Exact scientific discovery requires more than heuristic search: candidate constructions must be turned into exact objects and checked independently. We address this gap by extending TeXRA with an independent Lean 4 verification layer, turning it into a human-driven, AI-assisted discovery protocol with specialized Synthesis, Search and Verification agents for exact scientific discovery. The protocol combines symbolic synthesis, combinatorial and linear-programming search, exact reconstruction of numerical candidates, and formal verification. We apply it to nonadditive quantum error-correcting codes with prescribed transversal diagonal gates within the subset-sum linear-programming (SSLP) framework. In the distance-2 regime where logical states occupy distinct residue classes, the resulting Lean-certified catalogue contains 14,116 codes for K ∈{2, 3, 4} and up to six physical qubits, realizing cyclic logical orders 2 through 18, from which we extract closed-form infinite families. We also construct a residue-degenerate ((6, 4, 2)) code implementing the logical controlled-phase gate diag(1, 1, 1, i). At distance 3, we resolve the transversal-T problem for ((7, 2, 3)) codes within the complementary binary-dihedral BD16 setting: among the 12 candidates surviving the SSLP filters, 10 admit exact realizations and 2 are excluded by no-go proofs. All accepted constructions, families, and no-go results are formalized and checked in Lean, showing how AI-assisted workflows can bridge search, exact reconstruction, and formal proof in the physical sciences.