Quantum Error-Correcting Structure of Topological Vortex Logic: The [13, 7, 3]₃ CSS Family and its Algebraic Geometry

This paper establishes the algebraic and geometric structure of the [13, 7, 3]₃ ternary CSS quantum stabilizer code and its dimension-indexed family, arising from the Topological Vortex Logic (TVL) framework introduced in "T³ as a Closed Information-Processing Environment" (Zenodo DOI: 10. 5281/zenodo. 19682633). The 26 stable winding states of TVL on the three-torus T³ form 13 antipodal pairs w, −w. These pairs are in natural bijection with the 13 points of the projective plane P² (F₃), and the Z₃ charge structure of TVL induces a parity-check matrix making the antipodal pairs the physical qutrits of the [13, 7, 3]₃ ternary CSS quantum Hamming code (Ketkar et al. 2006). This paper studies the structural properties of this code and its dimension-indexed family [ (3ᵈ−1) /2, (3ᵈ−1) /2 − 2d, 3]₃ for d ≥ 2. Eight structural results are established: (1) Family construction and self-orthogonality: The CSS construction is valid at every level d ≥ 2, with H·HT = 0 over F₃ proved for all d. (2) Double distinction of d=3: The TVL code is the smallest family member with positive logical dimension and the unique member among d ≤ 6 with prime physical qutrit count (n=13; the next occurs at d=7, n=1093). (3) PSL irreducibility at d=3 and d=4: PSL (3, F₃) acts absolutely irreducibly on the 7-dimensional logical space (proved via Burnside's criterion). PSL (4, F₃) acts absolutely irreducibly on the 32-dimensional logical space (proved via Schur's lemma). Conjectured for all d ≥ 3. (4) Stabilizer weight enumerator: The distribution A₀=1, A₉=104, A₁₂=624 is derived entirely from the line-incidence geometry of P² (F₃), with each sub-case verified independently. (5) Closed-form weight enumerator: A (ω) = i^ (d mod 2) · 3^ (nd/2) for all d ≥ 2, with |A (ω) |² = 3^ (nd) an exact integer. The formula is specific to q=3: the magnitude equality |1+ (q−1) ω| = |1−ω| = √3 holds uniquely at q=3. (6) Non-standard quadratic form: The weight-mod-3 function on Cd/Cd^⊥ defines a non-degenerate quadratic form of Witt class +1 for d ∈ 3, 4, 5. At d=3 this differs from the standard sum-of-squares form on F₃⁷ (Witt class −1) ; at d=4 both forms coincide at Witt class +1. (7) Shell stratification as level sets: TVL's face, edge, and corner shells (|w|²=1, 2, 3) correspond precisely to the level sets Q₃=1, 2, 0 of the standard quadratic form on the parity-check space F₃³. (8) Polar duality and parabolic nesting: The q₃=0 sector of TVL is the polar of the corner point (1, 1, 1) with respect to the absolute conic. The q₃=0 line is σ-invariant. The family nesting d→d+1 corresponds to maximal parabolic subgroup actions in PSL (d+1, F₃). The paper applies the automorphism-group-action framework of Grassl and Rötteler (2013) to the ternary projective Hamming family and identifies three structural features not studied in that context: the closed-form weight enumerator, the family-wide Witt-class result, and the q=3 exceptionality identity. Four Python verification scripts with accompanying README are included as ancillary files, reproducing the computational results cited in the paper. Published in conjunction with the TVL framework archived at: 10. 5281/zenodo. 19682633 (Concept DOI, always resolves to latest version). Changelog v1. 0. 4 (May 2026) No changes to mathematical content, proofs, or verification scripts. 1. Self-referential family name removed: three occurrences of "the Paper E family" replaced with "this code family" or the explicit family expression in §5 and Appendix A. v1. 0. 3 (May 9, 2026) No changes to mathematical content or proofs. 1. Appendix A moved to before the bibliography (was incorrectly placed after). 2. Appendix heading corrected to "Appendix A" in table of contents and rendered heading. 3. verifyREADME. md: theorem numbers corrected (closed-form enumerator is Theorem 14; Witt class is Theorem 15). v1. 0. 2 (May 8, 2026) Substantial mathematical additions. Verification scripts unchanged. 1. New §5 (σ-structural properties of the family) with five new results: Q-Scope theorem (self-orthogonal iff q ∈ 2, 3) ; σ-isometry lemma; eigenspace symplectic orthogonality proposition; maximum Jordan block size theorem (= 3^ν₃ (d) ) ; σ-fixed Gram rank-1 proposition. 2. Appendix A added: complete proof of the maximum Jordan block size theorem (Lemmas 1–5 with full algebraic detail). 3. Abstract updated to reflect §5. 4. Page count: 13 → 17. 5. Internal labels removed from the closed-form enumerator and family-wide Witt class theorem titles. v1. 0. 1 (May 2, 2026) No changes to mathematical content, proofs, or verification scripts. 1. QuantumErrorCorrectingStructureTVL. pdf: Two display equations in §6 reformatted (overfull lines fixed: coset distribution and Q₃ level sets now use aligned layout). DOI on title page updated to Concept DOI (10. 5281/zenodo. 19983025). 2. verifyREADME. md: Heading "Paper E" replaced with paper title. Companion paper DOI now includes paper title. This record DOI added explicitly. Duplicate Requirements section removed. Grammar corrected in Witt-class contrast description. Weight enumerator remark section named precisely. License section removed (covered by record-level metadata). v1. 0. 0 (May 2, 2026) Initial release.