Compact spatial topology restricts the eigenmodes of primordial tensor perturbations, and the resulting discreteness can render the primordial stochastic gravitational-wave background (SGWB) anisotropic. Here we treat the CMB tensor $B$-mode covariance as a transfer-filtered measurement of that ultra-low-frequency anisotropy. Writing the normalized angular tensor-power measure as $F(k,\widehat{k})=1+Q(k,\widehat{k})$ and its nonmonopole moments as $q_{LM}(k)$, we obtain an explicit kernel that maps $q_{LM}(k)$ onto the off-diagonal covariance $\delta C_{\ell m,{\ell }^{\prime }{m}^{\prime }}^{BB}$. The kernel factorizes into tensor transfer functions and a spin-weighted Gaunt coefficient and obeys the parity rule $L+\ell +{\ell }^{\prime }$ even for $BB$ and odd for $TB/EB$. It is an exact source-response representation of the full compact covariance rather than an additional observable. For a cubic three-torus the geometry pins down a common cubic angular subspace and orientation across frequency bands, although the amplitudes of the allowed multipoles still depend on the radial shell and source spectrum. The same topology-restricted template can therefore be read out either through the CMB $B$-mode kernel or through the anisotropy response of PTA/LISA/Taiji/TianQin searches. Using CAMB transfer functions and an invariant anisotropic-template statistic, we contrast this tensor channel with the scalar $T/E$ covariance. Independent direct angular-shell sums and $q_{LM}$-Gaunt contractions agree at $L_q^{\max }=2{\ell }_{\max }$ to relative Frobenius residuals of $1.4\times {10}^{-14}–3.0\times {10}^{-14}$. The scalar sector holds most of the practical CMB topology information; a fixed-template scan places the combined full-sky $S/N=1$ transition between $L/{\chi }_{*}=2.34$ and 2.36, while the $B$-mode channel remains subthreshold but isolates the primordial SGWB contribution. We use these results to set out a cross-frequency template for future topology searches.

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• Cosmic microwave background
• Cosmology
• Gravitational wave detection
• Gravitational wave detectors