I remember reading these papers when they were first widely available. I am no great mathematician (and certainly not adequate to adjudicate the controversy in Corollary 3.12), but I remember having a sense of how the proof might work.
In essence, Mochizuki is trying to arrive at a more abstract understanding of numbers, whereby manipulating the fundamental attributes of numbers changes the relationships inherent in basic operators. The way I conceptualize it is roughly analogous to the following: 1 + 1 = 2 is true, but only within certain bounds (i.e. base 10 numbers) otherwise 1 + 1 = 10 could be true (in binary). So changing a basic property of numbers can alter the truth value of an operator. The angles in a triangle always sum to 180 within euclidian planar space, but space can be warped in ways where triangles with both angles summing to more than or less than 180 degrees are possible. In my (limited) read of Mochizuki, he is conceptualizing numbers where our generally understood operators (especially multiplication) perform their generally understood functions as being mailable in much the same way that non-euclidian space is malleable. Our operators are one special case in a larger family of cases, much as euclidian geometry is one special case within a wider family of cases.