The hypothesis of >>47 does not hold.
spec index digits group
e 10^12 6167766 RRLRR(LRL[4+4*k]RLR[6+4*k])*
sqrt5 10^9 156740731 (RRLLLLRR)*
sqrt7 10^9 171773357 (RRLRLRR)*
sqrt3 10^9 190649183 (RLR)*
sqrt2 10^9 191387843 (RLLR)*
phi 10^9 208987641 (RL)*Sqrt3 has more switches (2N/3) than Sqrt2 (N/2) but grows at a slower rate. Sqrt7 also has more switches (4N/7) than Sqrt2 and slower growth.