Pi_as_Dimensional_Mapping_Constant_v1.0

We define a one-parameter family Π(n) of dimensionless ratios as the quotient of surface measures of consecutive unit spheres: Π(n) = Sn Sn−1 = √π Γ(n/2) Γ((n + 1)/2) , where Sn denotes the surface measure of the unit n-sphere Sn ⊂ Rn+1. In particular, Π(3) = π. The formula is classical in n-sphere geometry; our contribution is to organize it as a "cross- dimensional mapping ratio" and to clearly separate mathematical definition, geometric intuition, and future physical applications. We do not prove the irrationality or transcendence of π, nor do we present any physical predictions .