2633 — Complex Shears Preserve a Common Hermitian Form Only on a Real Line
A coefficient family of nontrivial shears
preserves one fixed nondegenerate Hermitian form exactly when all nonzero coefficients lie on one real line in . Every such form is indefinite: it has determinant . The two coefficients are the smallest hostile family and force every common invariant Hermitian form to be degenerate.
Every complex shear nevertheless preserves the complex bilinear alternating form under transpose. Determinant-one complex symplectic preservation is not Hermitian passivity.
A nonconstant holomorphic coefficient family on a connected open domain cannot lie on one real line. It therefore cannot preserve one fixed nondegenerate rank-two Hermitian form. Any theta/Tate conservative structure must be derived from boundary currents, be parameter-dependent, or enlarge the realization.
Scope
This is an exact rank-two algebraic theorem. It does not derive a theta metric, boundary current, larger conservative realization, passivity, KYP positivity, or RH.
Durable verification
- Packet:
research/kitaev/complex-shear-families-preserve-a-common-hermitian-form-only-on-a-real-line.md - Checker:
research/kitaev/checkers/check_complex_shear_common_hermitian_form.py - Results:
research/kitaev/results/complex-shear-common-hermitian-form.json - Sequence authority:
seqclaim-c8f2b7b4ea2788adc6d11878 - Epistemic graph event:
ev-000000003953-8532d7f0-49bf-427f-8cc9-425a39d81744
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