Primitive Denominator Cancellation Spectra and Local Densities for Unimodular Word Forms and Quadratic Swap Returns

The paper develops an arithmetic refinement of our matrix-word and quadratic-observable investigations, rooted in the Cohn–Markoff and Christoffel setting. Starting from a primitive integer vector and a unimodular matrix state, it reconstructs a canonical symmetric positive-definite rational form of determinant one and proves an exact formula for its least integral denominator. It then completely classifies which denominator-cancellation factors can occur for nonsymmetric nonnegative unimodular matrices on the positive primitive cone, proving the impossibility of odd prime factors congruent to 3 modulo 4 and of a second factor of 2, with a precisely stated parity exception. Every admissible factor occurs infinitely often, and its natural density is given by an explicit product of local factors with an effective counting error. The paper also relates word palindromes, swap quadratic forms, a classical Fricke commutator condition, and prescribed return words under additional hypotheses. Its main novelty is the exact cancellation spectrum and local-density law linking primitive lattice arithmetic to unimodular word states, while keeping classical Markoff identities distinct from the new arithmetic classification.