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, developed in "Topological Vortex Logic: Generation and Colour Structure from T³/Z₃" (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. Eleven structural results are established: 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. 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). PSL irreducibility (iff d=3): PSL (d, F₃) acts absolutely irreducibly on the logical space Ld = Cd/Cd^⊥ if and only if d=3 — proved at d=3 via the Bardoe–Sin submodule structure (one composition factor at d=3, hence irreducible), with an independent Burnside-criterion confirmation (the 7-dimensional logical module spans the full matrix algebra). For every d ≥ 4 the action is reducible: Ld is a uniserial F₃SL (d, F₃) -module with exactly d−2 composition factors, by the Bardoe–Sin submodule structure of the GL (n, Fq) permutation module on PG (n−1, q). The earlier converse-of-Schur argument fails in defining characteristic, where dim EndG = 1 does not imply irreducibility. The logical space exists only on the self-orthogonal range q ∈ 2, 3, where the same d−2-factor structure holds in both fields — the binary factors being the trivial module and the exterior powers Λᵏ V of the natural module — so that Ld is irreducible exactly for (q, d) ∈ (3, 3), (2, 3): the TVL code and the Steane code [7, 1, 3] (Proposition 7). σ-structural properties of the family: the shift automorphism σd is an isometry of F₃^ (nd) for the weight form; its eigenspaces are mutually orthogonal under the symmetric weight form (except for the reciprocal-pair exception, result 5, below) ; its maximum Jordan block size on the logical space Ld equals 3^ (ν₃ (d) ) (the 3-adic valuation of d) ; for d = 3ᵏ the σ-fixed Gram matrix has rank 1; and the construction is self-orthogonal over Fq if and only if q ∈ 2, 3 (the Q-Scope theorem, via a Gauss power-sum identity). Reciprocal pairs on the logical space: the eigenspace-orthogonality above has exactly one kind of exception — a non-self-reciprocal irreducible factor q ≠ q* on Ld — occurring if and only if some divisor n of d, coprime to 3, satisfies −1 ∉ ⟨3⟩ (mod n) ; the smallest such d is 8. Both members of every such pair survive to Ld for all d, with the exact multiplicity given in closed form (the smallest surviving multiplicities are 408, 8050, 61318 at d = 8, 11, 13). 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. 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. Quantum weight enumerators: the Shor–Laflamme stabilizer and normalizer weight enumerators of the family are given in closed form. The stabilizer enumerator Ad (z) generalizes the d=3 line-incidence count to hyperplanes in PG (d−1, F₃) ; the normalizer enumerator Bd (z) is its F₉ MacWilliams transform; and the minimum distance dₘin = 3 is recovered directly. Non-standard quadratic form: the weight-mod-3 function on Cd/Cd^⊥ defines a non-degenerate quadratic form of maximal Witt index for every d ≥ 3 (anisotropic kernel ⟨1⟩ for odd d, hyperbolic for even d). At d=3 it is non-isomorphic to the standard sum-of-squares form on F₃⁷ (the weight form has discriminant −1, the standard form +1) ; at d=4 the two coincide. 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. 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. Published in conjunction with the TVL framework archived at 10. 5281/zenodo. 19682633 (concept DOI, always resolves to the latest version). Note on this record. Every result in this version is established by analytical proof or citation, with each central result additionally re-derived by an independent method; the deposit is paper-only from v1. 0. 9 (earlier versions' ancillary scripts remain attached to their own records). The revisions reflect that standard: open characterization questions were resolved one by one, and the result shown to be overstated — the PSL-irreducibility characterization — was corrected in the open (v1. 0. 5), sharpening the paper's central claim that [13, 7, 3]₃ is the unique ternary code in its family with an irreducible logical space; and an incomplete proof step — the off-diagonal case of the general self-orthogonality theorem — was replaced by a complete argument (v1. 0. 8), with the result, already established independently, unchanged; and the weight-form theorem statement was sharpened to maximal Witt index; the d=3 irreducibility argument and the F₉-structure characterization were rewritten to rest on citation and analytic argument in place of the earlier computations (v1. 0. 9). Changelog v1. 1. 0 (reserved — 10. 5281/zenodo. 21212326, not yet uploaded) — adds the reciprocal-pair characterization on the logical space as a new headline result, with its survival bound proven in closed form. The characteristic polynomial of the cyclic-shift action on Ld has a non-self-reciprocal irreducible pair if and only if some divisor n of d (coprime to 3) satisfies −1 ∉ ⟨3⟩ (mod n) ; pairs first occur at d = 8 (the two factors of x⁴+1, each of multiplicity 408) and are absent at, e. g. , d = 9, 10, 12, 14, 15. The criterion is strictly finer than the hidden-F₉ condition 4 | d, and sharpens the eigenspace-orthogonality proposition by locating exactly where its exceptional case occurs. The underlying factorisation theory of xⁿ−1 into self-reciprocal factors and reciprocal pairs is classical and is cited (Massey 1964; Yucas–Mullen 2004; Lidl–Niederreiter) ; the contribution is the specialisation to the logical space. Survival multiplicities are given in closed form for every d: an appendix derives the exact multiplicity M (d, n) − 2·3^ (ν₃ (d) ) together with an unconditional lower bound M (d, n) > 2·3^ (ν₃ (d) ) at every trigger dimension, so both members of each pair survive to the logical space for all d (the smallest surviving multiplicities are 408, 8050, 61318 at d = 8, 11, 13). Elevated to its own headline contribution in the introduction and conclusions (one of eleven structural results the paper now establishes, up from ten). The result carries two independent decorrelated verifications, and a prior-art check was performed before inclusion. v1. 0. 10 (July 6, 2026) — 10. 5281/zenodo. 21212009 — ten fixes, none touching the paper’s central results. A false subgroup claim is corrected: the cubic symmetry group Oh (order 48) does not itself embed in PSL (3, 3) ; it acts on the 13 points through its quotient Oh/±I ≅ S₄ (order 24, the central inversion fixing every projective point), whose image sits at index 234, not the previously stated 117 — matching the index already used correctly in the companion paper. |SL (3, 3) | is correctly labelled (the value shown was previously mislabelled as |GL (3, 3) |). The eigenspace-orthogonality proposition is qualified: distinct irreducible factors give orthogonal eigenspaces except when one is the reciprocal polynomial of the other (first occurring at d=8), in which case the eigenspaces pair nondegenerately instead. "Symplectic form" is corrected to "symmetric form" throughout §5, describing the σ-invariant inner product (the genuine stabilizer/normalizer symplectic duality in §6 is unaffected and unchanged). The prime-occurrence condition is correctly named a base-3 repunit prime (OEIS A028491), not a Wagstaff prime. An off-diagonal Gram-matrix entry is restated as the correct sum, Σppipj. A false claim of parallel line-pairs in the projective plane is removed (no two distinct lines in PG (2, 3) are parallel). A cross-reference in the introduction is corrected to point to the polar-duality section, matching the body result it describes. The charge descent to projective points is stated correctly as holding only in the charge-0 sector. The minimum-distance citation is corrected to the paper that actually reports a qutrit-hardware toric code (Iqbal et al. , Nature Communications 16, 6301 (2025) ), replacing a citation to an unrelated qubit-based paper. v1. 0. 9 (July 3, 2026) — 10. 5281/zenodo. 21184713 — erratum to the weight-form theorem (result 8), with the d=3 irreducibility proof and the F₉-structure characterization rewritten to rest on citation and analytic argument in place of computa