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

Zhihuan Zhou1,*, ShengYue Wang1, Zhuang Miao1, Chaoqian Ai1, and Hongchao Zhang2,†

  • *Contact author: zhihuanzhou1@163.com
  • †Contact author: zhanghongchao852@live.com
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Phys. Rev. D 114, 063511 – Published 8 September, 2026

DOI: https://doi.org/10.1103/hv6b-kccr

Abstract

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, ΔΩm=0.037 within Lambda cold dark matter (ΛCDM), is exactly invariant under the sound-horizon rescaling α≡rsmod/rsΛCDM, 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 cosθ=−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 α*=0.992 (0.8%  rs reduction) brings the joint fit to H0=70.3±0.3  km s−1 Mpc−1, still 3.2σ 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 {α,βdamp,w(z)} already spans 94% of the Ωm response direction; nonetheless BAO and SN constrain Ωm through independent channels and disagree, while the residual H0 deficit, anchored by the local distance ladder, resides in the absolute distance scale, which w(z) reshapes but cannot rescale.

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Phys. Rev. D 114, 063511– Published 8 September, 2026
  • Received 12 June 2026
  • Accepted 14 August 2026
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