A Unified Field-Native Cryptographic Verification Pipeline with Four Algebraic Engines and Machine-Checked Certificates

We present a complete cryptographic verification pipeline operating natively in four algebraic fields — GF(2) for symmetric ciphers, Z_3329 for lattice-based post-quantum cryptography, Z[w] for cubic congruence verification, and F_p for elliptic curve validation — eliminating Boolean encoding entirely. A specification front-end (CFL) compiles human-readable cryptographic properties through first-order logic to field-native solvers at 9,357 verifications per second. The GF(2) engine achieves 15-4,210x speedup over CryptoMiniSat5 on AES structures with 100% agreement. The Z_3329 engine — the first native solver for ML-KEM's arithmetic field — verifies production-scale (n=256) Kyber NTT transforms in 13ms without binary encoding. A cubic engine using Eisenstein decomposition computes exact solution counts for x^3 + y^3 = z^3 (mod p) via a unified formula verified across 2,556 primes with zero errors, feeding into elliptic curve point counting. All four engines produce machine-checked Coq proof certificates — the first such certificates for non-Boolean algebraic solvers. The complete pipeline executes in compiled C with no external dependencies beyond coqc.