Constraints on boson mass (m₂₂) from combining the soliton size scaling (Equation (1)) with an opacity (column) requirement. The vertical axis shows a characteristic physical scale; we compare the observed effective radius range of LRDs (green band, r_e ∼ 30–100 pc) with the soliton core radius predicted for M_s = 10⁸–10⁹ M_⊙ (blue band; dashed boundaries follow ﹩{r}_{c}\propto {m}_{22}^{-2}﹩). The red region marks parameters for which even a maximally loaded core (f_g ≈ 1) confined within r_e ≈ r_c with ξ ≈ 1 cannot reach N_H ≈ 10²⁴ cm^−2,. The opacity boundary is evaluated for a representative M_s ∼ 10⁹ M_⊙ (typical of LRD hosts). The orange shaded area illustrates sensitivity to baryon loading uncertainties by adopting conservative values (f_g = 0.1, ξ = 3). The viable region is the overlap between the observed size band and the nontransparent (Compton-thick-capable) region; the region outside the red shaded area corresponds entirely to the nontransparent (Compton-thick capable) regime. The “viable region” is specifically the parallelogram defined by the overlap between the observed size band and this nontransparent regime. Under the direct-mapping assumption (r_e ∼ r_c), this favors m₂₂ ∼ few, and we adopt m₂₂ = 2 as a fiducial baseline for subsequent figures.