Lining up in Threat Zones

If that wasn't obvious enough, Figure 7a, point 2., makes it absolutely clear that "line up" and "Slide sideways" are equivalent.
Unfortunately, the rule is mostly clear.
Unfortunately, because it allows some pretty egregious moves.
On the other hand, I have never (yet?) seen it abused in actual play.

I’m with you here, for the most part, but disagree with your final statement. Both of the moves in the later diagrams zendor uses are permissible BUT you can’t do them by sliding sideways as a group for 1 pip - you would need to move as individual elements, costing you 2 pips to line up the TZ’’s in the first example and 4 pips in the second example.

P

Arnopov makes a good point...and Figure 7b even reinforces the options given in Figure 7a.

In Figure 7b we have a group column with its front element only in a TZ, and its move options are:-
　　1. Halt
　　2. To line-up with Blade-Y or pivot and line-up with Blade-X (sliding sideways without making contact)
　　3. To move straight towards and/or slide sideways (to) line-up with Blade-Y without making contact
　　4. To slide/pivot (and move) towards Blade-X and line-up (sideways) without making contact
　　5. To move into front-edge contact with either Blade-Y or Blade-X
Only option 5 ends in contact...options 2, 3 and 4 all involve sliding sideways without making contact.

In addition, Figure 11 shows being lined-up without making contact.

So it appears that ‘lining-up’ is not solely confined to being in front-edge contact with the enemy.
Sliding sideways to distantly line-up is entirely separate from the ‘free sideways slide’ when in contact.
The only thing they have in common is the word ‘sideways’...but that doesn’t make them the same thing.

Hi Zendor, I’m sorry. I’m not a native speaker🖖🏼. I tried to say “exactly to the left or right”🙄😜😋 Regards, Ronald

Further I will proceed from the solutions that the whole group can make a sideways slide being in the opponent's TZ.
I understand that not everyone agrees with this interpretation, but I need to base further logic on some bases.

I wanna show you the next diagram. What do you think, is it possible for Ps element to make a contact with a diagonal movement?

I don't think it is an illegal move for that reason. 

It can then slide to its right in an attempt to conform until its movement is stopped by the Ax, at which point, the Bd must either conform or fight as if overlapped.  (Or it could just move forward diagonally until it is clear of the Ax and then move straight forward.)

I do wish I could give you a nice simple unambiguous answer Zendor.
Unfortunately such a simple explanation is not possible, so be prepared
for yet another one of my own personal long-winded interpretations.

Basically I think that no, the Ps in your picture cannot contact the Bd.
The reason it cannot do so is because it is already ‘lined-up’ in a TZ.

Figure 11 shows what being lined-up is...it’s when lines extending an
element’s sides matches a distant enemy element’s sides. And Figures
7a and 7b shows the possible options for moving into a lined-up position
when in a TZ...but I can find no examples of moving out of a lined-up
position while they are still in a TZ.
(It’s as if being lined-up in a TZ locks troops into that configuration)

The FAQ has an example of this:-
Q: I have an element of Blade that starts its move in the threat zone of a Spear
    that is on the end of a line of three enemy Spear. My Blade is lined up with the
    opposing spear. As I move my Blade forward, can I do so at an angle to catch
    multiple enemy Spear elements in my Threat Zone? Is this move allowed?
A: No, you must stay lined up with Spear in front. Please reference diagram 7b
    for the proper ways to respond to a threat zone.


Alternative moves
Could the Aux move out of the way?: no, as it can only leave a TZ by
moving directly straight back, and the Ps is in the way.
Could the Ps interpenetrate the Aux?: no, because the Ps must start
lined-up behind the Aux in order to interpenetrate it.

The only way I can see of doing it is if the Aux first moves sideways
to line-up with the Bd (see Figure 7a, option 2), then the Ps would be
lined-up behind the Aux and could interpenetrate it.
(But the Ps still needs enough room to do so...and there isn’t any)