Research proof index

The single-term coordinate chain rule for a holomorphic outer g, anti-holomorphic version, pointwise at u: ∂̄_I (g ∘ f) = deriv g (f u) · ∂̄_I f. As in dWirtingerCoord_comp_holomorphic_apply, the ∂g/∂f̄ channel vanishes and ∂g/∂f collapses to deriv g (f u).

∂̄_I z̄^J = δ_IJ. The conjugate of dWirtingerCoord_coordProj, read off through dWirtingerAntiDir_star_comp rather than recomputed. Used to assemble the coordinate-difference rule dWirtingerAntiCoord_coordDiff (§C).

For an anti-holomorphic function the anti-holomorphic Wirtinger derivative equals the complex Fréchet derivative of g at star u along the slot-I real coordinate direction, ∂̄_I (g ∘ star) = fderiv ℂ g (star u) (Pi.single I 1). Dual of dWirtingerCoord_eq_complex_fderiv_apply.

The single-term coordinate chain rule for a holomorphic outer g, pointwise at u: ∂_I (g ∘ f) = deriv g (f u) · ∂_I f. From the two-term dWirtingerCoord_comp_apply: for holomorphic g the anti-holomorphic coefficient dWirtingerAntiDir g 1 (f u) vanishes and dWirtingerDir g 1 (f u) collapses to deriv g (f u), both off the ℂ-linearity `cline...