Practical Note IV: Testing a Mass-Scaled Quantum-Horizon Echo in NGC 4151 Accretion Lags

This practical note tests a fixed mass-scaled quantum-horizon echo-delay law in the public Swift multiwavelength monitoring data of NGC 4151. The tested delay is not fitted to the lag data. It is computed from the black-hole mass and redshift using the two fixed branches $\Delta t_{\mathrm{obs}}(M;\alpha,z)=(1+z)\alpha\frac{r_S(M)}{c}\ln\frac{r_S(M)}{\ell_P}$, where $r_S(M)=\frac{2GM}{c^2}$ and the physical branch coefficients are $\alpha=2/3$ and $\alpha=4/3$. The physical origin of the two branches is the two-screen black-hole relation $R_{\mathrm{qh}}^3=r_S\ell_P^2$, which gives the one-way branch $\Delta t_{1\mathrm{w}}\simeq \frac{2}{3}\frac{r_S}{c}\ln\frac{r_S}{\ell_P}$ and the two-way branch $\Delta t_{2\mathrm{w}}\simeq \frac{4}{3}\frac{r_S}{c}\ln\frac{r_S}{\ell_P}$. For NGC 4151, using $M_\bullet=4.57\times10^7 M_\odot$ and $z=0.003262$, the predicted observer-frame delays are $\Delta t_{1\mathrm{w,obs}}=8.844\ \mathrm{h}$ and $\Delta t_{2\mathrm{w,obs}}=17.689\ \mathrm{h}$. The UV/optical lag-wavelength structure shows two near-horizontal lag layers close to these fixed predictions. The selected one-way layer has mean lag $\langle\Delta t_{1\mathrm{w}}\rangle=9.336\ \mathrm{h}$, and the selected two-way layer has mean lag $\langle\Delta t_{2\mathrm{w}}\rangle=18.480\ \mathrm{h}$. Their observed ratio is $\frac{18.480}{9.336}=1.979$, close to the predicted two-branch ratio of $2$. Inverting the same fixed delay law gives timing-mass estimates of approximately $M_{\mathrm{echo,1w}}\simeq4.82\times10^7 M_\odot$ and $M_{\mathrm{echo,2w}}\simeq4.77\times10^7 M_\odot$, consistent with the adopted reverberation-mapping mass scale of NGC 4151. The note compares the fixed quantum-horizon delay layers with disk-only lag-wavelength curves. A pure disk-only no-offset model does not naturally reproduce the two near-horizontal layers, while an offset-dominated disk-plus-offset fit effectively behaves as a common-delay model. The main result is therefore not a single lag coincidence, but a two-branch timing structure: the same response channels show both the one-way and two-way predicted delay layers. This record includes the PDF technical note, figures, selected lag-layer tables, model-comparison outputs, and reproducibility materials used to generate the reported NGC 4151 result.