformations -- phantom ($w<-1$) for BAO, quintessence ($w>-1$) for SN at $z<0.5$ -- an anti-alignment quantified by $\cos \theta =-0.97$ in $w(z)$ space. A full MCMC analysis of DESI DR2 BAO, Planck plik\_lite, and Pantheon+ bears this out: the optimal ${\alpha }^{\ast }=0.992$ ($0.8\%$ $r_s$ reduction) brings the joint fit to $H_0=70.3\pm 0.3\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, still $3.2\sigma$ below SH0ES, with the inter-dataset tension reduced but not removed. The obstruction reflects not a shortage of model freedom but an irreducible disagreement between probes. The deformation space $\{\alpha ,{\beta }_{\mathrm{damp}},w(z)\}$ already spans $93\%$ of the ${\Omega }_m$ response direction; nonetheless BAO and SN constrain ${\Omega }_m$ through independent channels and disagree, while the residual $H_0$ deficit, anchored by the local distance ladder, resides in the absolute distance scale that $w(z)$ reshapes but cannot rescale.

13 pages, 5 figures, 4 tables