I've read about the possibility of inverting the Mersenne Twister after 624 numbers of output. 624 matches the state size of my implementation of the Twister. Coincidence? If the generator only output 623 numbers, i.e. less than the state size, might inversion still be possible with really clever maths? Or is this mathematically and logically impossible?
This leads me to think about a general case. I got this embryonic analogy from solving simultaneous equations. Consider each member within the state array as a degree of freedom. And consider each number output as a degree of freedom. And if (for some generator) there is a group operation over the whole state such as an addition of all of the elements, this acts as a constraint. Can the following be true?
$F_o$ = degrees of freedom output,
$F_s$ = degrees of freedom within state,
$C_n$ = number of constraints on the state.
Is the random number generator “impossible to uniquely invert” if $F_o < F_s - C_n$?