Correlation of the Gibbons--Hawking 1977 Euclidean Action Formalism with the Closure of Singularities and Infinities in the VP Hypothesis

This study examines the structural correlation between the Gibbons--Hawking 1977 Euclidean action formalism and the closure of singularities and asymptotic divergences within the \textit{Universe in Terms of Planck Pressure} (VP) hypothesis. Its central aim is to demonstrate that the finiteness of the Euclidean partition function in the VP framework does not arise as an independent auxiliary assumption, but follows from the same regularity, finite-mass, and asymptotic-subtraction conditions that elsewhere in the theory are used to close the central singularity and infinity problems. In this sense, the work seeks to connect Euclidean semiclassical gravitational thermodynamics with the internal consistency requirements of the VP constitutive structure. The work is developed from a unified Euclidean generator in which the total action is decomposed into core, shell, matching, exterior, and asymptotic subtraction contributions. Within that construction, the Euclidean partition function is reformulated as a coupled functional over geometry and medium-state variables, rather than as a purely geometric object. The formalism is then constrained by the constitutive chain \[ A=\chi^2, \qquad \phi = 1-A, \qquad P = P_{\infty}+P_{\mathrm P}\phi, \qquad n=|\Psi|^2=A, \] which makes the static metric sector, the local clock field, the normalized pressure deficit, and the order parameter mutually dependent. On this basis, the paper derives the core contribution to the Euclidean action, analyzes its stationary points through radial Euler--Lagrange equations, and performs algebraic and variational consistency checks via the Jacobian matrix of the constitutive constraints, the Hessian of the core potential, and the principal symbol of the field equations. The principal result is twofold. First, if the VP regular-center conditions are satisfied, then the Euclidean core contribution remains finite. This means that the same regularity branch that removes central singular behavior in the Lorentzian formulation also guarantees finiteness of the Euclidean action. Second, if the asymptotic subtraction layer is imposed in the exterior region, then the full Euclidean action remains finite and its external vacuum limit reproduces the standard Gibbons--Hawking result in the Schwarzschild sector. The consequence is that the GH77 formalism is not rejected in the VP theory; rather, it is recovered as a special exterior-vacuum limit of a broader statistical theory in which geometry and medium-state structure are jointly present. The relevance of the work is methodological as well as conceptual. Methodologically, it proposes a way to reformulate the singularity-closure question in terms of the finiteness of a Euclidean statistical action. Conceptually, it attempts to bridge gravitational thermodynamics, variational regularity, and constitutive microphysics inside one framework. This is potentially important for any extension of the VP hypothesis toward a fully dynamical effective action, because it identifies which sectors are already mathematically closed and which remain open. What is new in this work is therefore not a mere restatement of the Gibbons--Hawking result, but a reinterpretation of that result inside a different constitutive architecture. The novelty lies in showing that, within the VP proposal, Euclidean action finiteness, regular-center structure, asymptotic subtraction, and finite mass accounting become parts of one and the same closure mechanism. The presented study thus positions the GH77 thermodynamic formalism as a consistency test of the VP internal structure and, simultaneously, as an exterior limiting case of a larger geometry--medium statistical theory.