There are two thermodynamic definitions for free energy as a measure of the ability of a system to do positive or negative work. First, is the energy taken in or given out by a reversible process at constant pressure and temperature, which is known as the Gibbs free energy developed by Josiah Willard Gibbs (1839-1903) in his theory of chemical thermodynamics. Second, is the Helmholtz free energy introduced by Hermann Ludwig Ferdinand von Helmholtz (1821-94) where and when he also discovered the law of conservation of energy in the year 1847. However, this law is meaningless for all open thermodynamic systems. It is meaningful only for isolated and closed systems. Confusions about free energy arise wherever and whenever the study of thermodynamics processes failed to distinguish among these three types of systems. An open system always allows both matter and energy going in and out the control volume. A closed system only allows energy to cross but an isolated system never allows neither matter nor energy to cross.

Classically, the control volume implied by a particular isolated system is topologically equivalent to a sphere of genus zero thus distinguishes an ‘inside’ and an ‘outside.’ However, at the subquantum domain this control volume is that of a Klein bottle where and when the ‘inside’ and ‘outside’ can only be relatively but not absolutely distinguished. Then by this nonvanishing genus topology both matter and energy are in situs confined onto a one-sided Möbius surface. In this sense, matter and energy never need to cross any system boundary at superluminal speed, luminal speed, subluminal speed, or at absolute rest. This triple point of open-closed-isolation completely linked every quantized space-time point of the space-time continuum as squares of energy described using second order singular symmetric Hadamard matrices embedded within a sieve of Diophantus of infinite order and dimension.