arXiv:2610.26529v1 Announce Type: new Abstract: For a nondegenerate Euclidean tetrahedron T, let {\Sigma}(T) denote the sum of its six interior dihedral angles. We establish the exact range [6 arccos(1/3), 3{\pi}) for the orthocentric and isogonic classes. The orthocentric part gives a short alternative proof of the sharp lower bound; the main new result is the corresponding isogonic theorem. Starting from classical insphere-contact geometry, we derive a reciprocal half-angle identity and a sharp defect inequality controlled by the scale-invariant dispersion of the reciprocal tangent-length parameters. This quantitative refinement shows that any nonzero dispersion forces a positive dihedral-angle-sum gap, and that near equality forces the normalized reciprocal parameters to be close to equality. Finally, the common kite subfamily realizes every value in the interval and proves sharpness of both endpoints.