The Complementary System Completeness Conjecture

In this paper, we first propose a novel, syntactically defined classification of number-theoretic problems: Simple System-Dependent Problems (SSDP) and Complex System-Dependent Problems (CSDP). We then formalize the Complementary System Completeness Conjecture, which asserts that there exists a finite family of consistent, recursively axiomatizable formal systems that interpret Q, such that every SSDP is decidable in at least one system within the family. This conjecture does not violate Gödel's incompleteness theorems, and proposes a potential novel framework that could offer a path to address long-standing open number-theoretic problems that are undecidable in standard PA.