Invariant Laws of Thermodynamics and Validity of Hasen\"{o}hrl Mass-Energy Equivalence Formula m $=$ (4/3) $E$/c$^{\mathrm{2}}$ at Photonic, Electrodynamic, and Cosmic Scales

According to a scale-invariant statistical theory of fields$^{\mathrm{1}}$ electromagnetic photon mass is given as $m_{em,k} =\sqrt {hk/c^{3}} $. Since electromagnetic energy of photon is identified as $amu=\sqrt {hkc} $, all baryonic matter is composed of light (photons) $E_{em} =Nm_{em,k} c^{2}=M_{em,k} c^{2}\thinspace [Joule]$ or equivalently $M_{em,k} c^{2}/8338\thinspace [kcal]=Namu=M_{a} [kg]$ where 8338 is De Pretto number$^{\mathrm{1}}$. Besides particle \textit{electromagnetic} energy one requires \textit{potential energy} associated with Poincar\'{e} $^{\mathrm{2\thinspace }}$ stress for particle stability leading to rest enthalpy$^{\mathrm{1}} \quad \hat{{h}}_{o} =\hat{{u}}_{o} +p_{o} \hat{{v}}=\hat{{u}}_{o} +\hat{{u}}_{o} /3=(4/3)m_{em,k} c^{2}$ in accordance with Hasen\"{o}hrl$^{\mathrm{3}}$. The 4/3 problem of electrodynamics (Boyer, T. H., Phys. Rev. Lett.\textbf{ 25}, 1982) is also related to Poincar\'{e} $^{\mathrm{2}}$ stress thus the potential energy$p_{o} \hat{{v}}=\hat{{u}}_{o} /3$. Hence, the factor 4/3 is identified as Poisson polytropic index $b=c_{p} /c_{v} $ and total particle rest mass will be composed of \textit{electromagnetic} and \textit{gravitational} parts $m_{o} =m_{em} +m_{gr} =(3/4)E_{o} /c^{2}+(1/4)E_{o} /c^{2}$. At cosmological scale, respectively 3/4 and 1/4 of the total mass of closed universe will be electromagnetic (\textit{dark energy}) and gravitational (\textit{dark matter})$^{\mathrm{1}}$ in nature as was emphasized by Pauli (\textit{Theory of Relativity}, Dover, 1958). Also, Poincar\'{e}-Lorentz \textit{dynamic} versus Einstein \textit{kinematic} theory of relativity will be discussed.\\ $^{\mathrm{1}}$ Sohrab, S. H.,\textit{ ASME J. Energy Resources and Technology} \textbf{138}: 1-12 (2016). $^{\mathrm{2}}$ Poincar\'{e}, H., \textit{Rend. del Circ. Mat. Palermo }\textbf{21}: 129-176 (1906). $^{\mathrm{3}}$ Hasen\"{o}hrl, F., \textit{Annalen der Physik }\textbf{321}: 589-592 (1905).