The relationship between difference and ratio and a proposal: Equivalence of temperature and time, and the first spontaneous symmetry breaking

The mathematical form of the energy-entropy relationship is that of the relationship between the difference and the ratio of any two ‘entities’ x and y creating a geometrical 3D projection x = y · z, i.e., a surface of the shape of a hyperbolic paraboloid being a particular form of a quadric. The significance of this relationship is discussed here in a realm beyond thermodynamics. Somewhat intuitively, yet still in strict mathematical analogy to the relation between its most fundamental state variables, I propose for any isolated universe the cosmological state variables, absolute temperature and absolute time, to be equivalent much in the same way as energy and entropy are equivalent for an isolated ensemble of similar objects. According to this principle, the cosmological constant is inversely proportional to the squared product of absolute time and absolute temperature. The spontaneous symmetry breaking into a time component and a temperature component of the universe takes place when the first de facto irreversible movement occurs owing to a growing accessed total volume. The relationship between difference and ratio in mathematics and physics In mathematics the concept of ‘a group’ is used to formulate, in a most general way, any kind of mathematical operation, for example, addition, multiplication or rotation. Generally, a mathematical group consists of ‘elements’ a, b, c, ... and an instruction that attributes to each pair of elements, say a and b, a new element a b  in such a way that three group axioms G1, G2 and G3 strictly apply. Abelian groups are commutative for which group axiom G4 applies as well: For any three elements and associativity applies a b c