Tom's move for 3rd and 5th row parallel ladders

There is a version of Tom's move that works for parallel ladders on the 3rd and 5th rows. It requires a large amount of space: 

Basic idea
By playing at "1", Red can connect to the edge. Verifying this requires a lot of steps, but here is the basic idea:  Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by edge template V2m. The latter template is itself based on Tom's move at "x". It works, for example, like this:  Now Red is connected by the (ordinary) Tom's move.

Proof that Red is connected
Red has two main threats. Via a ziggurat:

 And via edge template IV2b: 

Blue must play in the overlap: 

Intrusion at p, q, r


Now Blue must play in one of the 3 shaded cells. ​If Blue plays in the left 2 of those 3, then Red connects via IV-2-b. ​ Otherwise, Red connects via Tom's move.



Intrusion at s


Red is connected by edge template IV1d.

Intrusion at t


Now Red is connected by edge template V2m. If Blue plays 4 on the first row instead, Red connects by Tom's move: 

Intrusion at u
 Red is connected by Tom's move.

Intrusion at w
Red responds with



We will assume that Red simply defends the pink ziggurat, and therefore we will not need to consider any Blue intrusions there.

Red's main threats are



and



Blue must play in the overlap:



If Blue plays at a, then Red plays 5, after which Red connects via

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:e4 7:f4 B 2:g5 4:f3 6:e5 6:d6 E *:f5 *:h3 S red:area(i4,g6,j6,j4)" />

or

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:e4 7:e5 9:f5 11:h3 B 2:g5 4:f3 6:e6 8:d6 10:f6 S red:area(i4,g6,j6,j4)" />

If Blue plays at b, then Red plays 5, after which Red connects via

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:f4 R 5:e4 B 6:(e5 d6) R 7:h2 B 8:h3 R 9:g4 S red:area(i4,g6,j6,j4) E *:(g1 f3 f5 h4)" />

or

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:e4 7:e5 9:f5 11:h3 B 2:g5 4:f4 6:e6 8:d6 10:f6 S red:area(i4,g6,j6,j4)" />

If Blue plays at c, then Red responds with

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:h3 B 2:g5 4:e6 S red:area(i4,g6,j6,j4)" />

This forces Blue to defend towards the top, after which

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:h3 7:e4 B 2:g5 4:e6 6:(f2 f3 g2) S red:area(i4,g6,j6,j4)" />

connects directly or via red 1.

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:i4 5:h3 7:e4 9:e5 11:f5 B 2:g5 4:e6 6:(f2 f3 g2) 8:f4 10:d6 S red:area(i4,g6,j6,j4)" />

If Blue plays at d or e, then Red plays like this:

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:(h3 i3) R 5:e5 B 6:e4 R 7:f2 B 8:f4 R 9:g4 E *:(f5 h4) S red:area(i4,g6,j6,j4)" />

If Blue plays at f, then Red plays like this:

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:h4 R 5:f5 E *:(f2 e4) +:f4 S red:area(i4,g6,j6,j4)" />

Now Red threatens to play in one of the cells marked "*", and the only overlap (apart from the two captured cells below 5) is at "+", so Blue must play there. Then Red responds like this:

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 B 2:g5 R 3:i4 B 4:h4 R 5:f5 B 6:f4 R 7:h2 E *:(g1 e4) +:(g4 i3) S red:area(i4,g6,j6,j4)" />

Now Red is connected left by double threats "*" and right by double threats "+".

Intrusion at x
Red can respond here:

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6" />

Red's main threat is as follows,

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 5:f5 B c3 d5 e3 c5 2:e6 E *:e4 *:f2 S d6 e4 f3 f2 g2 g4 f4 f5 f6" />

connecting back via one of the cells marked "*".

If Blue plays either of the two of those cells adjacent to the edge, then Red plays the

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 5:f5 B c3 d5 e3 c5 2:e6 E *:(d6 f6 i4) S blue:(d6 f6)" />

same move anyway, connecting down via the other of those two cells or Tom's move.

If Blue plays one of the five upper-right of those cells <hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6 S blue:(f3 f2 g2 g4 f4)" /> then Red still connects via Tom's move,

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 5:e4 7:f5 9:i4 B c3 d5 e3 c5 2:e6 6:d6 8:f6 E *:(f3 f4 g4) S blue:(f3 f2 g2 g4 f4)" />

since at least two of the three cells marked "*" are still available to Red 1 to connect to Red's left group.

This leaves only two remaining tries for Blue:

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 B c3 d5 e3 c5 2:e6 S e4 f5" />

If Blue plays the right of those two,

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 5:e4 B c3 d5 e3 c5 2:e6 4:f5 E *:d6 *:h4" />

then Red connects via double threat.

Thus Blue instead plays the left of those two. ​ Red responds as follows.

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 1:g3 3:e5 5:f2 7:f3 B c3 d5 e3 c5 2:e6 4:e4 6:g2 S d6 e6" />

Now, it does not matter which of the two shaded cells Blue 2 is at, since


 * if Blue plays the other too then the order in which they were played doesn't matter, and
 * if instead Red plays the other, then whichever one Blue played will be dead.

So this is equivalent to Red having a clipped version of Edge template IV1d.

In particular, Red is connected down.

Intrusion at v or y or z
If Blue plays one of the three highlighted cells, then Red responds as shown below.

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:e5 E *:d6 *:e6 S blue:(f5 f6 g6)" />

This captures the two cells marked "*", forcing Blue 4, and then Red 5

<hexboard size="6x14" coords="hide" edges="bottom" visible="area(f1,d3,d6,n6,n4,l2,h1)" contents="R e2 d4 d3 B c3 d5 e3 c5 R 1:g3 3:e5 5:f2 B 4:e4 E *:d6 *:e6 S blue:(f5 f6 g6)" />

connects by double threat or edge template IV2e or edge template IV2p.