Cauchy Problems for VP Theory: an Eight-Stage Closure, Variational Core, Physical Branches, and the Cosmological VEPT/VPR-G. Time-Matrix Projection

This work presents VP theory (Visata per Planką; Universe in terms of Planck pressure) as a coherent mathematical and physical research program aimed at describing gravity, inertia, temporal structure, and cosmological dynamics through the constitutive state of the vacuum medium. The paper Cauchy Problems for VP Theory: an Eight-Stage Closure, Variational Core, Physical Branches, and the Cosmological VEPT/VPR-G. Time-Matrix Projection develops a rigorous eight-stage closure framework designed to test whether the VP system satisfies the deterministic standard of the Cauchy problem, preserves constraint propagation, admits a consistent variational core, and supports physically meaningful branches from the weak-field regime to cosmology. Particular emphasis is placed on the variational core, the selection of physical branches, and the cosmological VEPT/VPR-G projection through the VP time matrix, with the aim of analytically explaining the origin of discrepant observed Hubble parameters. The work is relevant to mathematical relativity, modified gravity, numerical evolution, quantum-fluid analogues, early-universe theory, and precision cosmology.