A Narain Lattice from the Ternary Hamming Code of PG(2,3): A Mixed-Shift Incidence Criterion, a Norm-Six Reflection Identity, and Two Obstructions

The [[13,7,3]]₃ ternary CSS Hamming code — the qutrit member of the nonbinary quantum Hamming family — is a stabilizer code with seven logical qutrits built on the thirteen points of the projective plane PG(2,3). Because its stabilizer is not maximal isotropic, the code determines an ensemble rather than a single conformal field theory, one theory for each maximal isotropic completion. This paper selects a polarized completion and exhibits the resulting Narain lattice explicitly: an even self-dual lattice of signature (13,13), sitting inside an ensemble of 32,717,081,600 completions. In the principal code-to-Narain papers cited in the paper, the nonzero-logical worked examples are the Shor and Steane codes, both with one logical qudit; the present paper carries out the corresponding construction for this seven-logical-qutrit code. The low nonoscillator lattice-primary multiplicities reproduce the weight distribution of the classical Hamming code. The chiral current algebra is unenhanced: neither chiral side contains additional non-Cartan lattice currents, because such a current would require a weight-three pattern in the ternary simplex code, whose minimum distance is nine. Two positive results and two scoped obstructions follow. A mixed-shift incidence criterion. The paper classifies the mixed minimal-lift family of order-three shifts built from a weight-nine word of the simplex code and a weight-three word of the Hamming code. This is not a classification of every order-three shift of the lattice. It distinguishes the general non-anomalous condition α·β ≡ 0 (mod 9) from the stronger same-charge, ℓ = 0 gluing α·β ≡ 0 (mod 27). The support overlap takes the values zero, two and three, and never one. The stronger same-charge gluing is admitted precisely at overlaps zero and two; the overlap-three family also fails the general anomaly condition. The complete signed inventory is 208, 1,872 and 624 shifts in the three overlap classes. A norm-six reflection identity. Separately, for the first nonzero-ℓ, norm-six branch, the charge-matched orbifold lattice is exactly the image of its parent under the ordinary reflection in the primitive shift vector. The norm-six condition is necessary as well as sufficient, and the glued cosets are matched by their charge rather than all being identified with the charge-zero coset. No direct equal-signature heterotic route. The direct cyclotomic Construction-AC route used here has equal left- and right-moving ranks and therefore signature difference zero, whereas a heterotic Narain lattice requires signature difference sixteen. This closes only that specified direct route; it is not a general obstruction to heterotic code constructions with additional gauge-lattice data. No isolated fixed points on the associated thirteen-torus. A symmetric geometric order-three orbifold of the associated torus cannot have isolated fixed points, because an order-three real orthogonal action without invariant directions requires even real dimension, whereas this torus has dimension thirteen. The order-three classes of PSL(3,3) acting on the thirteen coordinates likewise have fixed coordinates rather than a fixed-point-free action. The novelty boundary is explicit. The class-wide code-to-Narain construction, the completion count, the enumerator control of the spectrum, the minimum-distance current criterion and the general level-matching condition are cited prior results. The paper contributes the explicit seven-logical-qutrit instance and its low-level tables, the PG(2,3) incidence formulation of the mixed minimal-lift family, and the norm-six reflection identification. For the latter two formulations, the paper states only that no precedent was found in the checked sources, not an unqualified first-discovery claim. The two obstructions are instance-specific consequences that delimit the direct continuations considered. The object constructed is a toroidal bosonic conformal field theory. No heterotic embedding is supplied, no geometric orbifold is constructed, and no twisted sectors localised at fixed points occur. The algebraic structure of the underlying code is treated in the companion work on its quantum error-correcting structure, cited by concept DOI 10.5281/zenodo.19983025; the present paper takes the code as given and studies the Narain conformal field theories and shift structures it determines.