A recurring issue in strong-field tests of General Relativity is the question of how one should compare parameter estimates inferred from genuinely independent observational sectors. In the case of stationary black hole spacetimes, the Kerr hypothesis predicts that all sufficiently accurate observations of a given object should ultimately correspond to the same underlying pair of parameters $(M,a)$. However, the observational sectors used to infer these quantities are physically quite different in character: gravitational-wave measurements probe the dynamical evolution of compact systems, imaging observations probe null geodesic structure near the horizon, while orbital and accretion-based methods constrain yet other aspects of the geometry. Consequently, even before discussing possible deviations from General Relativity, one already encounters a fairly nontrivial statistical problem concerning the compatibility of sectoral parameter estimates.
The framework currently under consideration approaches this problem through a covariance-weighted construction on the combined parameter space. For each observational sector $k$, one introduces a parameter vector
$$
\theta_k =
\begin{pmatrix}
M_k \\
a_k
\end{pmatrix},
$$
together with an associated covariance matrix $\Sigma_k$. The sectoral estimates are then embedded into a stacked parameter vector, and compared against a common best-fit parameter $\bar{\theta}$ representing the null hypothesis of a shared Kerr spacetime. This leads naturally to the quadratic statistic
$$
T^2 = r^T C^{-1} r,
$$
where $r$ denotes the residual vector and $C$ is the covariance matrix associated with the stacked estimator. Geometrically, the statistic may be interpreted as a covariance-weighted squared distance on the residual subspace, closely related to the Mahalanobis distance appearing in multivariate statistics.
Under the standard Gaussian approximation for the inferred parameter distributions, the resulting statistic follows an asymptotic $\chi^2$ law with
$$
\nu = (K-1)p
$$
degrees of freedom, where $K$ denotes the number of observational sectors and $p$ the dimension of the parameter vector. This provides a direct statistical interpretation of sectoral disagreement in terms of exceedance probabilities relative to the null hypothesis of a common spacetime geometry.
One feature of the construction that appears conceptually useful is that the covariance matrix functions not merely as a weighting prescription, but effectively induces the geometry on the parameter space itself. In this picture, consistency testing becomes a problem of measuring distances inside a statistically curved space determined by the observational uncertainties. This perspective also clarifies why naive comparisons between overlapping confidence regions can sometimes obscure nontrivial inconsistencies once covariance structure is taken into account systematically.
Preliminary Monte Carlo studies indicate that the statistic reproduces the expected $\chi^2$ behaviour rather accurately under the null hypothesis. Introducing controlled sectoral biases shifts the distribution toward larger values of $T^2$ in a quantitatively stable way, suggesting that the framework is reasonably sensitive to inconsistencies comparable in scale to the observational uncertainties themselves.
Several extensions remain under consideration. The most immediate limitation of the present framework is the Gaussian approximation implicit in the covariance description. In realistic inference problems, posterior structure may become significantly non-elliptic due to parameter degeneracies, low signal-to-noise effects, or modelling assumptions. It therefore seems natural to investigate whether the covariance-based construction can be generalized to formulations involving posterior samples or more explicitly information-geometric approaches on the statistical manifold.
At present the framework should probably be regarded primarily as a structural proposal rather than an observational analysis. Nevertheless, the broader idea of treating consistency of spacetime geometry itself as a quantitative statistical object still appears mathematically and conceptually interesting.