A Pseudoscalar Representation Mapping from Parent-Group Vibrational Normal Modes to Symmetry-Adapted Magnetic Structures

Conventional approaches classify symmetry-allowed magnetic configurations but do not by themselves establish a direct, mode-resolved correspondence with parent-lattice vibrations. Here, we formulate a universal determinant-induced pseudoscalar twist for all 32 crystallographic point groups, establishing an exact representation-to-geometry correspondence between parent vibrations and magnetic order. Within the paramagnetic gray group $G\timesΘ_{\mathcal{T}}$, the spatial twist determines symmetry-defined magnetic geometry, while time-reversal parity independently specifies magnetic character. Each parent phonon irrep $Γ$ maps to $Γ_{\mathrm{mag}}=Γ\otimesΓ_{\mathrm{ps}}$, preserving multiplicities and yielding the projection identity $P_{\mathrm{mag},mn}^{(Γ\otimesΓ_{\mathrm{ps}})}=P_{\mathrm{ph},mn}^{(Γ)}$ under the common Cartesian realization. This establishes the \textit{Template Principle}: parent vibrational modes furnish real-space templates whose symmetry-enforced nodal manifolds are inherited exactly. Applied to monolayer $\mathrm{Cd}_2\mathrm{N}_3$, the framework identifies the ferrimagnetic ground state from the parent $A_{2u}$ sector, confirmed by first-principles calculations, alongside cluster magnetic octupoles and antiferromagnetic manifolds. It further provides an \textit{a priori} parent-group criterion for screening symmetry-allowed linear magnetic responses.