I ran a few things under a pedestrian-only (no chance of it being driven on) sidewalk slab and now, of course, the "tunnel" I made isn't supporting concrete. It's perhaps a 12 inch (30cm) section in the middle of the slab of concrete with perhaps 18 inches (45-50cm) to either side still on the compacted adobe clay soil it's been on for roughly fifty years now. It's clearly quality concrete, too.

So the question is simple, I think. Is there a rule of thumb for how much I can undermine a four inch (9-10 cm) thick concrete sidewalk before it's unreasonably susceptible to cracking, or at what point do I have to finagle back some support, if that's possible.