We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideal. Introducing a new robust coding technique, we establish the relative consistency of $\mathfrak{ a}_g=\mathfrak{ d}<\mathfrak{ c}=\aleph_2$ alongside the existence of a $\Delta^1_3$-wellorder of the reals and a co-analytic witness for $\mathfrak{ a}_g$.
Categories: math.LO