The community will undergo maintenance on October 16th at 10:00 PM PDT and will be unavailable for up to one hour.
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: