[PHYSICS BUG][HIGH] Fisher matrix uses O(ε) forward difference, not O(ε⁴) five-point stencil

[JasperSeehofer]

File: master_thesis_code/parameter_estimation/parameter_estimation.py:336

compute_fisher_information_matrix() calls finite_difference_derivative(), which implements a first-order forward difference:

179667\frac{\partial h}{\partial\theta} \approx \frac{h(\theta+\varepsilon) - h(\theta)}{\varepsilon} \quad [O(\varepsilon)\text{ error}]179667

The O(ε⁴) five-point formula:

179667\frac{\partial h}{\partial\theta} \approx \frac{-h(\theta+2\varepsilon)+8h(\theta+\varepsilon)-8h(\theta-\varepsilon)+h(\theta-2\varepsilon)}{12\varepsilon}179667

is already implemented as five_point_stencil_derivative() in the same class but is never called from the Fisher matrix computation. The class docstring incorrectly claims a five-point stencil is used.

Fix: In compute_fisher_information_matrix(), replace the call to self.finite_difference_derivative() with self.five_point_stencil_derivative().

Cost: ~4× more waveform evaluations per Fisher matrix. Accuracy improves by O(ε³).

References:

• Vallisneri (2008), Use and abuse of the Fisher information matrix, arXiv:gr-qc/0703086
• Cutler & Flanagan (1994), PRD 49, 2658

Physics Change Protocol required.

[JasperSeehofer]

Paper triage: This is the highest-priority open physics issue. The five-point stencil method five_point_stencil_derivative() already exists in the class but is never called from compute_fisher_information_matrix(). Switching requires 4× more waveform evaluations (56 vs 14) and will change all Cramér-Rao bounds, requiring a full re-run of the simulation campaign. Tracked as PHYS-3 in TODO.md.