Bosonic ND boundary-changing conformal weight ($h_{\mathrm{ND}}=1/16$)

For one free boson, the mixed ND open-string sector has strip vacuum energy $E_0=1/48$. The strip-to-plane transformation relates this energy to the plane Virasoro algebra zero mode by $E_0=h-c/24$, with central charge $c=1$. Hence its ground conformal weight is $h=1/48+1/24=1/16$. For $d$ independent ND coordinates the weights add to $d/16$. At $d=24$, the plane matter ground weight is $3/2$, while the strip oscillator vacuum energy is $1/2$. The plane physical condition $L_0^{\mathrm{plane}}-1=0$ is therefore equivalent to ${\alpha }^{\prime }{p}_{\parallel }^2+N+1/2=0$. The apparent different constants are coordinate/constraint conventions, not different mass spectra.