Effective distance-duality deformations as a diagnostic oflate-time calibration tensions

The Hubble tension between the SH0ES-calibrated luminosity-distance scale and the CMB/BAO-inferred expansion history can be recast as an effective violation of the distance-duality relation (DDR), η(z)≡ DL(z)/[(1+z)²DA(z)]. Using the Fisher-bias formalism, we explore how an overall distance recalibration and a broad redshift variation can accommodate different cosmological responses. These diagnostic profiles guide a compact model of the distance deformation. We adopt a three-parameter tanh parameterization and perform full MCMC sampling with Planck, DESI-DR2 BAO, Pantheon+ with SH0ES, and cosmic chronometers. The tanh and constant-DDR models give Δχ²tanh/Λ CDM=-35.1 and Δχ²const/Λ CDM=-31.5, respectively; the additional redshift-dependent contribution is therefore Δχ²tanh/const=-3.6. Redshift dependence is marginally preferred (~1.4σ), while neither the Bayesian evidence nor the deviance information criterion provides meaningful support for the extension. We thus find no significant evidence for redshift-dependent DDR violation, while the dominant constant offset remains degenerate with a recalibration of the supernova absolute magnitude MB in the absence of an external calibration. The combined analysis returns H₀=68.44±0.30\,km\,s⁻¹\,Mpc⁻¹ and Ωₘ=0.301±0.004. Finally, at the background level, the adopted SN-sector deformation is strongly degenerate with the specific evolving-dark-energy model considered here.