The Ratio and the Arrow: Time Asymmetry in the Euler Reflection Formula

The Euler reflection formula Γ(z)Γ(1−z) = π/sin(πz) has been used exclusively in its product form for three centuries. This paper examines the complementary ratio form Γ(z)/Γ(1−z), which is non-commutative under z ↔ 1−z and therefore carries directional information that the product destroys. I prove that the product is time-symmetric while the ratio is time-asymmetric, and that neither alone contains the complete information of the reflection formula. Both are required to recover the individual Gamma values. At z = 1/4, the product yields π√2, the foundation of time-reversible physics, while the ratio yields G* = Γ(1/4)/Γ(3/4) ≈ 2.959, a transcendental algebraically independent of π. A quadratic equation built from G* produces 1/α = 137.036 as its larger root, connecting the fine structure constant to the non-commutative branch of the reflection formula. I identify π, the lemniscate constant ϖ, and G* as three evaluations of a single object: the Gamma function at its half-integer and quarter-integer points. I argue that the apparent time-reversibility of fundamental physics is inherited from the exclusive use of the commutative product rather than from any property of nature itself.