In the current phase of the Kerr Trisector Closure (KTC) project, our work has focused on formalizing the statistical closure test and implementing it in a way that is directly applicable to real data. Since the conceptual structure of KTC is already established, the emphasis has been on robustness, diagnostics, and interpretability of inconsistencies between sectors.
Each observational sector yields an independent estimate of the Kerr parameters $\Theta = (M,a)$. We denote these as $\hat\Theta_{\mathrm{insp}} = (\hat M_{\mathrm{insp}}, \hat a_{\mathrm{insp}})$ for the inspiral sector, $\hat\Theta_{\mathrm{ring}} = (\hat M_{\mathrm{ring}}, \hat a_{\mathrm{ring}})$ for the ringdown sector, and $\hat\Theta_{\mathrm{img}} = (\hat M_{\mathrm{img}}, \hat a_{\mathrm{img}})$ for the imaging sector. Each estimate is accompanied by a covariance matrix $\Sigma_k \in \mathbb{R}^{2\times2}$, with $k \in {\mathrm{insp},\mathrm{ring},\mathrm{img}}$.
Under the Gaussian (Laplace) approximation, each sectoral posterior is modeled as $p_k(\Theta) \approx \mathcal N(\hat\Theta_k,\Sigma_k)$. The three estimates are combined into a single stacked estimator $\hat\Theta = (\hat\Theta_{\mathrm{insp}},\hat\Theta_{\mathrm{ring}},\hat\Theta_{\mathrm{img}}) \in \mathbb{R}^6$ with a full covariance matrix $C$. In the simplest case of statistically independent sectors, $C$ reduces to the block-diagonal form $C = \mathrm{blockdiag}(\Sigma_{\mathrm{insp}},\Sigma_{\mathrm{ring}},\Sigma_{\mathrm{img}})$.
Assuming General Relativity is correct (spoiler alert: obiously it is), all three sectoral estimates should correspond to a single Kerr parameter pair $\Theta^\ast$. This is expressed through the linear model $\hat\Theta = A\Theta^\ast + \varepsilon$, where $A = \mathbf 1_3 \otimes I_2$ and $\varepsilon$ is a zero-mean noise vector with covariance $C$. The best-fit common Kerr parameters are obtained via generalized least squares, $\bar\Theta = (A^\top C^{-1}A)^{-1}A^\top C^{-1}\hat\Theta$.
The closure residual is defined as $r = P\hat\Theta$, where $P = I_6 – A(A^\top C^{-1}A)^{-1}A^\top C^{-1}$ projects onto the $C^{-1}$-orthogonal complement of the model space. By construction, $r$ measures only inconsistencies between sectors and is insensitive to the overall best-fit Kerr parameters.
The Kerr Trisector Closure statistic is then defined as $T^2 = r^\top C^{-1} r$. Under the null hypothesis of perfect Kerr consistency and within the Gaussian approximation, $T^2$ follows a chi-squared distribution with $\nu = (K-1)p = 4$ degrees of freedom, where $K=3$ is the number of sectors and $p=2$ the number of parameters. Values of $T^2$ significantly larger than expected indicate a breakdown of cross-sector consistency.
An important aspect of the implementation is the diagnostic power of the closure statistic. For independent sectors, the total inconsistency can be decomposed as $T^2 = \sum_k (\delta\Theta_k)^\top \Sigma_k^{-1}\delta\Theta_k$, where $\delta\Theta_k = \hat\Theta_k – \bar\Theta$. This allows us to identify which sector dominates a potential inconsistency, rather than merely detecting its existence.
At this stage, the framework is fully implemented and tested on synthetic data. The next step is to apply the closure test to realistic forecast scenarios, incorporating expected uncertainties from multi-band gravitational-wave observations and horizon-scale imaging. This will allow us to assess the sensitivity of the Kerr Trisector Closure to sub-percent deviations in mass and spin across observational regimes.
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