The Bott–Thurston cocycle is a $$2$$ -cocycle on the group of orientation-preserving diffeomorphisms circle. We introduce and study formal analog cocycle. continuous $$A$$ -automorphisms algebra $$A((t))$$ Laurent series over commutative ring with values in $$A^*$$ invertible elements . prove that central extension given by equivalent to 12-fold Baer sum determinantal when $$\mathbb Q$$ -algebr...