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so it says my units are incompatable but they are unitless
H2, H3, and H4 are all unit'd, because they all incorporate T0 in their definitions, and that has a unit of K. You also use τ, which you defined as T/T0, and those are also unit'd.
You can always evaluate each summand one after the other to see that all of them are of different dimension - not even two of them have the same unit: