Post-Quantum Cryptography via Octonionic Yang-Mills: A Hierarchy of Hardness from P to Undecidable

We propose a new family of post-quantum cryptographic hardness assumptions based on the perturbative beta function of a Yang-Mills gauge theory over the octonions O, with structure group G_2 = Aut(O). We construct the Lagrangian, derive the Feynman rules, establish the extended Schafer identity, and prove beta_O^(1) = 7/(64 pi^2) = ||phi||^2 / (6 (4 pi)^2 CG_2^2), where ||phi||^2 = 168 = |PSL(2,7)| and CG_2 = 4. We define the problem SymbolicBeta(k): given k in N, return beta_O^(k) in the Goncharov-Brown canonical basis. We establish a complexity hierarchy scaling with the security parameter: SymbolicBeta(k = 1, 2) in P; SymbolicBeta(k >= 3) is PSPACE-hard (unconditional for k <= 12 via Brown 2012); SymbolicBeta_infinity is algorithmically undecidable via reduction from Richardson 1968. We construct five complementary robustness layers: (i) factorial Feynman growth; (ii) symbolic PSPACE-hardness via succinct PIT; (iii) non-representability in any recursively decidable function class; (iv) lattice undecidability (Cubitt-Perez-Garcia-Wolf 2015); (v) absence of holographic AdS/CFT solver. To the best of our knowledge, this is the first PQC family whose complexity class scales with the security parameter. As a byproduct, the perturbative beta function of any realistic non-supersymmetric Yang-Mills is not symbolically representable, providing a complementary perspective on the Yang-Mills mass-gap Clay problem as a lattice + continuum-limit construction.