= Solution
Use the free-boson <operator product expansion>
$$
X^\mu(z,\bar z)X^\nu(w,\bar w)\sim-\frac{\alpha'}2\eta^{\mu\nu}\log|z-w|^2.
$$
Write $E_p=:e^{ip\cdot X}:$ and $\epsilon=z-w$. The needed <Wick contractions> are $\partial X^\mu(z)\partial X^\nu(w)\sim-\alpha'\eta^{\mu\nu}/(2\epsilon^2)$ and $\partial X^\mu(z)E_p(w)\sim-i\alpha'p^\mu E_p(w)/(2\epsilon)$. For $V_\zeta=\zeta_\mu:\partial X^\mu E_p:$, the cross double contraction in the <worldsheet stress tensor> has a cubic pole: either of its two derivatives can contract with the derivative in $V_\zeta$, while the other contracts with $E_p$. Including the factor $-1/\alpha'$ gives
$$
T(z)V_\zeta(w)\sim-\frac{i\alpha'}2\frac{p\cdot\zeta}{\epsilon^3}E_p(w)+\frac{1+\alpha'p^2/4}{\epsilon^2}V_\zeta(w)+\frac1\epsilon\partial V_\zeta(w).
$$
A <primary operator> has no cubic pole. Hence \b[the first vertex is primary precisely when]
$$
\boxed{p\cdot\zeta=0,\qquad h=1+\alpha'p^2/4.}
$$
The antiholomorphic <conformal weight> is $\bar h=\alpha'p^2/4$. Primarity alone puts no <mass-shell condition> on $p$.
For $W_\zeta=\zeta_{\mu\nu}:\partial X^\mu\bar\partial X^\nu E_p:$, the holomorphic cubic pole is $-i\alpha'p^\mu\zeta_{\mu\nu}:\bar\partial X^\nu E_p:/(2\epsilon^3)$. The antiholomorphic <operator product expansion> gives the analogous pole involving $p^\nu\zeta_{\mu\nu}$. Thus the <transversality of derivative-exponential primary vertices> requires
$$
\boxed{p^\mu\zeta_{\mu\nu}=0,\qquad p^\nu\zeta_{\mu\nu}=0,\qquad (h,\bar h)=(1+\alpha'p^2/4,1+\alpha'p^2/4).}
$$
There is no trace-free requirement for primarity. A physical <integrated string vertex operator> must additionally have <conformal weights> $(1,1)$; for this vertex that gives $p^2=0$.
Now keep the <fermionic signs> in the real chiral <Majorana fermion> calculation. Contracting the external $\psi(w)$ through $:\psi(z)\partial\psi(z):$ gives
$$
:\psi(z)\partial\psi(z):\,\psi(w)\sim-\frac{\partial\psi(z)}{\epsilon}-\frac{\psi(z)}{\epsilon^2}.
$$
Multiplying by $-1/2$ and Taylor expanding $\psi(z)$ at $w$ produces
$$
T(z)\psi(w)\sim\frac{\psi(w)}{2\epsilon^2}+\frac{\partial\psi(w)}\epsilon.
$$
This proves that $\psi$ is a <primary operator> with \b[weight $h_\psi=1/2$]. To find the <central charge>, in $T(z)T(w)$ the two double-contraction pairings contribute, before the overall factor $1/4$,
$$
-\langle\psi(z)\psi(w)\rangle\langle\partial\psi(z)\partial\psi(w)\rangle=\frac2{\epsilon^4},\qquad \langle\psi(z)\partial\psi(w)\rangle\langle\partial\psi(z)\psi(w)\rangle=-\frac1{\epsilon^4}.
$$
The single <Wick contractions> give $2T(w)/\epsilon^2+\partial T(w)/\epsilon$, so
$$
T(z)T(w)\sim\frac1{4\epsilon^4}+\frac{2T(w)}{\epsilon^2}+\frac{\partial T(w)}\epsilon.
$$
Comparing the leading term with $c/(2\epsilon^4)$ proves \b[$c_\psi=1/2$], as in the <free chiral Majorana fermion conformal field theory>.
With four embedding bosons and the ordinary reparameterization <bc ghost system>, cancellation of the <worldsheet Weyl anomaly> in each chirality requires
$$
4+\frac{N_f}2-26=0,\qquad\boxed{N_f=44.}
$$
Thus there are \b[44 real left-moving and 44 real right-moving fermions], equivalently 44 two-dimensional nonchiral <Majorana fermions>. These are <internal free fermions in a four-dimensional bosonic string>. Adding internal matter to the bosonic string does not introduce local <worldsheet supersymmetry> or its superghosts.
For the requested polynomial <string vertex operators>, work in the unprojected <Neveu–Schwarz sector>. A full model also chooses spin structures and a consistent projection, which the central-charge condition alone does not specify. If the holomorphic and antiholomorphic oscillator contributions to the <conformal weights> are $\ell_L,\ell_R$, physicality requires
$$
\ell_L+\frac{\alpha'p^2}4=\ell_R+\frac{\alpha'p^2}4=1,\qquad M^2=\frac4{\alpha'}(\ell_L-1),\qquad \ell_L=\ell_R.
$$
Therefore only equal levels at most one can contribute to the requested nonpositive <bosonic string mass spectrum>.
With zero fermion insertions, level zero gives the <tachyon vertex operator> $E_p$, a scalar with $M^2=-4/\alpha'$. Level one gives the <massless closed-string vertex operator> $\zeta_{\mu\nu}:\partial X^\mu\bar\partial X^\nu E_p:$, with the transverse conditions above and the <string-state gauge redundancy> that removes longitudinal polarizations. The physical transverse <polarization tensor> decomposes into a symmetric trace-free <graviton>, an antisymmetric <Kalb–Ramond field>, and a scalar <dilaton>. In four spacetime dimensions these have respectively two, one, and one physical polarizations; the two-form can be dualized to a scalar.
With two fermion insertions there are two possibilities. One fermion in each chirality gives
$$
C_{ij}:\psi^i\bar\psi^j E_p:,qquad (\ell_L,\ell_R)=(1/2,1/2),\qquad \boxed{M^2=-2/\alpha'.}
$$
These are \b[$44^2=1936$ scalar tachyons] before any projection. There is no spacetime derivative or spacetime polarization to impose a vector transversality condition.
Two different fermions in the same chirality form an antisymmetric <fermion bilinear>, a weight-one current $J^{ij}=: \psi^i\psi^j:$; equal-species undifferentiated bilinears vanish. <Closed-string level matching> then requires a weight-one bosonic derivative in the opposite chirality. The resulting <string vertex operators> are
$$
A^{ij}_\mu:\psi^i\psi^j\bar\partial X^\mu E_p:,qquad \widetilde A^{ij}_\mu:\partial X^\mu\bar\psi^i\bar\psi^j E_p:,qquad i<j.
$$
They have \b[$M^2=0$ and are gauge vectors], with $p\cdot A^{ij}=0$ and $A^{ij}_\mu\sim A^{ij}_\mu+p_\mu\lambda^{ij}$, and likewise for $\widetilde A$. Each chirality supplies $\binom{44}{2}=946$ gauge vectors. The <fermion bilinears> generate the two $\mathfrak{so}(44)$ current algebras associated with the <Special orthogonal Lie algebra>, giving the unprojected <gauge group> $SO(44)_L\times SO(44)_R$. The tempting vertex $:\psi^i\psi^j E_p:$ alone fails <closed-string level matching> because its weights are $(1,0)$ before the common momentum contribution. Additional derivatives raise the levels above the nonpositive-mass range; a product of a left and a right bilinear has four fermion insertions and is outside this requested sector. This completes the <zero- and two-fermion vertices of an internally fermionized bosonic string>; a specified global projection can remove some of the listed states.
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