The Euler Universe: A Relational Phase Framework for Cosmology — Construction, Philosophy, and Stress-Testing of the Spiral U(θ) = e^((λ+i)θ)

We develop and stress-test a toy framework in which the cosmic phase object is the logarithmic spiral U(θ) = e^((λ+i)θ) — the unit circle when λ = 0 — with time identified with phase angle, c with the rotation rate, and cosmological redshift reinterpreted as accumulated phase, 1+z = e^(Δθ). Examined as a possible alternative to metric expansion, it is found wanting in instructive ways. The static (λ = 0) reading is contradicted by four independent observations — Type Ia light-curve time dilation, the CMB blackbody spectrum, the Tolman surface-brightness test, and supernova acceleration — and further constrained by baryon acoustic oscillations, structure growth, and nucleosynthesis. The spiral generalization…