$B$${}^0$ − $\bar{B}$${}^0$ entanglement for an ideal experiment for the direct $\mathit{\text{CP}}$ violation $\varphi$${}_3$ $/$ $\gamma$ phase

$B$${}^0$ − $\bar{B}$${}^0$ entanglement offers a conceptual alternative to the single charged $\mathit{\text{B}}$-decay asymmetry for the measurement of the direct $\mathit{\text{CP}}$-violating $\gamma$ $/$ $\varphi$${}_3$ phase. With $\mathit{\text{f = J / }}\mathrm{\Psi }\mathit{\text{K}}{}_L\mathit{\text{ , J / }}\mathrm{\Psi }\mathit{\text{K}}{}_S$ and $\mathit{\text{g = (ππ)}}{}^0\mathit{\text{ , (}}\rho {}_L\rho$${}_L$)${}^0$ , the 16 time-ordered double-decay rate intensities to $\mathit{\text{(f, g)}}$ depend on the relative phase between the $\mathit{\text{f-}}$ and $\mathit{\text{g-}}$decay amplitudes given by $\gamma$ at tree level. Several constraining consistencies appear. An intrinsic accuracy of the method at the level of $\pm$1${}^{\circ }$ could be achievable at Belle-II with an improved determination of the penguin amplitude to $\mathit{\text{g}}$ channels from existing facilities.