Jaimini's first innovation over Pārāśara is the chara-kāraka system. Where Pārāśara's natural-kārakas are fixed (Jupiter is always putra-kāraka, Venus always kalatra-kāraka), Jaimini's chara-kārakas are CHART-SPECIFIC — they are assigned based on which graha has the highest, second-highest, third-highest etc. degree within its rāśi. The Ātma-kāraka (AK) is whichever graha has the highest degree in its sign; the Amātya-kāraka (AmK) is the next highest; and so on.
The opening sūtra of Adhy. 1 · Pāda 1 — ātma-kārakaṃ sarva-uccatamaṃ graham — is one of the most quoted sūtras in jyotiṣa. It establishes that the chara-kāraka system is degree-based, not nature-based.
There are two schools on the count: 7-kāraka (where Rāhu is excluded — Saturn typically becomes a kāraka) vs 8-kāraka (Rāhu included as the 8th, with degrees measured 'reverse-from-30' due to Rāhu's retrograde nature). Both schools have classical and modern adherents. Our engine surfaces both — /jaimini-karakas shows the side-by-side.
Kārakāṃśa-lagna is the navāṃśa-sign occupied by the Ātma-kāraka. Adhy. 1 · Pāda 4 declares this the focal interpretive lagna for analyses of dharma and personal evolution — ātma-kārakāt navāṃśe lagnaṃ kārakāṃśam.