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

We identify a geometric obstruction to resolving the Hubble tension by combining early-time sound-horizon reduction with late-time smooth dark energy. Within $\mathrm{\Lambda }$CDM, the BAO–SN matter-density gap $\mathrm{\Delta }{\mathrm{\Omega }}_m=0.037$ is exactly invariant under the sound-horizon rescaling $\alpha \equiv r_s^{\mathrm{m}\mathrm{o}\mathrm{d}}\mathrm{/}{r}_s^{\mathrm{\Lambda }\mathrm{C}\mathrm{D}\mathrm{M}}$, 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 ${\alpha }^{\ast }=0.992$ ($0.8\mathrm{\%}$ $r_s$ reduction) brings the joint fit to $H_0=70.3\pm 0.3\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$, 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 $\{\alpha ,{\beta }_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}},w(z)\}$ already spans $93\mathrm{\%}$ of the ${\mathrm{\Omega }}_m$ response direction; nonetheless BAO and SN constrain ${\mathrm{\Omega }}_m$ 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.