Thanks, Marc. Indeed, I fretted all the time I was writing it, over whether this work might be *too* ambitious. Details can run away from one, if the scope gets too big. In the end, I could not escape the conclusion that if free will exists, it cannot be a property of random observer choice. This being so:
Only a binary decision that is square integrable can share a reciprocal relation *continuously* with a continuous range of variables in a finite interval of time (Buridan's principle is limiting). The linear relation M = 4P is not physically meaningful in the first degree, because it is not dynamic; M - 4P = 0. The second degree equivalent, M = Pq^2, q = 2, accounts for the full range of possible binary decisions in any finite interval where M = P. Tracing the relation back to the cosmological initial condition, the rest state of the universe is a "fourity" of possibilities in any locally bounded interval.
The units of M and P are dimensionless to the extent that M = P is unitary. In the reciprocal relationship M = Pq^2, the fundamental dimensionality M = 4P (P = M/4) gives us the division algebras we know to exist (R, C, O, H) as a complete algebraically closed range of computational fields relating binary choice to a continuous range of variables limited by a finite time interval.
Physical units are derived from empirical observations in a bounded measure space. Since four pure physical states imply the Hawking-Penrose singularity theorem, and insofar as four are the minimum necessary and sufficient for dynamic interaction between physics and mathematics (i.e., between measure and model), maybe we can move a little closer to Hawking's question of what "puts the fire in the equations" with number-theoretic arguments alone. Just thinking out loud here.
All best to you and your essay, too!
Tom
