Residual mass-shell action in light-cone string gauge ($L_0={\alpha }^{\prime }{p}^2+N-a$)

Fixing nonzero longitudinal oscillators leaves the center-of-mass symplectic term, transverse oscillator terms, and a zero-mode multiplier imposing the mass-shell condition. On a patch with $p^{+}\ne 0$, further choosing $x^{+}$ as time solves that constraint for $p^{-}=[p_{\perp }^2+(N-a)/{\alpha }^{\prime }]/(2p^{+})$. This distinguishes partial oscillator gauge fixing from the subsequent time choice.