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Appendix A
The difference in value between b(min) derived from the Schrödinger equation and b calculated empirically from Earth-surface F-sub-G is 8.4E-19 meters and proves to be the mean factor 1.079 or 7.9%.
The concept of continuous time t leads to exponential growth-decay:
a = a-sub-0 x e^ (rate x time) where e = lim (n infinity) (1 + 1/n)^n = 2.718.
Applying the expression for time itself, then
T(new) / T(old) = e^(r x t) = e^(0) = 1.
For time t itself, the rate r = 0 and there is no change in continuous time t so that one “second” of “time t” does not change. Time t is absolute.
Differential equations for centuries, e.g. the Schrödinger equation, assume time t is a real and continuous variable.
In the transformation t = cB, we need to treat continuity of time as a slight-contiguity of space.
In that case, we find 3 x 10^8 met/sec to be a large enough frequency (number n) to continue using the calculated value of e = 2.718 = lim as ninfinity of (1+1/n)^n.
But in a directional spatial sequential model, then space itself advances or grows per some rate different from r = 0.
As a one dimensional chalk line curves in a two dimensional blackboard, and as a two-dimensional earth-surface curves in three-dimensional space, then a change in 3-dimensional space needs to take place in a mathematical dimension higher than 3.
Postulating the higher number of dimension (vertices) to be 5 as in the Fibonacci infinite sequence, and considering physical events B = 1 / b = sec^2, then we calculate the following for one second of time t:
(V-sub-S / V-sub-0)^1/5 = e^(rV x t)^1/5 = e^(.618^(5-3))^1/5 = 1.079
or a 7.9% decrease in physical events B (increase in one-dimensional size b) from the continuous-time model used to calculate b(min).
Appendix B
Another Calculation for difference b-empirical - b(min) = 7.9%:
In two dimensional physics,
F = ma = m x met-sec^-2 becomes F-sub-2 = m x a-sub-2 = m x met –sec^-3/2
and one physical event B would no longer have units of sec^2; instead, sec^3/2.
The uncertainty principle then has a transformed h-bar, and now
b(min) = (h-bar)^1/2 x c^1/2 = 1.779E-13 meters.
Similarly, E-sub-B / m = 9.8 / c^3/2 J-kg^-1 per square boundary = 1.886E-12 J-kg^-1 per boundary and
b = E-sub-B / m / 9.8 = 1.924E-13 meters.
Then we again have the mean factor
= 1.079 or 7.9%
between b(min) and b (from F-sub-G) in two-dimensional space exactly the same as in three-dimensional space.
Appendix C
A General Fibonacci Calculation:
The Fibonacci infinite sequence was referenced in Appendix A,
F(n) = F(n-1) + F(n-2) with seed values F(0) = 0 and F(1) = 1.
Ratios converge, and
lim(n infinity) F(n+1) / F(n) = φ = (1 + 5^1/2) / 2 = .618 … and
lim(n infinity) F(n-2) / F(n) = γ = .382 … and so on.
Writing an example expression for spatial dimension ≥ 3 per Appendix A
∫∫∫∫∫∫∫∫dV = ∫∫∫∫∫dV(0) x exp(rV x t)
where rV = r-sub-V = φ
then
dx /dx(0) = exp(φ ^ (D(n+1) – D(n))^1/(n+1).
Except we are now doing math in another dimension, and while e = 2.718 in three dimensions, the base of natural logarithms should change in higher or lower dimensional space.
For example, in the case of 5 dimensions: e e^1 / γ^1.
We quickly find dx /dx(0) = 1.08.
A different example, for the case of spatial dimension < 3:
The base e must change as a function of the power of B, i.e. in three dimensional space B ~ sec^2 while in two-dimensional space B ~ sec^3/2.
The difference in power of physical events B
2 – 3/2 = 1/2 and the two-dimensional e = 2.718^1/2.
Then we quickly find dx /dx(0) = 1.08 similar to the previous mean calculations for the difference between b(min) and b-empirical.
The (Fibonacci) calculations hold true for any spatial dimension n moving through n+1 with a dimensional adjustment for e.
