You may recall that in my last discussion of your work, I commented that your data must have both positive and negative values for it to be independent of its origin. You replied that unknowable data can supply the negative values if necessary. I now agree with that.
I also expressed concern about the use of the exponential that seemingly results from the relativity fo that origin of the data values. What I did not understand is how the data could be exponential and the fundamental equation not be a wave equation, for exponentials are solutions of wave equations and not the Schroedinger equation unless the potential term V(x) is identically zero.
Well in my mind I believe this issue has been resolved. The key step in your mathematical derivation is from eq(2.4) to eq(2.5). In eq(2.4) the summantion over the beta terms equals zero because the sum is over indices where the delta function equals zero, and the delta function is a multiplier in thise terms. Thus the function called f in equation 2.5 must also be equal to zero. It follows that the function V(x) in eq(2.17) is also zero. So your have just derived the wave equation. That is expected since you cast the data in the form of exponentials.
In my mind that is a snooker. I wonder if you snookered yourself as well as the rest of us. But the snooker does not stop there. In deriving the fundamental equation, you take the derivative of the values of the data points called xi and xj. In no place do you define what space is. You just define the values of data. Now we could assume that you meant that the values were defined at points in space given by the indices i and j. But you go on to claim that the values can be taken in any order. Now obviously you cannot take values of space coordinates in any order. One cannot define spatial gradients in such cases.
Schroedingers equation requires spatial gradients. What you did is to confuse the values of the data xi with their location in space x. That is a snooker. My question is whether you dileberately mislead us or were you yourself misled.
In summary, you defined the data values to be exponential functions of the location of their zeropoints and then obtained a wave equation for the dependence of the values on the zero points.
However, I will grant you that most of your hueristic thinking was in the right direction. I base my opinion on the recent work of Michael Hall with Marcel Reginatto published in the Journal of Physics A and discussed in the 27 April 2002 issue of New Scientist. They took probability theory that contained both known data and unknown data and derived both the Schroedingers equation and the Heisenberg Uncertainty Principle. As far as I can tell they did not make any mathematical errors or confuse any variables.