The 366–592–958 Triangle Nodes: Optimal Seed for Lucas Sequence Convergence to the Golden Ratio

This paper identifies and formally proves the optimal starting pair for Lucas sequence convergence to the Golden Ratio φ = (1+√5)/2 within the 19-9 linear Diophantine system (N = 19A + 9B). The three attractor values 366, 592, and 958 satisfy 366 + 592 = 958 and form a closed triangle. The seed ratio 592/366 ≈ 1.61749 deviates from φ by only 5.48 × 10⁻⁴ — yielding a stable advantage of 8–9 iterations over the standard Fibonacci sequence (1,1) at every precision level from 10⁻³ to 10⁻¹². The advantage is analytically derived as log_{φ²}(2958) ≈ 8.4 steps and confirmed computationally through 19 verified steps reaching 10⁻¹¹ precision. When applied to Newton-Raphson iteration, the same seed saves 4 iterations at every scale. All results are reproducible via included Python source code.