(General) Conceptual Substratum As A New Foundational Metamathematical Cognitive Mechanism In Artificial Mathematical Intelligence

We describe (essential features and an axiomatization of) a new metamathematical (cognitive) ability, i.e., functional conceptual substratum, used implicitly in the generation of several mathematical proofs and definitions, and playing a fundamental role in Artificial Mathematical Intelligence (or Cognitive-computational metamathematics). Furthermore, we present an initial (first-order) formalization of this mechanism together with its characterizing relation with classic notions like primitive positive definability and Diophantiveness. Additionally, we analyze the semantic variability of functional conceptual substratum when small syntactic modifications are done. Finally, we describe cognitively natural inference rules for (mathematical) definitions inspired by functional conceptual substratum and we show that they are sound and complete w.r.t.\ standard calculi.