Classical-Group Q-Scope and the Canonicalization–Isometry Principle

To a classical group G over a finite field F_q — 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-perp contained in 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: s^2 = 1 for the linear, orthogonal, and symplectic families (which holds iff q is in {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, PO+, PO-, and Sp are self-orthogonal if and only if q is in {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 a characteristic-free 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 F_q. 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 sigma 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. Version 1.0.0. Part of the TVL preprint series — see Related identifiers.