Based on the scattering theory of Guo-Åberg-Crasemann (GAC), which has no artificial assumptions, high harmonic generation (HHG) is studied by using first-principles. The HHG spectra of different rare atoms are also calculated. Using the properties of ordinary Bessel functions and the Einstein photoelectric law in the strong-field case, we reveal a new cutoff law qcħω = (9 -4√2) Up + (2√2-1) Ip ≈ 3.34 Up + 1.83 Ip of HHG based on a mathematical deduction method and a graphical method, which accords well with the Popmintchev’s experimental result published on Science in 2012. This cutoff law also agrees well with our own calculation using the HHG transition rate formula derived from the GAC scattering theory. Thus, we have four pieces of independent evidence for the same cutoff law of HHG. The cutoff orders predicted by this theory are higher due to the absorption of the extra photons. These photons only participate in the photon-mode up-conversion and do nothing in the photoionization process.