You seem to be missing something.
I used "traffic jams" as a pictorial graphic of Dr. Dick's work. They are not literal; they are metaphorical.
Your rule about setting functions to zero and keeping them that way is not broken.
You seem to be missing that there may exist a "cloud" of functions that inter-relate so as to give different perspectives on each other. Your rule is not broken for individual functions.
The system I am describing, note, is outside time.
The delta function is easily represented by a simple graphic: draw a horizontal line, cross it in the center by a vertical line.
The lines are made up of discrete arbitrary points that bunch closer together towards the center where they intersect. I guess they could bunch anywhere in any way, thus give Stafford's unknown data in graphic form.
Track a match between any pair of points selected one from each line. Draw the curve of that superposition as the pair progress through both lines as if moving (one along the horizontal, one along the vertical). Stafford's "constant velocity" would be represented by this "constant matched pair track" through two zones of his "unknown data".
This constancy seems analogous to "rest mass" or inertia.
Rotate the superposition path to create a new vertical or horizontal line (rotation changes the density pattern of the points on the path).
Perhaps "taking a probability of finding a matched-pair superposition with another chosen point on a new superposed line" would be similar to what I graphically describe as rotation.
Perhaps probability plus relativity thus gives what is called "mass".
Making a match (quantization) plus probability may give what is called "energy". After all; the more energy something has, the more possibilities it has for interactions, for being involved in "matches"; the more massive is its matchability. So you have its mass-energy being probability-matchability.
Transforming mass to energy would be transforming probability into matchability I guess.
Bosons: apparantly it is quantization that accounts for the integer "spin" of bosons; and it is probability that accounts for "spin".
As there are many ways to get a match + a probability; bosons may occupy the same quantum state )overall match-potential + probability-potential; I'm guessing.
Fermions: apparantly it is relativity that accounts for the exclusivity of fermions; and for their half-integer spin. That is: probability gives rise to "spin', relativity gives rise to the half-integer aspect here for fermions.
So quantization gives bosons; relativity gives fermions. Probability (spin) combined with quantization gives integer spin for bosons; probability combined with relativity gives half-integer spin for fermions.
Returning to the crossed lines graphics:
Use this new line as either a vertical or horizontal line to be superposed with another such line. And so on. This system fits Stafford's delta-function explanations superbly; and apparantly readily leads to his deductions about physics laws. It appears consistent with textbook mathematics.
Supersonic flow is only one way of looking at superposed traffic jams. Delta functions happen to provide a way of looking at them.
Delta function collections represent the interdependence of definitions. This fits excellently with the philosophy text analysis by John Hospers of the broadening and narrowing involved in the intersection of logically consistent categories that give definitions.