Reaffirming the $d_{x^2-y^2}$ superconducting gap using the auto-correlation angle-resolved photoemission spectroscopy of Bi$_{1.5}$Pb$_{0.55}$Sr$_{1.6}$La$_{0.4}$CuO$_{6+\delta}$

Knowledge of the gap function is important to understand the pairing mechanism for high-temperature ($T_\mathrm{c}$) superconductivity. However, Fourier transform scanning tunneling spectroscopy (FT STS) and angle-resolved photoemission spectroscopy (ARPES) in the cuprates have reported contradictory gap functions, with FT-STS results deviating strongly from a canonical $d_{x^2-y^2}$ form. By applying an "octet model" analysis to autocorrelation ARPES, we reveal that a contradiction occurs because the octet model does not consider the effects of matrix elements and the pseudogap. This reaffirms the canonical $d_{x^2-y^2}$ superconducting gap around the node, which can be directly determined from ARPES. Further, our study suggests that the FT-STS reported fluctuating superconductivity around the node at far above $T_\mathrm{c}$ is not necessary to explain the existence of the quasiparticle interference