Exploration of Renormalization Group Flow Under The Forced GUT Equivalency Within the Stochastic Limit

We validate, through an example, the direct correspondence between the irreversibility of renormalization-group (RG) flow and entropy production thermodynamics imposed by Newell. Using the local RG framework of Osborn and Jack, we identify a scheme-invariant potential \( \tilde a(\mathbf g) \) and a positive-definite tensor \( \chi_{ij} \) satisfying an exact gradient formula, \( \partial_i\tilde a=\chi_{(ij)}\beta^j \). Mapping this structure onto the GENERIC formalism of Grmela and Öttinger reveals that RG evolution is a purely dissipative process in coupling space, governed by \( \dot g^i=M^{ij}\partial_j S \) with \( S=-\tilde a \). Numerical integration of a three-coupling gauge--Yukawa model confirms a strictly monotonic \( \tilde a(\sigma) \), verifying \( \dot{\tilde a}=\beta^i\chi_{ij}\beta^j\!\ge\!0 \) to machine precision. The result validates the thermodynamic interpretation of the four-dimensional a-theorem and confirms the imposed validity of RG irreversibility, validating the Newell's framework thermodynamics integration.