A Quantitative Framework for Testing the Hubble Tension in a Bianchi Type I Cosmological Background

The Hubble tension is usually formulated as a disagreement between two determinations of a single scalar parameter, H 0, within an isotropic FLRW model. We develop a quantitative framework treating the tension as a consistency test of the scalar FLRW compression of cosmological data in a homogeneous, anisotropically expanding Bianchi type I background. Beyond synthesizing established results on Bianchi I kinematics, null geodesics, and optical propagation, our original contribution is a worked weak-shear, axisymmetric calculation mapping a specified shear history into a low-redshift luminosity-distance quadrupole. The calculation explicitly separates the direction-dependent redshift--affine-parameter mapping from the Jacobi-focusing contribution, propagating the resulting distance quadrupole through an analytic polar-cap toy window. For freely decaying shear, we obtain A D(z) = -B H0 + (2q 0-1)B H0z/2 + (5-q 0-18q 0 2+6j 0)B H0z 2/12 + O(z cubed, B H0 2), where B H0=(H 0-H 0)/H 0 and j 0 is the mean jerk parameter. A representative BBN limit, sigma 0 10-23, implies B H0 9.5 times 10-12 and a distance-modulus quadrupole below 2.4 times 10-11 mag at z=0.15. The early-Universe bound used is adopted from prior work; the novelty lies in propagating it through the derived Sachs--Jacobi mapping into limits on the luminosity-distance quadrupole and catalogue-window bias. By contrast, a 1% directional shift requires sigma 0 about 2.5 times 10-5, while matching the Planck 2018--SH0ES 2022 separation requires sigma 0 about 1.8 times 10-3. Thus, the minimal shear-only model cannot resolve the tension, though the framework supplies a falsifiable programme for testing late-time anisotropy with SNe, BAO, and standard sirens.