Cardiophenomenology aims at refining the neuro-phenomenological approach created by F. Varela as a new paradigm, jointly based on Husserl's a priori dynamics of the living present and an experiment on anticipatory time-dynamics of visual motor perception. In order to do so, we will situate the paradigm of neurophenomenology at the cardio-vascular level, focusing on the emotional dynamics of lived experience and thus refining the dialogue, more precisely, the generative mutual constraints between first- and third-person analysis. In this article we present the theoretical hypothesis of cardiophenomenology, which places the bodily-emotional heartsystem at its core, as an intrinsic part of the cognitive system. The latter therefore needs to be enlarged in order to include an enactive embodied cardiac-affective dimension. Here we present five main arguments for the necessary inclusion of the bodily-emotional heartsystem in the cognitive system: first, two pragmatic operational arguments (experiential and methodological), then three theoretical ones (cognitive, homological and ontological-laden). A direct methodological-pragmatic consequence is the actual operativity of the generative mutual constraints based on an experiential (lived)-experimental (organic) continuity of the embodied cardiac-affective fold inherent in the subject.