Background. All life strives to be well, but not all life is well. This suggests that cognition aimed at improving and protecting well-being might share a common core across all life forms: core cognition Objective. In this first of a two-part theoretical article, we systematically specify the evolutionary core cognition of well-being from the perspective of general living agents. In Part 2 we apply this to identity development and the theoretical approaches to well-being. This first part aims to identify the strategies and conditions for the creation and protection of generalized well-being and describes associated behavioral ontologies. Results. We defined a set of key terms that, together, specify core cognition. This set comprises quite naturally concepts like agency, behavior, need satisfaction, intelligence, authority, power, and wisdom, which are all derived from the defining properties of life. We derived coping and co-creation as two essentially different, but complementary, behavioral ontologies. Copingis for survival and targeted problem solving and aims to end the need for its activation. Co-creation is for thriving and problem prevention and aims to perpetuate its activation. Co-creation can explain the growth of the biosphere. While both strategies are essential, the successful interplay of their strengths leads to the dominance of one of them: co-creation. Absence of success leads to a dominance of coping: a coping-trap and a strong urge to curtail behavioral diversity. We summarize the key terms of core cognition and the ontologies in two tables with defined terms.