This framework formalizes two foundational principles of the Theory of Entropicity (ToE): the Obidi Conjecture (OC), which posits that entropy is a real and dynamical physical field underlying all observable phenomena, and the Obidi Correspondence Principle (OCP), which requires that all empirically established laws of physics arise as limiting or coarse-grained approximations of entropic dynamics. The purpose of this formulation is to elevate these principles from conceptual formulations into mathematically stated axioms and theorem-like structures. The framework also establishes their relation to the Obidi Action (OA) as the local variational generator of entropic dynamics, and to the Vuli-Ndlela Integral (VNI) as the global entropy-constrained selection principle governing admissible physical histories.
Let M be a differentiable manifold representing the domain of physical events.
Let S be a real-valued field on M, called the entropic field, assigning to each event x in M a local entropic magnitude S(x).
Let physical configurations be denoted by phi, where phi may include matter fields, effective geometric degrees of freedom, gauge-like structures, and observational states.
Let the total physical dynamics be governed by an action functional of the general form:
Obidi Action: A[phi, S] = integral over M of L(phi, partial phi, S, partial S, coupling terms) dmu
where dmu is the invariant measure on M and L is the local entropic Lagrangian density.
Let the admissible dynamical histories be selected globally by the Vuli-Ndlela Integral, in the standard form you have established:
Z_ToE = integral over entropy-admissible configurations of exp[i S_classical / hbar] times exp[-S_G / k_B] times exp[-S_irr / hbar_eff]
with admissible domain restricted by the entropy condition:
Lambda(phi) > Lambda_min
where Lambda(phi) is the entropy density functional.
This structure gives ToE both a local differential formulation through the Obidi Action and a global selection formulation through the Vuli Ndlela Integral.