Direct galaxy--galaxy morphology chirality: A Wigner function shapelet framework for three-dimensional parity tests

Galaxy images encode internal morphology through phase relations that are not retained by low-order shape summaries. We formulate a three-dimensional parity test that directly couples the Wigner function shapelet (WFS) modes of two galaxies before compressing either image into a local pseudoscalar. Because images at different sightlines are defined on distinct tangent planes, we introduce canonical left-handed pair frames and compare the endpoint modes by parallel transport along the shortest great circle. In the Hermitian quadratic pair-covariance sector, independently tracking spatial parity, pair exchange, and rotation charge yields two classes of unordered-pair pseudoscalars: Type I observables require no line-of-sight factor, whereas Type II observables use a parity-even, pair-exchange-odd line-of-sight factor. We further decompose the directly measured parity-odd covariance as $P_{\mathrm T}=P_{\mathrm G}+P_{\mathrm K}$, where $P_{\mathrm G}$ is the Gaussian exchange contribution and $P_{\mathrm K}$ is the connected four-point image-mode contribution. Controlled numerical tests reproduce the geometric transformation laws to machine precision and recover the Type I/II response matrix with maximum cross-talk $5.735\times10^{-4}$, while separating Gaussian-only, connected-only, and mixed hierarchies. Midpoint and angular-bisector conventions converge as $\mathcal O(\vartheta^2)$ in the plane-parallel limit but define distinct wide-angle Type II templates with normalized overlap $0.7049$; their signal transfer agrees with the analytic prediction. In $9000$ realizations with $N_{\mathrm{gal}}=96$, $192$, and $384$, pair-independent errors underestimate the covariance, giving 95~{\%} coverage of $0.7913$ at $N_{\mathrm{gal}}=384$, whereas a 24-region spatial jackknife recovers $0.9442$. Once a line-of-sight convention is fixed, the framework is independent of the external coordinate system and provides a measurement-level basis for connecting mechanism-dependent physical responses to survey-specific observational transfer.