On the Novel Application of Araki Relative Entropy in the Formulation of the Spectral Obidi Action (SOA) of the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE), first proposed and further developed by John Onimisi Obidi, extensively uses the Araki relative entropy as a core mathematical component

The standard concept of a "Theory of Everything" in mainstream physics is a hypothetical framework that unifies all four fundamental forces. This established field uses the Araki relative entropy as a rigorous mathematical tool in quantum field theory (QFT) and statistical mechanics, primarily to study entanglement, causality, and thermodynamic properties in localized spacetime regions. 

The novel Theory of Entropicity (ToE), however, is a specific, alternative approach that uniquely elevates entropy to a fundamental field of nature, from which all physical laws and spacetime geometry emerge. 

In the Theory of Entropicity (ToE), the Araki relative entropy is used in several ways: 

• It serves as a measure of distinguishability between quantum states within ToE's quantum formulation.
• Its properties are utilized to maintain consistency with fundamental principles like the arrow of time and causality.
• It functions as a tool for understanding the relationship between information, entropy, and spacetime geometry, including in the context of holographic principles.
• It is fundamental to the "Obidi Action," a principle in ToE that leads to the "Master Entropic Equation," an entropic analogue of Einstein's field equations. 

While Araki relative entropy is an established tool in quantum information and QFT, the Theory of Entropicity (ToE) applies it in a novel, "field-theoretic" manner, hitherto unknown in modern theoretical physics. Thus, the Theory of Entropicity (ToE) bears an original innovation in its mathematical foundation.