ALMA CO(2-1) Gas Dynamics in NGC 315: A Multi-Method Benchmark for Supermassive Black Hole Mass Measurement

We present ALMA Cycle~7 \cotwo\ observations of the circumnuclear disk in NGC~315 at an angular resolution of $0\farcs230\times0\farcs175$, improving on past measurements and resolving the sphere of influence (SOI) of the supermassive black hole (SMBH), whose mass has previously been estimated of $M_{\rm BH}= \left(2.08^{+0.33}_{-0.15}\right) \times 10^9$~M$_\odot$ The high spatial resolution and sensitivity enable robust full-cube forward modeling of the molecular gas kinematics and a direct comparison of multiple independent gas-based dynamical modeling techniques. We apply standard Bayesian codes using both MCMC and nested sampling approaches, as well as a frequentist code to the same dataset, exploring systematic uncertainties associated with the stellar mass distribution, gas surface-brightness parameterization, and disk geometry. All methods yield consistent black hole masses, indicating that the inferred $M_{\rm BH}$ is not strongly method-dependent. Combining the ensemble of independent molecular-gas-based models, we derive an ensemble median black hole mass of $M_{\rm BH}/10^9\,\mathrm{M_\odot} = 2.02^{+0.04}_{-0.05}$(stat)$^{+0.05}_{-0.04}$(sys), where the comparable contributions to the full error budget arise from modeling systematics rather than formal fitting uncertainties. Our $M_{\rm BH}$ is consistent with the empirical $M_{\rm BH}$--$σ_\star$ and $M_{\rm BH}$--$L_{\rm bulge}$ scaling relations, and lies 32\% below an independent stellar-dynamical measurement, a discrepancy we discuss in the context of systematic differences between gas- and stellar-based methods. NGC~315 serves as a benchmark for quantifying molecular gas-dynamical $M_{\rm BH}$ systematic uncertainties and for future cross-comparisons of gaseous and stellar dynamical approaches.