We identify a geometric obstruction to resolving the Hubble tension by combining early-time sound-horizon reduction with late-time smooth dark energy. The baryon acoustic oscillation (BAO)-supernova (SN) matter-density gap, $\mathrm{\Delta }{\mathrm{\Omega }}_m=0.037$ within Lambda cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$), is exactly invariant under the sound-horizon rescaling $\alpha \equiv r_s^{\mathrm{mod}}/r_s^{\mathrm{\Lambda }\mathrm{CDM}}$, and late-time $w(z)$ deformations cannot eliminate it either; reconciling the two datasets requires opposite deformations; phantom ($w<-1$) for BAO, quintessence ($w>-1$) for SN at $z<0.5$; an antialignment quantified by $\cos \theta =-0.98$ in $w(z)$ space. A full Markov chain Monte Carlo analysis of DESI DR2 BAO, Planck plik_lite, and Pantheon+ bears this out; the optimal ${\alpha }^{*}=0.992$ ($0.8\%r_s$ reduction) brings the joint fit to $H_0=70.3\pm 0.3\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, still $3.2\sigma$ below SH0ES, with the interdataset 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 94% 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, which $w(z)$ reshapes but cannot rescale.

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• Cosmological parameters
• Dark energy
• Evolution of the Universe