The Halting Problem as Structural Necessity: A Cognitional Mechanics Account of Undecidability

This paper provides a structural account of the Halting Problem within the axiomatic framework of Cognitional Mechanics (CM). Rather than revisiting undecidability as a technical limitation or offering a variant of Turing’s original proof, the paper explains why undecidability necessarily arises in any system governed by non-commutative operations, bounded constructive depth, and unreachable regions of configuration space. Execution and halting verification are modeled as distinct operations over a formally defined Turing configuration space. Their non-commutativity, together with an intrinsic limit on operational depth, makes the global behavior required for a universal halting decision structurally inaccessible. From this perspective, undecidability is not an artifact of self-reference or diagonalization, but a direct consequence of the operational structure itself. The Halting Problem is thus reinterpreted as a canonical manifestation of structural necessity. This account situates Turing undecidability within a broader theoretical context, showing it to be a general feature of any computational or cognitional system satisfying the CM axioms, rather than a peculiarity of Turing machines or formal languages.