We study a fixed-core absorption problem for regular induced subgraphs. A set is q-modular if all induced degrees are congruent modulo q. Given a q-modular witness A and a retained core U subset A, we ask when deleting equal-trace q-tuples from A\U can make U into a 2q-modular witness. The main contribution is a finite absorption-or-obstruction certificate. We give an exact quotient formula for the deletion-tail obstruction in complement-orbit coordinates: the correct expression uses oriented differences n_B - n_{U\B}, not sums. Equal-trace q-tuples absorb exactly the span of their trace classes in F_2^U / 1_U. In particular, a connected graph of q-heavy two-point traces on U, together with one odd trace when |U| is even, absorbs every top-bit defect by deleting at most q(|U|-1) tail vertices. If fixed-core absorption fails, the obstruction is an explicit even parity cut of U. We also record the parity base, the terminal modular criterion, and a conditional modular-witness threshold theorem explaining the relevance to the Erdos-Fajtlowicz-Staton problem. The paper does not claim to solve that problem or to improve the general lower bound for F(n).