Pi_as_Dimensional_Mapping_Constant

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 . V2.0Corrected index-offset errors in the lookup table (Π(3)=π/2, etc.); unified the Gamma closed form; retracted the incorrect recurrence n/2 and the incorrect asymptotics √(2/(πn)); added parity formulas and correction record; added the convention S₀=2; stated the recurrence in the closed form 2π/n; corrected wording in the geometric interpretation; added references