Dear Emmanuel,
You write, "I don't understand what your eta is."
You can think of eta as being the "time-integral of energy". Hayrani Oz (Prof. Of Aerospace Engineering at Ohio State University) has used similar time-integrals of energy (what he calls enerxaction) very successful in his work for many years. Together we will be coauthoring a chapter in a Thermodynamics book coming out this July)
You write, " But I am not sure that we will find the solution to the unity of Physics in a new mathematical development"
I agree that going deeper into the current "rabbit's hole" of theoretical physics will not bring us closer to a 'unity of Physics' or discover 'what is' the Universe. I see all such efforts as 'metaphysical' in that these seek to answer 'what is'. And like all 'metaphysics' in the past, in my humble opinion such efforts ultimately fail.
My approach to physics is different. Though mathematics can provide us with 'logical certainty', it cannot give us the 'truth of what is'. We can only know our 'measurements' of 'what is'. I see 'measurement' as the essence of physics. The mathematical foundations of physics, therefore, should be mathematical identities pertaining to measurement. The Pythagorean Theorem that we use to measure distance is a good example of this. Not 'mathematical models' describing 'what is'.
In my essay I derive Planck's Law for blackbody radiation without using energy quanta. I show that this Law is actually a mathematical identity that describes the interaction of measurement. It simple gives us a mathematical way of calculating 'energy intensity' if we know the 'energy absorbed' at some given temperature. This is the reason why the experimental spectrum is indistinguishable from the theoretical.
You write, "... the unity of Physics must come from Physics itself"
My thinking is that what gives physics (and anything else for that matter) 'unity' is the human mind that comprehended physical experience as an integrated whole. I believe in the ancient Greek idea that "Man is the Measure of All Things". And I apply this principle to physics as well - highlighting of course, Man and Measure.
Do check my recent post, "If the speed of light is constant, then light is a wave". It's a very simple and elegant mathematical proof of this proposition. And please support my efforts to bring these results before the panel for review!
Best wishes,
Constantinos