π as a Structural Constant in Vacuum-Dominated Cosmology

Constant [Formula: see text] appears ubiquitously in quantum field theory and cosmology, often through rotationally invariant measures and the normalization of gravitational field equations. This work examines whether standard vacuum energy physics admits natural dimensionless ratios in which [Formula: see text] emerges as a structural constant, without introducing new dynamics. At the microscopic level, a Gaussian-regulated zero-point energy integral yields a dimensionless normalization identity equal to [Formula: see text], traced explicitly to the rotationally invariant momentum measure. At the macroscopic level, a spatially flat, vacuum-dominated FLRW universe implies a dimensionless ratio equal to [Formula: see text] by algebraic rearrangement of the Friedmann equation. These results are shown to be structural identities rather than independent predictions. Deviations from [Formula: see text] are quantified once matter and radiation are included, rendering the macroscopic ratio epoch dependent. The analysis clarifies the origin, meaning and limitations of [Formula: see text]-based relations within standard quantum field theory and cosmology.