Co-Designing Quantum Codes with Transversal Diagonal Gates via Multi-Agent Systems

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-guided multi-agent platform for exact scientific discovery. The platform couples symbolic synthesis, combinatorial and linear-programming search, exact reconstruction of numerical candidates, and formal verification in Lean. We apply this platform 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 platform yields a Lean-certified catalogue of 14,116 codes for Kin\{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, illustrating how AI-assisted workflows can bridge search, exact reconstruction, and formal proof in the physical sciences.