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 $A_D(z) = -B_{H0} + (2q_0-1)B_{H0}z/2 + (5-q_0-18q_0^2+6j_0)B_{H0}z^2/12 + O(z^3, B_{H0}^2)$, where $B_{H0}=(H_{\parallel 0}-H_{\perp 0})/H_0$ and $j_0$ is the mean jerk parameter. A representative BBN limit, $\Omega_{\sigma 0} \le 10^{-23}$, implies $\vert{}B_{H0}\vert{} \le 9.5 \times 10^{-12}$ 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 $\Omega_{\sigma 0} \approx 2.5 \times 10^{-5}$, while matching the Planck 2018--SH0ES 2022 separation requires $\Omega_{\sigma 0} \approx 1.8 \times 10^{-3}$. 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.