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We present a measurement of the ratio of partial branching fractions of thesemi-leptonic inclusive decays, $\overline{B} \to X_{u} \ell \overline{\nu}$ to$\overline{B} \to X_{c} \ell \overline{\nu}$, where $\ell = (e, \mu)$, usingthe full Belle sample of $772 \times 10^{6}$ $B \kern 0.18em\overline{\kern-0.18em B}$ pairs collected at the $\Upsilon(4S)$ resonance. The ratio ismeasured via a two-dimensional fit to the squared four-momentum transfer to thelepton pair, and the charged lepton energy in the $B$ meson rest frame, wherethe latter must be larger than $1$ Ge\kern -0.1em V, covering approximately$86\%$ and $78\%$ of the $\overline{B} \to X_{u} \ell \overline{\nu}$ and$\overline{B} \to X_{c} \ell \overline{\nu}$ phase space, respectively. We find$\Delta \mathcal{B}(\overline{B} \to X_{u} \ell \overline{\nu})/ \Delta\mathcal{B}(\overline{B} \to X_{c} \ell \overline{\nu}) = 0.0196(1 \pm8.4\%_{\rm stat} \pm 7.9\%_{\rm syst})$ where the uncertainties are statisticaland systematic, respectively. In addition, we report the partial branchingfractions separately for charged and neutral $B$ meson decays, and for electronand muon decay channels. We place a limit on isospin breaking in $\overline{B}\to X_{u} \ell \overline{\nu}$ decays, and find no indication of lepton flavoruniversality violation in either the charmed or charmless mode. Furthermore, weunfold the $\overline{B} \to X_{u} \ell \overline{\nu}$ and $\overline{B} \toX_{c} \ell \overline{\nu}$ yields and report the differential ratio in leptonenergy and four-momentum transfer squared.

Measurement of the Ratio of Partial Branching Fractions of Inclusive  $\overline{B} \to X_u \ell \overline{\nu}$ to $\overline{B} \to X_{c} \ell  \overline{\nu}$ and the Ratio of their Spectra with Hadronic Tagging