No m-cycles of the 3n−1 map for m ≤ 61

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. Those three facts are three constraints on one vector, and using them together rather than separately is a fourth ingredient: a cycle cannot keep all its local minima near the floor and still carry its odd steps, because the chaining limits how fast the minima can climb. With Rhin’s bound this gives: the 3n − 1 map has no m-cycle with 1 ≤ m ≤ 61 other than the two known ones. For m ≤ 2 that is a floor-dependent form of a theorem Simons proved without any floor; for 3 ≤ m ≤ 61 we know of no earlier statement. For m ≤ 52 no admissible cycle length lies below Rhin’s ceiling; for 53 ≤ m ≤ 61 the admissible lengths are excluded, the closest by 0.1 bits. At m = 62 one length remains, 83130157078217, with 0.3 bits of room, and a floor of 2^(55.25) removes it. The same tables give m ≤ 49 from 2⁴⁰, m ≤ 51 from 2⁴⁴, m ≤ 68 from 2⁵⁶, m ≤ 74 from 2⁶⁰ and m ≤ 89 from 2⁶⁸; run on the 3n + 1 side at Hercher’s floor the same machinery returns m ≤ 90 against his published 91. No published verification floor and no m-cycle theorem with m ≥ 3 is known to us for this map.