MRS-AUTH RESEARCH PHASE/ASSUMPTIONS — NO NIST/IETF CERTIFICATION post-quantum security with a novel cryptographic layer derived from the theory of linear Diophantine representation systems. The construction combines three components: (1) a post-quantum KEM (Kyber) for key encapsulation, (2) an MRS chain, a deterministic structured traversal of the solution space of N = 19A + 9B — as an entropy source fed through HKDF modelled as a Random Oracle. And (3) AES-GCM for authenticated encryption. The session key is formed as the XOR of the KEM-derived key and the MRS-derived master key. The central mathematical insight is that the minimal coefficient A₀ of any representation N = 19A + 9B equals the digital root dr(N), a consequence of the congruence 19 ≡ 1 (mod 9). This property is scale-invariant and extends canonically to three-dimensional systems pA + qB + rC under mild divisibility conditions. The resulting MRS chain is fully deterministic yet cryptographically opaque after HKDF compression, bridging structural predictability with entropic security. The MRS-AUTH-KEM achieves IND-CCA2 security under three standard assumptions: IND-CCA2 security of the underlying KEM, the Random Oracle Model for HKDF, and IND-CPA security of AES-GCM. The security reduction proceeds via a four-game hybrid argument, formalized in EasyCrypt. The adversary's advantage is bounded by 2 · negl(λ), and breaking the scheme requires simultaneously defeating both Kyber and the MRS chain providing a concrete, quantifiable security margin beyond existing hybrid constructions. Domain separation between HKDF contexts prevents cross-context collisions, and the scale-invariance of the digital root ensures that the anchor A₀ remains stable under all system extensions. RESEARCH PHASE — NO NIST/IETF CERTIFICATION