The Generative Unity of the Fundamental Constants

This short mathematical physics note presents a unifying theorem showing that the constants e, i, φ, and π arise as complementary projections of a single minimal symmetry: the quarter-cycle operator acting on the undivided identity I. Within the 4-phase generative operator algebra Ψ = (C, θ, M, n), with update rule G(Ψ) = (C+1 mod 4, θ+φ, αM, n+1), we establish five closure results:(i) algebraic closure yields the imaginary unit i,(ii) geometric closure yields π,(iii) recursive stability selects φ,(iv) exponential growth and global regulation yield e and Φ,(v) all four together form the smallest sufficient constant set for the generative operator. As a corollary, different physical regimes correspond to the dominance of different constants: classical (e, π), quantum (i), chaotic (α, δ), and coherent (φ). The result provides a minimal generative framework linking analysis, geometry, recursion, and physical structure.