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 $G\times {\mathrm{\Theta }}_{\mathcal{T}}$, the spatial twist determines symmetry-defined magnetic geometry, while time-reversal parity independently specifies magnetic character. Each parent phonon irrep $\mathrm{\Gamma }$ maps to ${\mathrm{\Gamma }}_{\mathrm{m}\mathrm{a}\mathrm{g}}=\mathrm{\Gamma }\otimes {\mathrm{\Gamma }}_{\mathrm{p}\mathrm{s}}$, preserving multiplicities and yielding the projection identity $P_{\mathrm{m}\mathrm{a}\mathrm{g},mn}^{(\mathrm{\Gamma }\otimes {\mathrm{\Gamma }}_{\mathrm{p}\mathrm{s}})}=P_{\mathrm{p}\mathrm{h},mn}^{(\mathrm{\Gamma })}$ under the common Cartesian realization. This establishes the $\mathit{\text{Template}}\mathit{\text{Principle}}$: parent vibrational modes furnish real-space templates whose symmetry-enforced nodal manifolds are inherited exactly. Applied to monolayer ${\mathrm{C}\mathrm{d}}_2{\mathrm{N}}_3$, 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 $\mathit{\text{a}}\mathit{\text{priori}}$ parent-group criterion for screening symmetry-allowed linear magnetic responses.