The Logarithmic Energy of Golden-Angle Fibonacci Points on S²: Non-Existence of the Linear-Coefficient Limit, a Log-Periodic Law, and a Closed Form for Its Mean

The minimal logarithmic energy of N points on the unit sphere satisfies E(N) = (1/2 − ln 2)N² − (1/2)N ln N + C·N + o(N), and the optimal constant C_opt = 2 ln 2 + (1/2) ln(2/3) + 3 ln(√π/Γ(1/3)) ≈ −0.05560530 was proven in October 2026. For the most widely used deterministic point family — golden-angle Fibonacci spiral points — we show by controlled computation to N = F₃₁ = 1,346,269 that the corresponding linear coefficient has no limit: C(N) converges to a log-periodic function of N with multiplicative period φ² (peak-to-peak 1.71×10⁻⁴), predicted with zero free parameters from the spiral’s local lattice traversing the closed geodesic of ℤ[φ] on the modular surface, and confirmed by the raw sums to ~10⁻¹⁰. The log-period mean admits the conjectured closed form C̄ = (1/4) ln 5 − (1/2) ln π − γ/2 + (1/2)·L′(1,χ₅)/L(1,χ₅) = −0.0447735589091889797…, verified to 60 digits against the geodesic integral; Fibonacci-N subsequences converge to two parity-dependent limits. This places the spiral above the Diamond ensemble constant c⋄ = −0.04922209… (higher energy) and quantifies its per-point deficit relative to optimal. This is a results report with a stated error model and a conjectured closed form, not a proof; every claim carries an epistemic tag (DERIVED / OBSERVED / PROPOSED / EXTERNAL).