Conserved energy without work or heat

By merely observing the common empirical temperature that two like subsystems in contact reach, as a recordable function of their initial empirical temperatures and specific volumes, we can deduce logically a conserved energy function that depends only on empirical temperature and specific volume. What makes possible this offbeat 18th-century inference, quite free of any 19th-century concepts of "heat" or measurable mechanical work, is the Invariance Law of nature, whereby any four like subsystems must settle down to the same common temperatures regardless of which two-by-two pairings are made-so to speak, an "independence of path" property. Mathematically, the Invariance Law imposes Abel-like functional equations on the admissible functions. Remarkably, recourse to such temperature-equalization data renders it impossible to identify one constant in the energy function. By observing how two quite different substances come into equilibrium, or by utilizing "adiabatic slopes" at two points for one substance, or by observing its expansion into a vacuum, we can determine all its unknowns. Much of conventional expositions of the First and Second Laws seems derivable from the equation of state and the First Law's energy function.