A degeneracy of a real symmetric 2×2 block requires TWO conditions to hold at once: $\Delta =0$ and $V=0$ — the square root vanishes only when both do. A single knob traces a one-parameter path through the $(\Delta ,V)$ plane, and a path generically misses the single point where two conditions hold: degeneracies have codimension 2 (three, for complex Hermitian, where Re and Im of the coupling must both vanish). This is the von Neumann–Wigner counting: crossings seen along one-parameter sweeps are infinitely fine coincidences, so what one actually observes is approach-and-repel, with the minimum gap set by whatever coupling refuses to vanish.

The exception is a symmetry. If the two states carry different quantum numbers of an operator commuting with $H$ (different parity, different angular momentum projection, different particle number), then $V=⟨1|H|2⟩=0$ identically — not by tuning but by theorem — and one condition remains: $\Delta =0$, reachable with one knob. Crossings between different-symmetry levels are allowed and common; crossings within a symmetry class are forbidden and avoided. Reading a level diagram is therefore symmetry spectroscopy: every avoided crossing announces a coupling, every true crossing announces a conserved quantum number distinguishing the pair.