We demonstrate a silicon-chip biphoton source with an unprecedented quantum cross-correlation up to gsi(2)(0) = (2.58 ± 0.16) × 104. The emitted photons are within single whispering gallery mode with self-correlations of gss(2)(0) = 1.90 ± 0.05 and gii(2)(0) = 1.87 ± 0.06 for signal and idler photons, respectively. We observe the waveform asymmetry of cross-correlation between signal and idler photons and reveal the identical and nonexponential nature of self-correlations of individual signal and idler photon modes, which is the nature of cavity-enhanced nonlinear optical processes. The high efficiency and high purity of the biphoton source allow us to herald single photons with a conditional self-correlation gc(2)(0) as low as 0.0059 ± 0.0014 at a pair flux of 1.95 × 105 pairs/s, which remains below 0.026 ± 0.001 for a biphoton flux up to 2.93 × 106 pairs/s with a tunable photon preparation efficiency in the single-mode fiber up to 51% and a raw heralding efficiency up to 3.4%. Our work unambiguously dem...