Dear Kevin.
It was a real pleasure to read your essay, and I fully agree with your main idea we can summarize in the following terms: (i) There are mathematical edifices being interpretations of set theory specified by group theory, knowing that group theory by definition covers all aspects of symmetry, and (ii) physical laws stricto sensu are experience oriented interpretations of mathematical edifices ultimately formalizable in terms of group theory. Through all your essay, you advance relevant examples usefully condensed in table 1. Perhaps it could be interesting to add a counter-example: The so-called Clausius "law" is not a physical law stricto sensu. It is a pseudo-mathematical expression, as evidenced by the pseudo-differential without real mathematical signification figuring in it, and so not a physical law but just a kind of stenographical transcription of an increasing tendency. Despite of statistical/probabilistic/information theoretical ways allowing to circumvent this problem - to circumvent and nothing more (see below) - thermodynamics touching to irreversible processes continues to generate some malaise within physics. Much has been written about "law like reversibility v/s de facto irreversibility"; this discussion beginning with Boltzmann, Loschmidt, Zermello ...is far from reaching its end. Anyway, for a physical law to be a law stricto sensu, it must be reversible, and so, symmetrical in prediction and retro-diction, and this would not be possible outside phenomena being formalizable by mathematical groups carrying all fundamental symmetries.
Just perhaps a little remark. Additivity is fundamental, I entirely agree with you on this point. But as you state very rightly that several Newtonian "principles" later were recognized as CONSERVATION LAWS, there is a group being more fundamental than additivity. I mean the Klein 4-group, a commutative transformation group F = {a,b,c,i; *} where a,b,c are transformations to be determined, i being the identity transformation and * an internal composition law so that any composition of transformations belonging to F is a transformation belonging to F. One shows easily - further details are in the semi-technical end note of my own essay - that within the Klein 4-group, a*b*c*i = i. In other terms, the Klein 4-group formalizes ultimately all systems remaining identical to themselves through all their possible transformations.
In my opinion, the Klein 4-group because of its absolutely fundamental aspect can be used to distinguish law like reversible physics and fact like irreversible phenomena added to physics, even if the latter can be approached in terms of statistical mechanics and/or information theory, both being formalizable additively as you say it in your table 1.
In a semi-technical end note of my own essay I touch briefly this point. Contrary to what common sense, intuition, and even simple grammar might suggest, irreversibility is not a direct negation of reversibility. In terms of group theory, these phenomena have nothing in common.
First an intuitive example. Consider an ideal watch without internal frictions etc. whose needles turn by their own inertia at a constant speed. This system, as long as nothing disturbs it, is reversible in terms of the spatial configuration of its needles; it will return to any configuration it occupies at a given moment. Under these conditions, the system (i) is characterized by an entropy variation equal to 0 and (ii) "remains the same" because it conserves its functioning mode. Now let us create an irreversible situation by projecting the system violently to the ground. This time the entropy variation is superior to 0, while the system - reduced to fragments - does not conserve its functioning mode. Nobody would seriously say that the fragments scattered on the ground are the "same" system as the ideal watch in operating condition. So reversibility PRESUPPOSES the conservation of the functioning mode characterizing the considered system, whereas irreversibility CONSISTS ON the transition [conservation of the functioning mode 竊' non-conservation of the functioning mode].
The intuitive expression "functioning mode of a system" is certainly vague, but it can be formalized by the Klein 4-group where the combination of all the 4 possible transformations gives always the "identity transformation". More details can be found in the end-note of my essay. But briefly speaking, the Klein 4-group formalizes ultimately all systems remaining the same through their transformations. Any physical law is in fine an interpretation I(F) of the intrinsically reversible Klein 4-group F. Irreversibility is the transition I(F) 竊' non-I(F). So "real" physical phenomena are superpositions of IDEAL reversibility and DE FACTO irreversibility. Hence a gas initially in disequilibrium, composed of molecules with their movements dictated by reversible Newtonian mechanics remains ideally reversible but describes de facto an irreversible transition. Now information theory, because of its additive structure, can measure in a probabilistic-exact way the lack of information we encounter regarding the gas system, HOWEVER this exact approach relates not to the system as such, formalizable in terms of I(F), BUT to the system as it is contaminated by superposed irreversibility generating factors so that I(F) 竊' non-I(F). In other terms, far from establishing irreversibility as a law like phenomenon, information theory, just measures the damage done by GIVEN de facto irreversibility to ideally reversible physical systems.
Well it is not sure that you will agree with all this. The reversibility v/s irreversibility issue is and remains controversial. But in this domain, group theory is good common basis for interesting discussions about controversial domains.
Best regards and happy Easter
Peter