Nonlinear Right Bi-Skew Commuting Maps: Parameter Rigidity and Semiprime Classification

We classify arbitrary, not necessarily additive, maps Φ:R→R satisfying the right bi-skew commuting identity Φ(a)b∗+ηbΦ(a)∗=aΦ(b)∗+ηΦ(b)a∗. For unital algebras with a second-kind scalar involution, all scalar parameters η are treated; solutions are symmetric right multipliers except at η=1, when a term from the left annihilator of the commutator ideal may occur. For first-kind involutions and η=±1, non-PI prime rings and prime PI-rings of central degree at least three admit only symmetric right multipliers. Orthogonal and symplectic degree-two exceptions on M2(F) for both signs show that the degree bound is sharp. Using central closure and orthogonal completion, we extend the classification to 2-torsion-free semiprime rings without assuming an identity. At η=1 every solution is additive; at η=−1 nonadditivity may occur only on the commutative central summand.