π³/15: The Geometric Root of π, Λ, and G

This work examines the possibility that the geometric constant π, the cosmological constant Λ, and Newton’s gravitational constant G admit a common boundary geometric origin.

The construction begins with a spherical radiative boundary whose topology, flux projection, phase space organization, and harmonic spectral summation combine into the dimensionless invariant

π³/15.

Under the boundary projection adopted here, this invariant is related to the cosmological horizon through

ΛR_H² = π³/15

and, in corresponding form,

Ω_Λ = π³/45.

If the horizon is further resolved holographically through

N = A_H / l_P²

with

l_P² = ħG / c³,

then G appears as the Planck scale closure parameter associated with that same horizon geometry.

In this picture, π encodes the spherical boundary structure, Λ its cosmological projection, and G its microscopic holographic closure.

The construction is geometric rather than dynamical, and suggests that part of the normalization of cosmological and gravitational parameters may be traced to a common radiative boundary structure.