No m-cycles of the 3n−1 map for m ≤ 58

Let g(y) = y/2 for even y and g(y) = (3y − 1)/2 for odd y, the 3n − 1 map on the positive integers, equivalently the shortcut 3n + 1 map on the negative integers. Its known cycles are 1, (5, 7, 10) and the eleven-element cycle at 17, and every start below 2⁵¹ reaches one of them. An m-cycle is a cycle with m maximal runs of odd elements. We transpose the template of Simons and de Weger for 3n + 1 m-cycles to this map, deriving its constants on this side rather than borrowing them: the odd step subtracts, so in the variable u = y − 1 an odd run is exact multiplication by 3/2, a run of a odd steps starts at y ≥ 2^(a) + 1, the cycle equation bounds the linear form Λ = o log 3 − K log 2 by m/(x_(min) − 1) with constant one, and successive local minima obey u_(i + 1) < u_(i)^(log₂3)/2. With Rhin’s bound this gives: the 3n − 1 map has no m-cycle with 1 ≤ m ≤ 58 other than the two known ones. For m ≤ 2 that is a floor-dependent form of a theorem Simons proved without any floor; the statement is new for 3 ≤ m ≤ 58. For m ≤ 52 no admissible cycle length lies below Rhin’s ceiling; for 53 ≤ m ≤ 58 the admissible lengths are excluded by the chaining, the closest by 9.8 bits. At m = 59 two lengths remain; they are the output of the template’s last step, not its input, and the floors that remove them are 2^(51.87) and 2^(55.18). The same tables give m ≤ 44 from 2⁴⁰, m ≤ 49 from 2⁴⁴, m ≤ 63 from 2⁵⁶, m ≤ 68 from 2⁶⁰ and m ≤ 82 from 2⁶⁸; on the 3n + 1 side at the floor of Simons and de Weger the same enumeration and tests return their Lemma 18 to the unit. No published verification floor and no m-cycle theorem with m ≥ 3 is known to us for this map.