The Cascade Algorithm: Classical RSA Security at the Quantum Barrier

The Cascade Primality Algorithm, derived from the Bounce Theorem, achieves deterministic primality testing in O((log log n)² · log n) operations with zero classification errors and no probabilistic component. At 1,400-bit integers this represents a computational reduction of 84 billion times relative to AKS, the current gold standard. We demonstrate two direct and unconditional improvements to RSA security: (1) The practical barrier to 8,192-bit and larger RSA key generation is removed. (2) The Carmichael number vulnerability — the only known structural weakness in RSA key generation — is unconditionally eliminated. We formally define RSA security at the quantum barrier and prove the Cascade Algorithm achieves it. DOI of foundation paper (Paper IV): 10.5281/zenodo.20084634.