A central question beyond the Landau paradigm is the nonperturbative critical theory of the symmetric mass generation (SMG) transition, where strong interactions gap Dirac fermions in $2+1$ dimensions without triggering spontaneous symmetry breaking or topological order. Sign-problem-free studies have previously found fermion mass generation without bilinear condensation in related models, but the fermionic scaling properties of Hamiltonian SMG transitions remain incompletely characterized. Using sign-problem-free projector determinant quantum Monte Carlo simulations of a bilayer honeycomb lattice model with $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)/{\mathbb{Z}}_2$ symmetry and linear sizes up to $L=21$, we find evidence for a direct continuous transition: the single-particle and bosonic gaps open consistently near $J_c\approx 2.6$, with correlation-length exponent $\nu =1.14\left(2\right)$ from the better-converged fermionic analysis, while all 19 symmetry-inequivalent local fermion-bilinear multiplets remain disordered. We obtain an unbiased estimate of the fermion anomalous dimension, ${\eta }_{\psi }=0.071\left(1\right)$, which differs from the large-$N$ prediction (${\eta }_{\psi }\approx 0.595$) and the variational Monte Carlo estimate (${\eta }_{\psi }\approx 0.62$), thereby constraining candidate descriptions of SMG criticality. A comparison with a related $\mathrm{Spin}\left(5\right)\times \mathrm{U}\left(1\right)/{\mathbb{Z}}_2$ model, which develops an intermediate excitonic phase, illustrates how the symmetry content controls competing bilinear orders.

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• Quantum criticality
• Quantum phase transitions
• Quantum Monte Carlo
• SU(N) symmetries