Order relations for the near geometric and near spectral geometric curves

We investigate order and spectral properties of the weighted near geometric curve $A{\star }_{t}B=A^{1\mathrm{/}2}(B^{1\mathrm{/}2}{A}^{-1}{B}^{1\mathrm{/}2}{)}^t{A}^{1\mathrm{/}2}$ on the cone of positive definite matrices. Our focus is on comparisons governed by the Loewner order, eigenvalue inequalities, and log-majorization. We establish a normalized eigenvalue comparison with the weighted arithmetic mean, give a sufficient condition for the corresponding Loewner inequality, and obtain conditional log-majorization relations. Motivated by the relationship between the weighted geometric and spectral geometric means, we introduce the near spectral geometric curve $A{\widetilde{\mathrm{♮}}}_{t}B=(A^{-1}\star B{)}^{t}A(A^{-1}\star B{)}^t$, where $\star ={\star }_{1\mathrm{/}2}$. Setting $D=A^{-1}{\star }_{1}B$, we prove that $A{\widetilde{\mathrm{♮}}}_{t}B=A{\mathrm{♮}}_{t}D$ and $B{\prec }_{\log }D$, and show that $D=B$ if and only if $A$ and $B$ commute. We further characterize commutativity and near-order monotonicity, prove midpoint isospectrality, and derive affine interpolation and constant-speed identities. Finally, we establish a Lie-Trotter formula and additional log-majorization inequalities. These results connect the two near geometric constructions with the geometric and spectral geometric means and clarify their order and spectral behavior.