I was perplexed after re-reading the manga lately. We discovered that Wrath was formed when King Bradley took a philosopher's stone, which granted him the Ultimate Eye and enhanced badassery but no regeneration abilities. However, when Ling Yao does the same, he may still regenerate, granting him the (near) immortality he wanted. Justifying this little discrepancy resulted in a tremendous "aha!" moment for me, so I wanted to share it with you. King Bradley was born into a harsh Darwinist boot camp, where he fought to be the strongest of his classmates. They exist only to become the next Fuhrer, which, unknown to them, entails becoming a monster. Bradley fights like he always has when he is forced to bear Father's Wrath. According to his tale, the spirits "warred inside him" until only one remained, one with no compassion and only hatred. A homunculus with just one soul is nearing the end of its existence.
