The whole thing about assumptions is just semantics and needn't be pursuued further. I take umbridge of your claim that I cannot follow the derivation of Noether.
But here is my latest problem with your work. Your derivation seems to fail in the limit of high time resolution. We can always think of the data stream as happening at such high time resolution that in any one time interval, only one data point is obtained. This is actually how quantum mechanical detection theory of light works. And so if your derivation applies to quantum mechanics, it certainly should apply to the detection of light, which is afterall where quantum mechanics came from.
So if each time interval has only one data point, call it Ai, where i now indicates a point in time, not space, then all the summations in your derivation disappear, as does the delta function. All data is knowable as the one point is unique, the uniqueness being obtained by taking small enough time intervals.
The three remaining constraint equations immediately yield exponential solutions. And the fundamental equation is a wave equation as V(x)=0.
So in this limiting case Schroedinger's equation is not derived. Yet this limiting case is completely general. Your results could be obtained if you require a summation over a sufficiently long enough time to get high statistical probabilities.
Perhaps you can say that the entire psi pattern can come from the one data point plus unknowable data. But then I have to ask- why do you need even one data point. Your derivation would then work if all you had was unknowable data. The result is the interesting concept that the fundamental equation does not require any data. It is true even if we sense nothing at all, or know nothing at all. Maybe so.???