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,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 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 HomSL(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.