Resonance Defects in RIG Two-Parameter Calculus: Implicit Squares, Sharp Stability, and Finite Topology

We study reflection-resonant two-parameter differentiation on finite commutative square-zero path algebras. For defects with independent coordinate linear parts and arbitrary higher terms, we reduce their joint Koszul cohomology to the squares of a finite implicit solution in the remaining path algebra. This reduction works over a localized integer ring without inverting 2. It gives a necessary and sufficient real stability criterion and a sharp threshold: perturbations starting in degree N cannot change the full-support block with fewer than 2N remaining positions. At exactly 2N positions, complementary path coefficients determine every Betti-number loss and the complete localized-integer torsion module. We identify monomial-defect blocks with augmented relative simplicial chains. Combining the two reductions realizes any finite simplicial homology in an explicitly graded reduced model, with arbitrarily high-order perturbations, independent linear parts, and pairwise conjugate single-direction operators of exact nilpotency order eight.