This five-star logic puzzle has 4 noun types, 12 nouns per type, 6 links, 216 facts, and 1 rule. The noun types are Throne, God, Sphere, and Parents. To simplify things, I merged Father and Mother into the Parents noun type. There are 864 marks and 6 grids. Because of the way the clues are given, the first seven gods are the males, and the first seven spheres are for the males. Likewise, the last five gods are the females (goddesses), and the last five spheres are for the goddesses.
The parents are slightly harder to work out. From clue 1, four of the immortals where the children of Cronus and Rhea. Clue 2 states one was born from the foam of the sea (no mom and dad), and another was born from the head of Zeus (father but no mother). Clues 3 states the remaining 6 had Zeus as the father. The mother was either Zeus's wife, or one of three mortal women: Leto, Maia, and Semele. So at least one mother had more than one child.
Clue 9 tells us Maia has a son who was a twin, and clues 6 and 13 tell us Zeus's wife had two sons, one being Poseidon. Therefore, Leto and Semele each have one child.
It is imperative that the nouns for each type are not rearranged.
Clue 1 states four of the immortals - the older generation - were children of Cronus and Rhea. Zeus and his wife are two of them. Clue 4 tells us he has a brother. Clue 6 tells us the goddess of war and Poseidon's mother are members, so one of them must be Zeus's wife. Clue 12 tells us Demeter, the goddess of the sky, was a member. This means Demeter is Poseidon's mother. Which means Demeter is the wive of Zeus. Clue 14 tells us Dionysus and Apollo are from different generations, so one of them is a member.
Clue 4 tells us the two centermost thrones (6 and 7) belong to Zeus and his wife in some order. Clue 4 also says Zeus sat next to his brother. Here is what we know about the old generation.
The parents of Poseidon are either Zeus/Wife1 or Zeus/Wife2.
Clue 9 gives us the eight members of the younger generation, along with their order in the throne room from left to right. Being in the younger generation means your parents are not Cronus and Rhea.
We already know from clue 6 that Poseidon is a son of Demeter (Zeus's wife). Clue 13 tells us Poseidon's brother is married to Athena, so Poseidon must be the unmarried son in clue 9.
If the program solves this puzzle, it can find a solution using level methods 1a and 2a, with help from the rule.
This is an extremely difficult puzzle to solve because of all the upfront analysis. But once the parents are known, the program can find the solution without assumptions.