Inflation is always semi-classical: diffusion domination overproduces Primordial Black Holes

We use the Hamilton-Jacobi (H-J) formulation of stochastic inflation to describe evolution inflaton during a period Ultra-Slow Roll (USR), taking into account field's velocity and its gravitational backreaction. demonstrate how this formalism allows one modify existing slow-roll (SR) formulae be fully valid outside SR regime. then compute mass fraction, $\beta$, Primordial Black Holes (PBHs) formed by plateau in inflationary potential. By accounting for as it enters plateau, we find that PBHs are generically overproduced before inflaton's reaches zero, ruling out free diffusion or even noise domination on dynamics. also examine local inflection point similarly conclude entering quantum dominated therefore surmise is always predominantly classical with effects subdominant. Both characterized very sharp transition between under- over-production regimes. This can seen either severe fine-tunning production PBHs, strong link fraction $\beta$ shape potential plateau's extent.