In “Claims” add “An Heir’s Total Claim Power is equal to the sum total strength of all their Claims.”

In “Transition of Power” replace

When The Old King Perishes, the Heir whose single strongest Claim is stronger than all other Heirs’ strongest Claims achieves victory. If multiple Heirs are tied for strongest, their next-strongest Claims are successively compared.

with

The Heir Apparent is determined by the following steps. If a single Heir has a higher Claim Strength among all Heirs, they are Heir Apparent. If two or more heirs are tied for having the highest Total Claim Power the Heir Apparent is determined by the following tiebreaks (with each step only applying to the heirs who were still tied after the previous step)
* Whoever has the highest strongest single claim
* Whoever has the highest second strongest claim
* Successively compare next highest claims until a Heir Apparent is determined or the still tied Heir’s claim list is exhausted
* Highest Age
* Highest Reputation

If after applying all of these tiebreak steps, two or more Heirs are still tied, there is no Heir Apparent. 

When The Old King Perishes, the Heir Apparent (if there is one) achieves Victory.