The φ-ODE: A Single Equation Encoding the Geometry of Expanding Radiating Bodies

We derive a single ordinary differential equation whose structural coefficientis the golden ratio φ = (1 + √5)/2, and whose solutions completely encodethe geometry of an expanding radiating shell: Archimedean spiral (˙r = const,background vacuum ΛI), Fibonacci spiral (¨r ̸ = 0, perturbed vacuum ΛII), orcontraction.The state variable is Φ(t) ≡ R(t + T)/R(t), the ratio of shell radii separatedby one natural period T. The equation isdΦdt =−Γ(t)Φ2 −Φ−1 ,where Φ2 −Φ−1 = 0 is precisely the equation whose positive root is φ. Thefixed point Φ = φ is a stable global attractor for all Φ > 0: every expanding body,regardless of initial conditions, converges to golden-ratio growth under the vaz˜aocoupling. The transition hypersurface Σc between the two vacuum phases is thelocus Φ = φ.We prove that the φ-ODE contains the second-order vaz˜ao dynamics as aspecial case: near the stellar emission peak z∗ ≈ 1.86, the vaz˜ao field satisfiesV′′ ≈ V/τ2, whose growing mode u(t) = et/τ has a constant ratio Φ = eT/τthat equals φ at the unique period T = τ lnφ. This period is derived, notassumed. With the DESI DR2 calibrated value τ = 4.50, the Fibonacci timescaleis Tφ = τlnφ ≈ 2.17 (in redshift units) and the universal pitch angle of thegolden spiral is ψφ = arctan(lnφ/π) ≈ 8.71◦.The φ-ODE provides the first physically motivated derivation of why thegolden ratio appears in the growth of structures with cosmological memory.