the Common Geometric Origin of π, Λ, and G

This work examines whether three constants usually treated as conceptually distinct, the geometric constant π, the cosmological constant Λ, and Newton’s gravitational constant G, may be related through a common boundary geometry. The analysis begins with a spherical radiative boundary where topology, flux projection, phase space structure, and harmonic spectral summation combine into the invariant factor π³⁄15. Projecting this invariant onto the cosmological horizon yields the relation ΛR_H² = π³⁄15 and the corresponding dark energy fraction Ω_Λ = π³⁄45. If the horizon is further treated holographically, the Planck scale resolution introduces Newton’s constant through ℓ_P² = ħG⁄c³, with the horizon cell count N specifying the microscopic closure of the same geometry. In this framework π characterizes the boundary geometry, Λ its cosmological projection, and G its Planck scale resolution. The proposal is not presented as a complete dynamical theory but as a geometric reconstruction suggesting that aspects of cosmological and gravitational normalization may originate in a common radiative boundary structure.