e (2.718...) and (3.1415...) pi
Here are some reasons and a conjecture.
As given before, pi is initially 3 in terms of quanta and "degenerates" to 3.14159... as a limit as the quanta become increasingly smaller (width of measuring string included).
But what about "e" ? Is it a degenerate number like pi ? If so, in what way ?
I conjecture that "e" degenerates from "2" in the probablility of the appearance of a new particle in any given unit space. Recall that the probability was given as 1/2 (a new particle is either there or it is not). This applied to a standard Euclidean manifold.
However, this standard Cartesian three space may be modified by Planck's constant which would allow for variations in length, mass and time such as to maintain a constant 1/R value (where R is the present radius of the universe).
Now, clearly the hypothetical "unit cube" of the Euclidean manifold might shrink to the limit of 0 or it might expand without limit.
Therefore, the probability is trapped in the analogous situation of a point at unit distance from the origin within an angle. If the point (probability) moves to the 0, it has less distance to move than if it moves out to, say, 1198. Assuming that one is measuring distance on a hypothetical Euclidean frame. But from the angle's spherical reference frame the distance to 0 or to infinity is equal.
I propose then that the unit cube is "stretched" greater than 1 unit while others are correspondingly compressed (inverse fractions) and that therefore the probability of finding a new particle in a hypothetical unit cube (unstretched) is less than 1/2.
Of course, if one unit cube is compressed and another expanded and the sum of the two is still 2 units ... then the probability is still 1/2. But this situation would not be a case of inverse fractional alteration. Thus, if one unit compresses to 1/3, the other must expand to 3 times yielding a net increase of 1+1/3 unit cubes ... and therefore a corresponding decrease in the probability of finding a new particle since 2 unit cubes now occupy the space of 3+1/3.
Specifically, I conjecture that it is 1/e. And that the further one removes oneself from the beginning of time, the more closely the probability approaches 1/e. If this is so, pi is married to e at the very core of existence.
I have no proof of this but if one should occur to me,