There used to be a lot of discussion in L19 about the true meaning of thickness. There is an old heuristic of counting 3 points for every height of a thick wall (note that the thickness of the wall doesn't directly contribute to this equation, and instead is normally discounted for the value of an attack by the opponent, assuming the wall can be defended).
I will add a theorist's perspective. If a group G is completely alive in empty space, they regardless of how the opponent approaches, unless they occupy the empty positions neighbouring G, then G has at least 50% chance of occupying any (dame) point adjacent to G. There is also good chance of making territory locally too. Even if the opponent is strong, they still can't afford to play too loosely near to wall, or else they can be captured.
If we estimate, a la Spight influence functions, the influence of strength decays by a factor of two with every move played, then under stone counting, the wall has a value of 1 + 1/2 + 1/4 + ... = 2 for each height. I can't really justify that the possibility of making territory doesn't change this value, but atm, I can't even tell you if it should increase or decrease this value.
If the opponent has a strong group nearby, then they compete for value, perhaps by cancelling out value somewhere along the way so it becomes 1 + 1/2 + 1/4 + 1/4 = 2 (sentegote tends to double the probability a boundary play occurs). This is no change at all. (note we haven't counted the value of the opponent's group)
If the opponent doesn't have a strong group nearby, then pushing the value of the wall up to 3 per height seems a good estimate. There is probably an argument for why it must be less than 4.
Now compared this to if G wasn't completely alive or could be cut. For every move that the opponent plays, it is more likely to be sente. If the group is killed, then though it still has lingering influence with threats to save it, this becomes a drastic reduction. For example, perhaps we should instead count influence as 0 + 0 + 1/4 if the capturer expects to answer threats two moves away from saving the group. Then for every move that the opponent plays nearby, if we assume the owner of G may ignore the threats (ignoring the sente reduction), we can add a value of (21/4)/(2^n) if the threat is n moves from capturing G to the opponent's moves.
Accounting for the sente reduction (the amount depends on the local temperature), then we should add less value. However, be careful as cuts can be double attacks if the player has another group H nearby that depends on G for support, increasing the value.
Overall, in summary, a good rule of thumb seems to be that having thickness can reduce the value even of strong moves by the opponent nearby by up to a factor of 2x. (and weak dead moves by as much as the global temperature). There were many weak assumptions in this derivation, but I think this is a good summary regardless. Playing near thickness is like playing in the centre. It is less valuable than the corner, but far from worthless.
_________________ Give me triangles strong enough and I can measure the universe.
When Venus transits, we can align our clocks to one event. By measuring the angle to flat Earth at two places far apart on Earth, we can compute the distance to Venus and the Sun.
