Three, Four, or Five: On the Arity of Mathematical Structures

This work studies mathematical structures carrying a canonical condition number κ=λmax⁡/λmin⁡ and proves a structural classification theorem for the number of independent canonical formulations (“arms”) associated with that condition number. For each admissible base point O, the arity is shown to satisfy arity(O)=3+εH(O)+εsmooth(O). Accordingly, every admissible structure has exactly three, four, or five independent arms: three universal arms, a fourth in the entropy-active regime, and a fifth in the presence of a canonical compact invariant. The framework also proves that compactness implies entropy activity, so five-arm cases occur only when the entropy-active arm is present. This record contains the abstract in advance of the full preprint, which is in preparation.. The full work, together with the broader project materials and updates, is available via the linked repository.