The connexion is 'an ever-changing but always consistently interlocking graphic representation of the way the 64 hexagrams interrelate'. What does 'always consistently interlocking' mean?

Any connexion generated by this program is called transcendental because the explicit information necessary to generate it, transcends to an implicit level in the very result.
The 'explicit information' consists of the hexagrams themselves, and their one-to-one correspondence with the China Labyrinth's hexagons. For a better understanding of the nature of a connexion, it is advised to read the China Labyrinth first.
The essence of a connexion is that each of the 64 hexagons differs from all others in terms of the pattern its neighbours form around it. Therefore each hexagram, can be identified by this very pattern.
A connexion may be 'wiped clean', leaving a bare connexion of 64 unmarked hexagons, without losing any of its information!

Every hexagram can always be identified by the pattern of its neighbours

That is what 'always consistently interlocking' means, and what the program essentially does.

The number of different connexions may very well surpass the number of atoms in the universe, and each shows a way the hexagrams 'interact'. The most direct form of interaction exists between two adjacent hexagrams. By the nature of a connexion such interaction is always between yin lines.
Lines are not static: old yang lines open up to become young yin and old yin lines close down to become young yang. The hexagram that connects to a young yin line has been a 'waxing influence' on the old yang, the hexagram that connects to an old yin line is a 'waning influence'. If changing lines appear in a divination, the connexion shows these influences.
Can any yin line connect to any hexagram? No.

By the nature of the China Labyrinth, a particular yin line can only connect its hexagram to eight other hexagrams. Here's an example. Suppose we want to connect a hexagram to the second line (the bottom edge) of the hexagram Chun.

Obviously the connecting hexagram must have a yin line in the fifth position (the top edge), but that's not all.
By the nature of the connexion the positions left and right between the two hexagrams are defined by Chun's lower trigram.
The bottom line, corresponding to the bottom-left side, is yang, therefore the position adjacent to it will remain empty. This position however borders on the top-left side of the hexagram we want to link up. Therefore the top-left side of this hexagram, corresponding to its top-most line, must also be yang.
A similar reasoning applies to the third line of Chun, corresponding to its bottom-right side. It is a yin line, so the position next to it will have a hexagram. This hexagram will also have the hexagram we want to link up as its neighbour. Consequently the hexagram we want to link up must have a yin line in the fourth place.
The conclusion is that only hexagrams with the upper trigram Ken will fit. And there are precisely eight of these.

Of course a hexagram can interact along any and all of its yin lines, so the number of them is a measure of its 'affinity', its tendency to do so.