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We present the first model-independent measurement of the CKM unitaritytriangle angle $\phi_3$ using $B^{\pm}\to D(K_{\rm S}^0\pi^+\pi^-\pi^0)K^{\pm}$decays, where $D$ indicates either a $D^{0}$ or $\overline{D}^{0}$ meson.Measurements of the strong-phase difference of the $D \to K_{\rmS}^0\pi^+\pi^-\pi^0$ amplitude obtained from CLEO-c data are used as input.This analysis is based on the full Belle data set of $772\times 10^{6}$$B\overline{B}$ events collected at the $\Upsilon(4S)$ resonance. We obtain$\phi_3 = (5.7~^{+10.2}_{-8.8} \pm 3.5 \pm 5.7)^{\circ}$ and the suppressedamplitude ratio $r_{B} = 0.323 \pm 0.147 \pm 0.023 \pm 0.051$. Here the firstuncertainty is statistical, the second is the experimental systematic, and thethird is due to the precision of the strong-phase parameters measured fromCLEO-c data. The 95% confidence interval on $\phi_3$ is$(-29.7,~109.5)^{\circ}$, which is consistent with the current world average.

First measurement of the CKM angle $\phi_3$ with $B^{\pm}\to D(K_{\rm S}^0\pi^+\pi^-\pi^0)K^{\pm}$ decays