Dear Thomas McFarlane,
The derivation of an 'operator equation' is based simply on the fact that physics tends to be written in terms of such equations. So, if we start with the goal of defining a 'physics' on a one and only substance, with no 'laws of nature' existing in some Platonic dimension, then the only possible operator equation must be that any 'operation on the substance' must be equivalent to 'the substance interacting with itself', since nothing else exists. But this is formal; both operation and substance are undefined at this point.
Then, since we cannot do physics without physical facts, I pull in two facts: Maxwell's energy of field proportional to square of field, and Einstein's energy-mass equivalence. These 'suggest' that the phi*phi term on the right is energy, hence mass, and this 'suggests' that the operator del is the directional derivative, so I explore those suggestions.
Possibly other suggestions could have lead to other areas of exploration. For example, the formal equation, before the terms are defined, may have 'suggested' to me a General Relativistic equation and I may then have explored the consequences of such. It's easier to derive results using the approach I took, but that does not necessarily preclude a 'tensor operator' interpretation.
The issue of math is trickier. I am saying that, unlike some theories that assume a 'mathematical universe' or 'God making the integers', if we can only get 'particles' out of our field, then these particles can be used to construct 'counters' [as shown in reference 3] and, per Kroneckar, 'man can do the rest'. So, since I show how to get 'particles' out of the field, and have shown elsewhere how to build counters from these particles, and men have done the rest, I feel free to use math. It may sound circular, but what I am aiming for is to have math arise from our field, and not have to assume some Platonic world of math outside of our original primordial field.
I hope this makes sense to you.
Those were good questions. Thank you for studying my essay. I would be happy to attempt to answer more questions, if you have them.
Good luck in the contest.
Edwin Eugene Klingman