arXiv:2610.29576v1 Announce Type: new Abstract: We introduce matrix $\Phi$-ribbons, extending the scalar $\Phi$-ribbons to the matrix case. For $\Phi$ in the subadditivity class, we establish data processing and tensorization properties of matrix $\Phi$-ribbons. We establish scalar and matrix $\Phi$-entropy inequalities and use them to show that both $\Phi$-ribbons characterize the corresponding $\Phi$-strong data processing inequality (SDPI) constants. We obtain exact evaluations of the SDPI constants for several sources. We provide examples showing that the matrix $\Phi$-SDPI constant can exceed the scalar counterpart. Finally, we derive inner bounds for $\Phi$-ribbons from prescribed independence structures and prove that these bounds are optimal when $\Phi(t)=t\log t$.