Mod 3 Congruence Families for twisted signatures of string manifolds in dimensions up to twenty-eight

In the paper, we establish congruences for twisted Dirac indices and signatures on closed string manifolds by specializing the two-parameter modular forms of Liu and Wang to the tangent bundle. Fourier coefficient comparisons yield uniform congruences modulo $3$ in dimensions $4k$, $1\le k\le 7$. Suitable parameter choices give explicit formulas involving only genuine twisting bundles in dimensions $8,12,16,20,24,$ and $28$, and recover the comparisons of Chen and Han in dimension $24$. The level-one Witten genus further strengthens selected Dirac congruences to moduli $729,27,27,$ and $9$ in dimensions $16,20,24,$ and $28$, respectively. The proofs use Fourier expansions in reduced $K$-theory and integer identities obtained by clearing powers of two from the denominators before reduction modulo $3$. We also construct non-string spin counterexamples in each of the six listed dimensions and identify the additional coefficient that obstructs extending the uniform argument to dimension $32$ without further information.