Solution 4 Black has three territories, but nothing even vaguely resembling a moyo, so the only significant operative Sector Lines are White’s. As shown, it’s quite apparent that, especially in the left center area, they’re very long and quite porous, with Black stones to support any invasion very near.
Based on those facts, plus the observation that the lower left corner is still wide open, an invasion rather than a reduction clearly seems appropriate.
Invading on the 3-3 point in the corner would be absolutely safe, but would almost certainly leave Black confined there and White thick outside, with a result that would be too good for White.
So Black reasoned that he would almost certainly be better served by instead invading on the lower left side with a Knight’s move Kakari against the White 4-4 stone. After that, the resulting fight could develop in many different ways, with details far outside our present interest, so we won’t go into those various possibilities.