Proposition: Conformal data do not fix a physical energy scale

Fix a VOA and its canonical rotation generator $L_0$. Additional assignments $H_{\alpha }=\alpha L_0$, $\alpha >0$, as physical Hamiltonians change the energies of its positive-energy vectors without changing the VOA, its mode bounds, or its declared sector record. Thus those data alone do not fix a dimensionful energy or mass.

Proof. The assignments do not alter the operator $L_0$, its canonical rotation action, or the VOA identities. They alter only the external conversion between dimensionless conformal energy and the proposed physical Hamiltonian. For a vector of conformal weight $h>0$, the assigned energy is $\alpha h$, which varies with $\alpha$.