Let q=p^e be a prime power and n>=2. We construct a nontrivial singular mixed identity for PSU_n(q) of variable length at most 16q+8p+12. The bound is uniform in the dimension and includes characteristic two without an exception. A product of two unitary transvections has order dividing one of q-1, q+1 or 2p. A nested commutator combines these alternatives, while separate free conjugators ensure formal nontriviality. Together with the lower bound of Bradford, Schneider and Thom, this gives Theta(q) for the length of a shortest mixed identity in the projective special unitary family.

This is a complete theorem for the stated unitary-family scope, not a solution of all remaining Lie-type families or shortest nonsingular identities in the associated Oberwolfach questions. The package contains the four-page English manuscript, portable LaTeX source, a written proof and scope audit, a bounded literature review and exact finite-field/free-product diagnostics. The finite tests supplement rather than replace the general proof.

Alper Ferudun, Mercury Software GmbH. AI-assisted, self-audited and unrefereed; no independent expert review, proof-assistant verification or absolute-priority certification is claimed. Licensed CC BY 4.0.
