No, there is nothing mysterious about the Dirac delta function; except to someone who does not understand it or how to make use of it! At no place in my work does a delta function set to zero become non-zero later on. The fact that you think so implies you did not pay one bit of attention to anything I said.
Any physicist who was even half way competent in the use of constraints would comprehend that the two things you are confusing are vastly different statements discussing completely different issues.
There are three different functions being discussed in the equations 1.18, 1.19 and 1.20. Those three different functions are F, P and Psi.
F is the function which expresses the constraint provided by the so called "laws of physics" (see the discussion prior to equations 1.1 and 1.2).
P is the function which expresses the probability of seeing some set of arguments.
Psi is the function from which P is developed (see the discussion provided with equations 1.3 through 1.6)
Equation 1.18 (F=0) is a statement of the constraint which will force the arguments to obey the "laws of physics"; i.e., a valid set of data is given by the roots of the equation F=0.
Equation 1.19 is the statement that FP=0 cannot be false (and has nothing to do with the values of the arguments)! That result follows directly from the definitions of F and P. If the data is valid (i.e., the arguments obey the "laws of physics") F must be zero and if the data is invalid (i.e., the arguments do not obey the "laws of physics" and F is not zero) then P is zero. So the product is zero independent of the what the arguments are!! Notice the comment immediately after equation 1.19: " It follows that the product, FP, is identically zero ---> even when the arguments are allowed to range over all possibilities".
Only a complete idiot would presume that I was saying that F was always zero. If F were always zero, then FP would be zero no matter what the probability P was and one would have the ridiculous statement that the probability of seeing invalid data could be non-zero!
The arguments are free to range over any and all values and FP will always be zero so long as F=0 constrains the data to a valid set and P=0 for any invalid set!! FP=0 requires neither F nor P to be zero; either function may be non-zero for any given set of data. F times Psi must be zero as P is nothing more than the square of the magnitude of Psi! This is nothing more than simple logic which anyone should be able to follow.
However, I will admit that I sent a copy of my manuscript to the Harvard Physics Department in 1987 and received a reply from the chairman (which I still have by the way) which says "no one at Harvard can read this." At the time I thought it was funny but I didn't take him literally. But now I wonder, maybe he meant exactly what he said! Since Paul had no training in physics I found it reasonable to lead him step by step through the logic. Perhaps, since you went to Harvard, I should give you the same leniency I gave him.
Back to the simple logic of my presentation: the issue discussed above is completely separate from the question "can a sum over delta functions provide the constraint impugned to F?" That is another question and the analysis of that question is covered in the discussion surrounding equations 1.21 through 1.24. Setting the sum over delta functions to zero simply provides the constraint that no two arguments x sub i and x sub j can be the same (if they were the same, the delta function goes to infinity and the sum blows up). The proof follows directly from the statement that, if F is written as a sum of delta functions then, it is always possible to conceive of a set of unknowables which will cause F=0 to constrain the knowables to what is seen: i.e., the constraint F=0 can be written with F as a sum over delta functions.
The statement that "F=0 can constrain the data to a valid set" is not at all equivalent to the absolute statement that F=0. To presume such a ridiculous thing is to completely overlook the logic of the presentation.
Now, if you go and look at the set of equations 1.25, you will see that I say, "thus it is that we come to place four fundamental constraints on the algorithm Psi which is to give us the probability of any given observation". Notice that the constraints are placed on the algorithm Psi, not on the sum over delta function. I am not setting F=0, I am setting F times Psi equal to zero which is 100% equivalent to setting FP=0. As I have already said above, that constraint must be valid for --> "any arguments whatsoever": i.e., the sum over delta functions is not set to zero!!!
The steps I go through here are not at all original with me. This is a common logical construct used over and over again in any discourse on mathematical physics involving delta function interactions. The fact that you need to be led step by step through such an analysis implies to me that you are totally unfamiliar with such arguments and have seriously over stated your qualifications. The issue is that I no longer trust your veracity.
Put this together with your confusion over indices of an array of arguments and the arguments themselves. That issue should be clear to a beginning graduate student. All I can make of this is that you are completely incompetent. Your analysis does not really give me a very good impression of the physics education available at Harvard
Admit that you are slow witted and I will forgive you and make an effort to explain what I am doing -- Dick