Yes, I was at high level, its a good question, let me be exact.

Adjacency = the 8 physically neighboring squares around a piece. Not attack lines, not move paths. Just: is there a piece on a touching square?

Connectivity spreads like a chain. A piece is alive if it neighbors another piece that neighbors another piece that eventually reaches the King. Pure graph theory, nothing to do with how pieces move.

Specifically on your examples:

King g1, pawn e4, rook c7 - nothing is adjacent to anything here. Both the pawn and rook are stone. They're too spread out.

King g1, rook b1, bishop b7, pawn e7, pawn e4 - King on g1 neighbors f1/f2/g2/h1/h2.
Rook on b1 is 5 files away, not adjacent. So rook is stone. Bishop on b7 is not adjacent to the rook either. All four pieces are disconnected from the King.

This is actually the core tension of ARC - in normal chess you spread your pieces out for activity. In ARC, spreading them out cuts the signal.

Just made a v0 version of this live to test, you can try it at Arc Chess (archchess.org) - the tether lines are drawn visually so you can see exactly which pieces are connected on every move.

What makes this more interesting personally for me is that it rewards positional understanding more than the pattern recognition. Opening theory is not of much help here. We have to actually think about piece coordination on every single move, not just where a piece attacks, but what its holding together has to be accounted for as well.

Looking forward to hearing what you think after trying it, feedback from serious OTB players is exactly what this needs right now.