A Tangent Configuration in the Projective Plane of Order Three: Symmetry Diamond, Orbits, and Commutants

This paper isolates a finite-geometric configuration arising from Topological Vortex Logic (TVL). The twenty-six nonzero stable winding labels of the three-torus model form thirteen antipodal pairs, identified with the thirteen points of the projective plane PG(2,3). The selected mod-three grading becomes the line H with equation x1+x2+x3=0, while the shell form becomes the quadratic form Q(x)=x12+x22+x32. The paper studies this thirteen-point configuration independently of any physical interpretation. Tangency: the zero set of Q is a nondegenerate four-point conic, and H is its tangent at [1:1:1]. Thus the distinguished line and conic determine a unique point of contact. Symmetry diamond: the linear group fixing the labelled functional is AGL(2,3), of order 432, and its determinant-one subgroup is ASL(2,3), of order 216. Their induced actions on the four points of H are S4 and A4. Retaining the quadratic form cuts the group to the six coordinate permutations, inducing S3; its determinant-one intersection is C3. The A4 and S3 branches are incomparable and meet in C3. Five-orbit decomposition: the form-preserving group partitions PG(2,3) as 1+3+3+3+3. The distinguished line is the union of the singleton tangent point and one three-point orbit; the conic is the union of that singleton and a different three-point orbit. Permutation-module commutants: on the standard three-dimensional module of the line, the S4 and A4 commutants are scalar. On a three-point orbit the S3 commutant is two-dimensional. On the nine-point complement it has dimension 18. On the whole thirteen-point module, the decomposition is 1⊕5⊕2⊕4, giving a commutant of dimension 41; the subalgebra preserving all five orbit blocks separately has dimension 9. The paper is deliberately scoped to finite projective geometry, permutation modules and commutants. TVL supplies the provenance and names of the point families, but no particle assignment, flavour model, mixing matrix or phenomenological prediction is asserted. The mathematical inputs from the TVL winding-state paper and its quantum-code companion are cited by concept DOI 10.5281/zenodo.19682633 and 10.5281/zenodo.19983025.