Orthodox h# with only king moving for black 1

a) 1.Kd5 Bg2+ 2.Kd4 Rh4#

b) 1.Kd4 Rh4+ 2.Ke5 d4#

c) 1.Ke5 d4+ 2.Kd5 Bg2#

And we start right away with doubled cycle. Depending on the position of the white knight the mating net it is clear which mating net would be weaved around bK. But the king has to make the first move, always the tempo move to the square, where he is mated in some other phase. This gives cycle of black moves. On the other hand, the mating nets to be prepared motivate the cycle of white moves (turning together with the black cycle).

By the way, although all solutions could be cut to three halfmoves by omitting the first moves, the problem would not be sound as h#1,5. There would be new cooks. Interesting!

1...e8Q 2.Kd6 Qg8 3.Ke7 c8S#

1...c8S 2.Kxe6 e8R+ 3.Kf7 Sd6#

1...c8R 2.Kxe6 e8Q+ 3.Kd6 Rc6#

Difficult cycle of white promotions appearing in solutions (QS - SR - RQ) is realized in the miniature form. The repetition of 2.Kxe6 is the only small blemish I could find.

1.Ke4 Rc8 2.Kd3 Bc2#

1.Kxf5 Bc2+ 2.Kg4 Rg8#

1.Ke6 Rg8 2.Kf7 Bb3#

1.Kd5 Bb3+ 2.Kc6 Rc8#

Black king makes 2x4 moves along foud propeller wings, i.e. they are slightly curved. Where is the cycle then? Of course, white moves are forming four-fold cycle.

1.Kxe5 Qe7+ 2.Kxd5 Rxd6#

1.Kxe4 Rxc4+ 2.Kxe5 Qe6#

1.Kxd5 Rc5+ 2.Kxe4 Qg4#

The pattern of black moves is here the same as in the 1st example above. However, the motivation is very different. While in the Mario's h#2 Black had to make tempo moves, here bK has to capture white pieces to open lines for guarding of potential flights in mating nets. Well done with strong white material.

1.Kf3 Qa7 2.Kg2 Qxh7 3.Kf1 Qxh1#

1.Kd3 Qb7 2.Kc2 Qxh1 3.Kb3 Qxb1#

1.Kf5 Qb5+ 2.Kg6 Qxb1+ 3.Kh5 Qxh7#

White minimal with wQ - and wK quite away from actions - makes choice of possible mating nets. Black king walks into them, while wQ makes necessary preparations. The skeleton of mutual guarding Qb1-Rh7-Rh1-Qb1 motivates cycle of the 2nd and 3rd white moves.

1.Kxe5 Re8+ 2.Kxd5 Rd1#

1.Kxd5 Rd8+ 2.Kxe4 Re1#

1.Kxe4 Rxf4+ 2.Kxe5 R8xf5#

Again the idea shown in the ZJ's helpmate of annihilation of piece standing on the line needed for guarding square in the mating net. The white material is lighter here. Which one do you like more?

1.Kxd5 Bg5 2.Kc6 Sd4+ 3.Kb6 Bd8#

1.Kxf3 Be3 2.Kg2 Rg5+ 3.Kh1 Rg1#

1.Kxf4 Sxh2 2.Kg3 Rg5+ 3.Kh4 Sf3#

Cyclic Zilahi is motivated by need to transport bK to mating nets, however white officers are standing in the way. But as always one of them is not needed for checkmate, everyting just falls into right place. Model mates are nice (and perhaps expectable) bonus.

a)
1...Rb4 2.Kf3 Bc4 3.Kg4 Be2#
1...Bb5 2.Kf2 Rc4 3.Kf1 Rf4#

b)
1...Rc4 2.Kxa6 Rb4+ 3.Kb7 dxc6#
1...Bc4 2.Kxa4 Bb5+ 3.Ka3 Ra4#

Skillful blend of two known schemes involving bK and white linemovers RB.

In a) White creates battery using Indian manoeuvres and bK enters initial squares of the rear pieces of batteries, with double-check mates, everything in the diagonal-orthogonal fashion.

In b) White lets bK pass a6/a4 with further move of the interfering piece to the b-file and single-check model mate.

Where is the cycle? - you may ask again. And as in the Torsten's h#2, there is the cycle of the 1st and 2nd white moves (Rb4-Bc4-Bb5-Rc4-Rb4), although it seems almost incidental. It is perhaps the sign of a good cyclic problem - that there are many other points of sufficient interest besides the formal cycle.