Cubic incompleteness: Hilbert's tenth problem begins at degree three

We present a preliminary result addressing a long-standing open question: Are cubic Diophantine equations undecidable? } We answer in the affirmative. By encoding Gödel's first incompleteness theorem via a Fibonacc-based Göodel numbering and the Zeckendorf representation, we reduce the arithmetic complexity sufficiently to construct an explicit cubic Diophantine equation whose solvability is independent of a model provided by Peano axioms. We present three results. First, a commonsense argument that relies on an unjustified generalization by necessity. Second, a constructive procedure that, given any Turing machine M, produces a cubic polynomial QM (u) Zu of total degree at most 3, such that the equation QM (u) = 0 has a solution in Nᵏ if and only if M halts on empty input. Third, we formalize a thesis concerning bounds.