THE 45-HECTARE PARABOLIC MODEL: Functional Zoning, Triangular Centroid (15,30) and Demographic Symmetry (22.5, 506) of Vatican City State

Vatican City State is 44 hectares physically (Lateran Treaty 1929) but 45 hectares conceptually. This 1-hectare difference is not an error but a functional constant b = 45 that unifies geography, mathematics and demography. We propose two complementary models: (1) The Parabolic Model f(x) = -x² + 45x + 0.25 with vertex f(22.5) = 506.5 matching UN WPP 2026 population 506, and f(15) = f(30) = 450.25 proving symmetry; (2) The Triangular Model g(x) = -2x + 90 with base 45 = b, height 90 = 2b, area 2025 = b² proving b² as integral from 0 to 45 of g(x)dx. The centroid of the triangle is at (15, 30), which divides Vatican into 15ha Public & Sacred Zone and 30ha Private & Governance Zone (residence of 506 souls). The centroid of the parabola is at (22.5, 202.5) = (b/2, b²/10). The 45-year window 1980-2025 validates b as spatio-temporal factor.