Measurement of the beam asymmetry <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Σ</mml:mi></mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>π</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>η</mml:mi></mml:math> photoproduction on the proton at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>γ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>9</mml:mn></mml:mrow></mml:math> GeV

We report measurements of the photon beam asymmetry $\mathrm{\ensuremath{\Sigma}}$ for the reactions $\stackrel{P\vec}{\ensuremath{\gamma}}p\ensuremath{\rightarrow}p{\ensuremath{\pi}}^{0}$ and $\stackrel{P\vec}{\ensuremath{\gamma}}p\ensuremath{\rightarrow}p\ensuremath{\eta}$ from the GlueX experiment using a 9 GeV linearly polarized, tagged photon beam incident on a liquid hydrogen target in Jefferson Lab's Hall D. The asymmetries, measured as a function of the proton momentum transfer, possess greater precision than previous ${\ensuremath{\pi}}^{0}$ measurements and are the first $\ensuremath{\eta}$ measurements in this energy regime. The results are compared with theoretical predictions based on $t$-channel, quasiparticle exchange and constrain the axial-vector component of the neutral meson production mechanism in these models.