From Frobenius seeds to self-orthogonal modules over symmetric algebras

We construct finite-dimensional symmetric algebras admitting nonprojective self-orthogonal modules from Frobenius algebras equipped with a module satisfying explicit stable and homological conditions. The resulting modules have complexity three, and their stable endomorphism algebras are isomorphic to the algebra of dual numbers over the ground field. The construction combines a variable input with a fixed auxiliary factor carrying the required twisting, and preserves the input through specified quotient and corner constructions. The proof combines derived induction, an obstruction to perfect factorization, and a cone construction over a Gorenstein triangular matrix algebra. For a symmetric algebra and a self-orthogonal generator, we describe the derived category of the opposite endomorphism algebra through a recollement with a two-term $2$-symmetric dg algebra. For our counterexamples, it is formal and quasi-isomorphic to an explicit four-dimensional dg algebra. Applications to cyclic coverings of a quantum exterior algebra yield connected examples with explicitly determined numbers of simple modules. The construction works over every field containing an element of infinite multiplicative order.