Life In 19x19

NB 20220514. My understanding is tesuji are ways to attack/defend in surprising ways. Mostly common they play inside what looks like an area of strength, attacking a shape from an unusual direction because one side is strong enough to kill or severely attack the other side. There are still rules and patterns in that perhaps you don't attack from the usual direction, but perhaps you attack from the 2nd most common direction (or 3rd if there is a double attack). Attacking (or or making territory) is such a tight rope that you can lose a lot if you fail, so consistency of plans is more important.

When both sides are weak, the order and choice of attack is even more important. Decisions about sacrificing arise.

Also, I still been able to find a clear answer to the question
"When is the opponent's vital point also your own?"
This is based on the proverb "The opponent's vital point is your own."
My understanding is that it often isn't when both sides are weak but not yet dead, particularly in a semeai, but when it comes down to the final killing move when there is just one weak group, then the vital point is the same for both sides. My point is that the first player has an advantage, and you can often attack the opponent more strongly than just playing a vital point if your first move threatens the vital point also from a different angle. Whereas that threat wouldn't work if the opponent had already played the vital point. But can this statement be made precise enough to be proved?

20220514

Tiger mouth: weak to peep and diagonal shortage of liberties.

one space jump: weak to cut, but a cutting move also damages the opponent's liberties by 2, so if the stones of the jump are strong enough, it is powerful.

many snapbacks can be seen as variations on the one space jump.

If I comment on concepts for 101weiqi along with my approximate time to solve

https://www.101weiqi.com/qday/2022/5/14/

1: (0s) two groups in a semeai and though both have cuts, B can capturing cuts independently. The important thing is to save the weak group rather than defend own cuts. This is possible by leaning on W's weakest stones by cutting a diagonal. B and W diagonals are adjacent but B has a connection and W doesn't. Both have 2 liberties, but W has two such groups one of which is near the edge.
2: (0s) All B's stones are strong except the hane (but that doesn't matter unless W captures 3 for an eye). W only has 3 liberties, but B has nothing to lean on and no useful threat on those liberties as they are next to W's other solid wall. B can push in where B is strong enough at the hane where W could have made a tiger mouth, destroying O19 with only one move and killing. We see that when breaking a tiger mouth, your own stones already have less liberties, but they gain 2 liberties by breaking the tiger mouth as well as access to space at and beyond those liberties.
3: (2s) One eye kills no eye. Attacking the outside would be sufficient, but though W has 3 liberties and B 4, trying to cut loses 2 since you have to cut two one space jumps. Instead, cut the other two one space jumps from inside the corner, which is W's weakest stone. This gains B an extra 2 liberties at 3-1 and 1-1. W's problem is that W doesn't have enough liberties to play at either and B already has an eye. Connect before cutting is a good principle when fighting in the opponent's area, especially if the opponent can also connect there.
4.(5s) The vital point is obvious, but takes reading to check. An extra large monkey jump works when W also has a cutting point. S14 is W's main attempt to cut (the vital boundary point if S15 is too slow to work), but S14 lacks liberties when T14 is the B's connection point. This is an example of an elephant jump that doubles as two one space jumps that W lacks liberties to cut both. B maintaining the S15 sente is also vital here because B doesn't have R13. (a sibling problem could be made if W had one more liberty with R15 but one less with S13. I think Q13 isn't necessarily for a partial kill. W needs a solid capture, so B needs to make every S and T point are miai from S16 down or be the first to get both (counting down from 16, 15, ...). S12 is the support required to make this possible, and S11 is too far without R13. A key point is that though S15 is too slow, it remains a vital point likely under W control to the point where W would like to throw in at T15 and force B to lose liberties connect both S15 and T15. This fails here as B S15 is double atari.
5. (70s) Approaching from the corner is too slow since an approach move is required, and L18 doesn't work since P19 is safe when B doesn't have an option of R19. Hence, P19 and Q18 becomes the only option, but does it work? We prefer ko to seki, especially as B gets to capture on the right, though W does manage to get L18. The point seems to be that Q18 is a weak point but B can't cut without lots of preparation. W never has incentive to connect unless directly threatened. And yet, it is possible to cut with enough preparation since B's inside play does threaten it, and the throw in is a direct threat. W must eventually capture the B's inside stones in order to maintain the threat of making Q18 eyespace. B also can't play Q18 as that is 4 dead stones. The situation is made more confusing since W can take time to play L18 rather than capturing since B has to approach. B must play accurately enough to throw in since W is also too weak to play at R19. The principle seems to be that when both sides have a weak point (Q18, S17), directly approaching might be too slow, since the opponent will just respond, retreating solidly. However, this is true for both, both are weak at the boundary R19, so attacking to push the opponent towards the boundary is helpful, if the attack works.
6. (7s) A standard tesuji. Knowing a sacrifice is necessary is important, but normally that is necessary when fighting get more on the other side in your opponent's area. It wouldn't work without M14 exposing the cut. Again we have one space jumps. The squeeze forces a defence of the cut (which isn't valuable at the boundary of B and W thickness), and this weakens L17 enough to clamp and connect. This also wouldn't work if W had P18. M17 is a more obvious forcing move that doesn't sacrifice, but it doesn't support connection enough to work. This also only works as W's cutting stones are weak enough. If M16 was 3 stones, there is a related tesuji, but B needs more support, getting to pro level go.
7. (30s but without full reading) Difficult as it only barely works. B getting both O18 and O17 is sufficient, and W getting O18 blocks off another eye at Q19. This is too far away from B and W doesn't have many weaknesses. B's only chance is O18 or P18. But P18 fails to O18 since W has enough liberties to make connection miai. Hence O18 is forced and B must continue at P18. B is close enough this threatens to connect, so W must block, but then B must use the chance to make an eye, since W is very low on liberties if W has to spend 3 moves capturing and connecting at O16 without any liberties. There is so much space, but that means that B has the chance to also control it with an eye by making miai of cutting or connecting.
8 (80s complete reading) Pretty much a forced line, though there are many tesujis involved. W can almost live in sente with S15 tesuji, but not really as it doesn't fix W's main cutting points. Key point is that B capturing P18 and cutting P16 will win the capture race, so W mustn't let this happen. Both B and W must commit to a side to defend. B's P17 is very low on liberties especially if aiming to cut P16. B can play simply at O18 and even connect O17, but W gets Q18 and Q17 in sente since B at O18 remains weak, finally living with T18. If B plays Q18, it turns out B can win the capture race with R18 despite S19 tesuji since Q17 is atari and W is too weak (no time to play R16 and it wouldn't help anyway). However, this means that B also doesn't have time to play Q17 since W getting R18 and R16 in sente allows S15 to live. Finally, we have a ko at the top that is flower viewing for B. The principles seem to be that with a weak stone with many cutting points in the opponent's area, don't expect to be able to save everything and still cut since the defender will get many forcing moves. The priority to break into the territory at the weakest point rather than save the stone. This could be O18 and O17 just for points in sente, but here reading that Q18 works to almost kill is even better since B is strong enough at L18 to continue attacking.

q8 20220513, I didn't solve this, the move inside is subtle, but it is needed to connect to the last point of the bulky five that B needs.

My current impression is that you imply all ideas for games are borrowed from more general ones, especially for beginners. What is complex for some may be simple for others, with different perspectives. Whereas maths tries to find absolute answers to popular abstract structures, and applied maths builds concrete ladders and models to guide understanding and decisions. We would like measures and metrics rather than fuzzy associations, but perhaps all new endeavours for understanding start fuzzily.