[FlorianPfaff]

Complex Watson fitting conjugated the wrong operand when forming scatter matrices from row-wise observations. For nine observations at [1, i]/sqrt(2) and one at its orthogonal conjugate, both weighted and unweighted fitting selected the minority direction.

Form both scatter matrices as sums of z zᴴ, matching the PDF convention. Their eigenvalues are unchanged, preserving the concentration calculation and existing weight normalization.

Validation

Compared with main f653ab8e621fce69eeb9c1c39a0c50fc9a7275c9 using identical frozen dependencies: Python 3.13.14, NumPy 2.4.5, CPU PyTorch 2.12.1, JAX 0.9.1 (x64), and SciPy 1.17.1.

• Before: 4 failed, 2 passed in the new regression file on each of NumPy and PyTorch. The direction overlap was zero instead of one. Existing targeted tests passed.
• After: all 6 new regressions pass on NumPy and PyTorch. Coverage includes both estimation entry points, weighted/unweighted observations, PDF ranking, unitary coordinate changes, independent sample phases, and an independent D=2 concentration-moment reference.
• Broader Watson/Bingham/mixture checks: NumPy 96 passed, 1 skipped; PyTorch 97 passed; JAX 20 passed, 77 skipped. Skipped testcase identities are identical before/after; the new JAX skips preserve the algorithm's existing backend restriction. Existing SciPy/NumPy/PyTorch deprecation warnings remain.
• Repository Ruff, focused Black, and git diff --check passed. Independent source/test review found no issues.
• Wheel and source distribution built. Installed-wheel smokes passed for NumPy and PyTorch in isolated processes outside the checkout, confirming installed-package origins, both fitting branches, direction/density behavior, and concentration.
• CUDA was unavailable on this CPU-only host.
• Existing PR CI passed all 10 backend/Python jobs plus lint, documentation, packaging, and minimal-install checks. All supporting PR workflows passed; no failures were found.

Two-file fix from current main. The separate mixture-moment issue remains a follow-up.

[github-actions]

✅MegaLinter analysis: Success

Descriptor	Linter	Files	Fixed	Errors	Max errors	Warnings	Elapsed time
✅ COPYPASTE	jscpd	yes		no		no	4.73s
✅ JSON	prettier	7	0	0		0	0.57s
✅ JSON	v8r	7		0		0	3.49s
✅ MARKDOWN	markdownlint	69	0	0		0	1.44s
✅ MARKDOWN	markdown-table-formatter	69	0	0		0	0.26s
✅ PYTHON	black	2042	0	0		0	36.95s
✅ PYTHON	isort	2042	3	0		0	1.69s
✅ REPOSITORY	betterleaks	yes		no		no	0.88s
✅ REPOSITORY	checkov	yes		no		no	26.24s
✅ REPOSITORY	git_diff	yes		no		no	0.08s
✅ REPOSITORY	secretlint	yes		no		no	3.88s
✅ REPOSITORY	syft	yes		no		no	2.12s
✅ REPOSITORY	trivy-sbom	yes		no		no	1.49s
✅ YAML	prettier	11	0	0		0	0.48s
✅ YAML	v8r	11		0		0	8.87s
✅ YAML	yamllint	11		0		0	0.27s

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