Why does the α formula use 360 degrees? The number 360 appears to be a human convention — the Babylonian choice for dividing a circle. This paper proves it is not arbitrary in the context of Pentagon Physics. The formula 360/φ² gives the golden angle (137.508°), and 360 is the smallest highly composite integer N such that N/φ² ≈ α⁻¹. The degree system is the natural integer encoding of angular closure where the series structure is manifest and convergence is rapid. A gradians test (400 units) produces the formula 400/φ² − ... where the coefficients are no longer composed purely of primes — the structure breaks. 360 participates structurally.