I suppose we have to cover the same ground again.
In your paper at eq. 1.18 you explicitly set F =0.
Then in eq. 1.22 you set F equal a summation over delta functions and still set it equal to zero.
You can do this because the summation over j and i explicitly excludes i=j. Those are the indicies I referred to.
In eq. 1.23 you indicate that the delta function would equal infinity if i=j., and in the following paragraph you elaborate that the infinity must be avoided.
Therefore when you later integrate you cannot integrate over the infinity to get a non-zero value.
But in actaulity, your whole result is because you got a non-zero value when you integrated over the delta functions, which you yourself constrain to always equal zero.
For more examples, in eq. 1.27 the middle term containing betas is just the same F function, which we know equals zero. Take out the beta term and you lose all your results.
Same thing in eq. 2.1 which is your general equation. Then in eq. 2.5 you even call it the term f again, but small f this time. Still equals zero anyway, but the delta functions are gone from faulty integration over them. This f function ultimately becomes the potential function V in Schrodinger's eq 2.17. So what tyou really derived is the wave function, which of course follow from the shift symmetry.
Otherwise, I like your derivation, especially the parts about treating unknowable data. I liked your ability to equate EM waves to summations over the psi functions. which you seem to call wave functions in the paragraph just below eq. 4.10. I was always of the impression that EM waves were the wave function of photons. But you apparently have removed the wave function concept another layer away from the photon.