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 quotient of sl₃ by its centre, irreducible in characteristic 3), and L carries an invariant non-degenerate symmetric bilinear form B and an invariant alternating trilinear form φ — each unique up to scalar — the latter being the Cartan form φ (X, Y, Z) = tr (X, YZ). 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 determined analytically, from Aschbacher's trilinear-form identification together with the G₂ order polynomial, and is confirmed independently by an orbit–stabilizer count inside O (7, 3) that uses only the classical order formula and Cartan–Dieudonné generation, so the confirming count does not itself invoke 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. Every structural claim is proved analytically in the body or follows by citation; the group order is additionally confirmed by an in-text orbit count. 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. The underlying group-theoretic fact — G₂ (3) acting on a 7-dimensional orthogonal space over F₃, with PSL (3, 3) inside acting monomially on the 13 points of PG (2, 3) — is due to R. A. Wilson (2012), who obtained it by reducing the 14-dimensional real G₂-lattice modulo 3; that construction makes no connection to quantum codes. The contribution here is the code-theoretic reading: the 7-space as the logical space of the [13, 7, 3]₃ code, G₂ (3) as a symmetry of its logical operators, and the invariant as the Cartan 3-form on the adjoint module. The paper is self-contained and analytic throughout the body: every proposition and the main theorem are proved in the text — the identification L ≅ psl₃ via Lübeck's classification; the form uniqueness via Schur's lemma, the Cartan-bracket structure, and a character-theoretic proof of the one Hom-dimension vanishing that completes the exact-sequence argument (Weyl-character factor avoidance with a socle-exclusion step, carried to the finite group by Steinberg's restriction theorem) ; and the stabilizer via Aschbacher's trilinear-form characterization with a reproducible orbit count kept as independent confirmation. 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. v1. 0. 1 (July 2026) — self-contained, fully analytic version. No theorem or numerical result changed. Every proposition and the main theorem are now proved analytically in the body — including the one Hom-dimension Homₒ₋ (₃, ₃) (K, psl₃) = 0 that v1. 0. 0 established by finite computation. The ancillary verification scripts and the computational appendices are removed; the orbit count confirming the group order is kept in-text. A prior-art reference (R. A. Wilson, 2012) is added, crediting it with the underlying group-theoretic result; the paper's contribution is stated as the code-theoretic reading (unchanged). Several proof steps are completed and tightened in the text: the Cartan form's non-degeneracy is proved (equivariance of the contraction plus irreducibility of psl₃, with an explicit nonzero witness) rather than asserted; the physical-obstruction argument derives the embedding into S₁₃ from the simplicity of G₂ (3) against the abelian phase kernel; the analytic Lemma's proof is organised into labelled steps with all internal cross-references repaired; a mis-stated direct-sum claim about sl₃ in characteristic 3 is corrected to a composition-factor statement; and a nonstandard “short-root” label for the A₂ module is replaced by the standard description (the label is kept only where it is correct, for the short-root ideal of g₂ in characteristic 3). The graph-automorphism remark now exhibits its realiser explicitly — the involution X ↦ −Xᵀ on psl₃, proved to preserve both invariant forms exactly and unique up to the scalar 2, whose partner negates the trilinear form — replacing an unexplained normalisation. v1. 0. 0 (June 2026) — initial release.