An Exceptional Finite Simple Group on a Quantum Code's Logical Space: G2(3) and the [[13,7,3]]_3 Ternary Hamming Code

This paper establishes that the logical space of the [[13,7,3]]_3 ternary CSS Hamming code — the quantum stabilizer code arising from the Topological Vortex Logic (TVL) framework and studied in "Quantum Error-Correcting Structure of Topological Vortex Logic: The [[13,7,3]]_3 CSS Family and its Algebraic Geometry" (Zenodo DOI: 10.5281/zenodo.19983025) — carries a hidden action of the exceptional finite simple group G₂(3). The logical space of the [[13,7,3]]_3 code is a 7-dimensional F₃-vector space L carrying a faithful action of the code's automorphism group PSL(3,3). As an SL(3,3)-module, L is the characteristic-3 adjoint module psl₃ (the short-root form of the Lie algebra of type A₂, which is irreducible in characteristic 3), and L carries a UNIQUE invariant non-degenerate symmetric bilinear form B and a UNIQUE invariant alternating trilinear form φ, the Cartan form φ(X,Y,Z) = tr([X,Y]Z). The joint stabilizer of the pair (B, φ) in GL(7,3) is exactly the exceptional finite simple group G₂(3), of order 4,245,696 = 2⁶ · 3⁶ · 7 · 13. The order is established directly by an orbit–stabilizer count inside O(7,3), using only the classical order formula and the Cartan–Dieudonné generation of the orthogonal group, so it does not depend on the characteristic-sensitive form-stabilizer theorem. The physical code automorphism group PSL(3,3) embeds in this G₂(3) at index 756. The G₂(3) is a symmetry of the LOGICAL space alone: the same prime-3 arithmetic that produces the group obstructs it physically, since ν₃(13!) = 5 < 6 = ν₃(|G₂(3)|) forbids any embedding of G₂(3) into S₁₃, hence any faithful monomial action of G₂(3) on the 13 physical qutrits. All structural claims are reproduced from scratch over F₃ by the accompanying verification script. This is a result in finite group theory and quantum coding. It is not a statement about G₂-holonomy geometry, with which the finite group G₂(3) shares only a name and a root system; no geometric or physical (octonionic / compactification) reading is claimed. Two Python verification scripts with an accompanying README are included as ancillary files, reproducing the computational results from scratch over F₃ (standard library and numpy only). Published in conjunction with the TVL framework: "T³ as a Closed Information-Processing Environment" (10.5281/zenodo.20806554, Concept DOI) and "Topological Vortex Logic: Generation and Colour Structure from T³/Z₃" (10.5281/zenodo.19682633, Concept DOI); and building directly on the code-theory paper "Quantum Error-Correcting Structure of Topological Vortex Logic" (10.5281/zenodo.19983025, Concept DOI). Concept DOIs always resolve to the latest version. Changelog v1.0.0 (June 2026) Initial release.