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 G₂ = Aut (O). We construct the Lagrangian, derive the Feynman rules, establish the extended Schafer identity, and prove betaO^ (1) = 7/ (64 pi²) = ||phi||² / (6 (4 pi) ² C₆䃒²), where ||phi||² = 168 = |PSL (2, 7) | and C₆䃒 = 4. We define the problem SymbolicBeta (k): given k in N, return betaO^ (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) ; SymbolicBetaᵢnfinity 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.
Aarao Melo Lopes (Thu,) studied this question.