Bias-aware Fisher forecasts for stochastic gravitational-wave background

Whittle-likelihood forecasting of the stochastic gravitational-wave background underpins planned mHz interferometers (LISA, Taiji, TianQin). A composite two-stage estimator, per-bin amplitude estimation then unweighted log-space template fitting, inflates the marginal-Fisher amplitude error bar on resolved-bin count by up to an order of magnitude at $11.5$-d baseline. This inflation arises from the composite implementation, not the determinant-approximate $\mathtt{\text{gauss\_D}}$ likelihood it approximates. Jointly maximised, $\mathtt{\text{gauss\_D}}$ is unbiased ($R=1.000$ within ${10}^{-4}$) and is recommended baseline with exact-Whittle likelihood. With exact Whittle as control and $\mathtt{\text{gauss\_D}}$ as working model, we calibrate bias vector and mean-squared-error matrix for power-law and smooth broken-power-law templates. First, for sharp signals of a $32$-point sound-shell scan, the composite estimator inflates amplitude error bar by $R=10.2$ at $11.5$-d baseline (median $\simeq 3$), follows $R^2-1\propto T_{\mathrm{s}\mathrm{e}\mathrm{g}}^2$ at shorter segments, and has a non-removable efficiency gap ${\mathbf{C}}_b$ up to $18\mathrm{\%}$ in ${\alpha }_{\mathrm{o}\mathrm{u}\mathrm{t}}$. Second, the usual single-parameter Whittle-bias diagnostic does not track multi-parameter inflation and errs both ways. Third, inflation is negligible on $30$-min segments, $R-1\le 1.5\times {10}^{-3}$ over measured bias-exponent range, vanishes below sensitivity floor, and tracks resolved signal strength with $\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}[\ln \mathrm{S}\mathrm{N}\mathrm{R},\ln (R-1)]=0.91$. Results are frequentist; interval diagnostics transfer to flat-prior credible intervals only in the locally Gaussian regime. The calibration is a strong-signal, stationary statement transferable only to a matching noise model.