Quantum Error-Correcting Structure of Topological Vortex Logic: The [[13,7,3]]_3 CSS Family and its Algebraic Geometry

Abstract

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: Stability, Root Systems, and Module Structure of Winding States on T³ with a Selected Z₃ Grading" (Zenodo DOI: 10.5281/zenodo.19682633). The 26 stable winding labels of the stipulated TVL split model form 13 antipodal pairs. Reducing these pairs modulo 3 identifies them bijectively with the 13 points of P²(F₃). Choosing one homogeneous representative for each point gives the columns of the 3×13 projective parity-check matrix H; its self-orthogonality yields the [[13,7,3]]₃ ternary CSS quantum Hamming code. The selected functional q₃ singles out a distinguished projective line through its kernel but does not construct the code. The 13 projective points index the abstract code coordinates; no hardware realization follows from this association. The paper studies this code and its dimension-indexed family [[(3^d−1)/2, (3^d−1)/2 − 2d, 3]]₃ for d ≥ 2. Sixteen structural result groups are established; v1.1.2 preserves the established core while synchronizing scope, repairing proof gaps, and adding the low-weight, fixed-space, Jordan, Singer and equivariant-structure developments summarized below: Family construction and self-orthogonality: The code is the d=3 member of the family [[(3d−1)/2, (3d−1)/2−2d, 3]]₃ for d ≥ 2, and the CSS construction is valid at every level because HdHdT=0 over F₃. Double distinction of d=3: The d=3 member is the smallest family member with positive logical dimension and the unique member among d ≤ 6 with prime physical length n=13. PSL irreducibility: PSL(d,3) acts absolutely irreducibly on the logical space Ld=Cd/Cd⊥ if and only if d=3; for every d≥4 the action is reducible and uniserial with d−2 composition factors, by the Bardoe–Sin module structure. σ-structural properties of the family: The signed coordinate shift σd is an isometry; distinct eigenspaces are orthogonal except for reciprocal pairs; its maximum Jordan block size on Ld is 3ν₃(d); the projective construction is self-orthogonal only for q∈{2,3}; and the uniform full-support Pauli normalizes the binary members but not the ternary members. Reciprocal pairs on the logical space: A non-self-reciprocal irreducible pair occurs exactly when some divisor n of d coprime to 3 satisfies −1∉⟨3⟩ modulo n, first at d=8; both members survive to Ld with a closed, always-positive multiplicity. Weight enumerators: The d=3 stabilizer distribution is derived from the line geometry of PG(2,3), and closed classical stabilizer and quantum normalizer weight enumerators are given for the full family; the normalizer enumerator is the F₉ MacWilliams transform and recovers distance three non-circularly. Low-weight normalizer species: The symplectic-weight-four, -five and -six layers are classified by projective support species, with exact local-code profiles, Pauli-pair multiplicities, purity, dimension thresholds and PSL(d,3)-orbit structure. The aggregate coefficients and their family stability are classical; the species-level decomposition is the content claimed here. For weight six, the planar and rank-four classifications, all multiplicities and pair data, and the single-orbit assertions are proved analytically in the paper through the projective-systems/code dictionary and explicit local-code normal forms. Equivariant Witt decomposition: Over F₃, the maximal-Witt-index decomposition can be chosen σd-equivariantly for every d, including the modular cases 3|d not covered by the usual averaging argument. Quadratic form on logical space: Weight modulo three defines a non-degenerate quadratic form of maximal Witt index on the logical space; at d=3 its coset distribution is 729:756:702. Independent quadratic forms and shell stratification: The weight-induced logical form has discriminant (−1)d, anisotropic kernel ⟨1⟩ for odd d and is hyperbolic for even d. Separately, the standard sum-of-squares form on F₃³ has TVL’s face, edge and corner shells as its level sets. Polar duality: The absolute conic Q₃=0 is the corner shell of PG(2,3), and the polar of a corner point is the σ-invariant line carrying the A₂ root system; the corrected trichotomy is tangent for corner points, secant for edge points and external for face points. Minimum-weight logical operators: For every d≥3, symplectic-weight-three logical supports are exactly incident point–line flags of PG(d−1,3), each carrying eight nonzero normalizer pairs and four projective Pauli directions, with B₃=(2/3)(3d−1)(3d−3) and a transitive PSL(d,3)×SL(2,3) action. Complete σ-fixed subspace and restricted Gram rank: For every d, dim Ldσd=r(d)−2. The restricted form degenerates exactly when 3 divides d, and has rank one precisely at the pure powers of three. Complete Jordan type: The full elementary-divisor type is type(F₃nd) minus twice the type of the regular F₃[⟨σd⟩]-module; the earlier maximum-block theorem follows as a corollary. Singer cycle: The projective Singer permutation has order (3d−1)/2, while its monomial lift has that order for odd d and twice that order for even d. In all cases the lift order is coprime to three, so the action is semisimple; for odd d the invariant logical subspace is a line. Parabolic structure of the family: Hyperplane nesting d→d+1 is governed by a maximal parabolic subgroup with unipotent radical Z₃d; the projective-parabolic Levi quotient is GL(d,3) for even d and GL(d,3)/{±I} for odd d. The paper applies the automorphism-group-action framework of Grassl and Rötteler (2013) to the ternary projective Hamming family. Its principal additions include the closed-form weight enumerators, the family-wide maximal-Witt-index result, the q=3 exceptionality identity, and the uniform-transversal Pauli dichotomy. Related TVL framework: 10.5281/zenodo.19682633 (concept DOI, always resolves to the latest version). Changelog v1.1.2 (July 31, 2026) — 10.5281/zenodo.21705393 — two reference fixes, with no change to any mathematical statement. The secondary citation to the three-torus results paper is removed from the stability preamble and from the bibliography, the framework paper alone supplying the stability model and theorem. This was announced in the v1.1.1 changelog, but the file deposited as v1.1.1 predates the change and still carries the citation. The most recent references are rearranged into the correct order. v1.1.1 (July 29, 2026) — 10.5281/zenodo.21661061 — scope-synchronization, correction and addition release, bringing the total to sixteen structural result groups. Scope synchronization to the companion framework paper (10.5281/zenodo.19682633): the exact stable-label → projective-quotient → CSS construction replaces the earlier orbifold and code-inducing-charge framing, and physical realisation is stated as an open question. Corrections: the 3×13 matrix is the parity-check / projective point-coordinate matrix, not an incidence matrix; PSL(d,3) is the projective special linear subgroup, the full collineation group only for odd d; an invalid projective notation and a residual q₃-descent paragraph are removed. Further corrections: the prime-length discussion (a later prime occurs at d = 13), the projective-parabolic Levi quotient, the d = 4 quadratic-form wording, the all-eigenvalue step in the Jordan upper bound, and the removal of a circular appeal to the known minimum distance. The polar-duality classification is corrected: only the six external points have secant polars, while the three internal points have polars disjoint from the conic, so the corner, edge and face shells correspond to tangent, secant and external polars. Uniform-transversal Pauli dichotomy: uniform transversal X and Z preserve the binary projective Hamming CSS codes but not the ternary ones; in particular they are not logical operators of [[13,7,3]]₃. Minimum-weight logical operators are classified for all d ≥ 3: their supports are exactly the point–line flags of PG(d−1,3), each carrying four projective Pauli directions, with B₃ = (2/3)(3d−1)(3d−3) and a transitive orbit. The classical collinear-triple fact is prior art. The shift-fixed subspace dimension and its restricted Gram rank are given in closed form for every d; the rank degenerates exactly when 3 divides d, and equals one precisely at the pure powers of three. The complete Jordan type of the shift on the logical space is determined for every d, not only its maximum block size; the earlier maximum-block theorem follows as a corollary. The Singer cycle, of order coprime to the characteristic for every d, acts semisimply throughout the family, with a one-dimensional invariant logical subspace for odd d. A prime-length corollary is folded in: if (3d−1)/2 is prime then d is prime, the converse failing at d = 5. Species decomposition of the low-weight normalizer layers: for each symplectic weight up to six the joint supports are classified, with the multiplicity, purity and orbit structure carried by each species. Every multiplicity is derived in closed form. The weight-six planar classes are proved from incidence moments and canonical local-code forms; the rank-four classes and their PSL orbits follow from monomial equivalence of the two-dimensional dependency codes, with the determinant-square-class argument preventing orbit splitting. The coefficients and their stability across the family are not new and are governed by classical theory, which is credited. Equivariant Witt decomposition: over F₃, the maximal-Witt-index splitting can be chosen equivariantly for the coordinate shift for every d, uniformly across the semisimple and modular shift cases. The published equivariant Witt theorem assumes the group order invertible; the odd-characteristic modular case treated here is not covered by that averaging t