To a classical group G over a finite field Fq — acting on its natural geometry (a projective space, a quadric, a Hermitian variety, or the Lagrangian Grassmannian) — this paper attaches a "canonical code": its columns are the points of that geometry in their natural projective (Plücker) coordinates, each normalized to a canonical representative. It then asks one arithmetic question about all of these constructions at once: for which q is the canonical code self-orthogonal (equivalently dual-containing, C⊥ ⊆ C — the condition that lets it serve as the parity-check side of a CSS quantum code)? The answer is uniform, with a single explanation. Self-orthogonality holds precisely when the scale factor used to normalize each orbit point is an isometry of the relevant form: s² = 1 for the linear, orthogonal, and symplectic families (which holds iff q ∈ 2, 3) and Nm (s) = 1 for the unitary family (which holds iff q = 2). This is the canonicalization–isometry principle, and it yields a complete dichotomy across every classical family and every dimension: the canonical codes of PSL, PΩ⁺, PΩ⁻, and Sp are self-orthogonal if and only if q ∈ 2, 3, and those of PSU if and only if q = 2, with a single low-dimensional exception in each non-symplectic family. The dichotomy is established over every finite field — every prime power q — not merely the primes. The symplectic case, the deepest, is settled by an invariant-form lemma together with a Schubert-cell argument on the Lagrangian Grassmannian; the other four families fall to an elementary first-nonzero decomposition whose only field input is a power sum over Fq. The ternary Hamming code [13, 7, 3]₃ is the d = 3, q = 3 instance of the linear family, which ties the dichotomy to the Topological Vortex Logic (TVL) programme, where that code arises; the correspondence is made precise (the Bridge Theorem). Two structural by-products are recorded: the Jordan and cyclotomic behaviour of an associated cyclic symmetry σ at all primes, and the maximal Witt index of the weight form for every member of the family. The result generalizes the classical Gleason–Pierce–Ward theorem (on self-dual divisible codes) to a self-orthogonal, cross-family, all-dimensions statement for these group-orbit codes. Individual codes in the family are classically known; the uniform mechanism and its q-dichotomy are the contribution. Changelog v1. 0. 1 (July 3, 2026) — proof-repair erratum to the invariant-form lemma; results unchanged. One step of the lemma's printed proof (that an equivariant endomorphism preserves torus weight spaces) fails over the finite field at exactly the q ∈ 2, 3 where the lemma is used, and the sentence calling the argument characteristic-free is withdrawn. The proof is repaired via the Siegel parabolic (for q > 2 and for q = 2 with k odd), with the q = 2 even-k case closed analytically; no stated result changes, with minor wording harmonization. v1. 0. 0 (July 1, 2026) — initial release; part of the TVL preprint series.
Vladimer Merebashvili (Fri,) studied this question.