Exact Cases and Bounds

This manuscript studies the minimum number $\mu_{\mathbb R}(d,r)$ of real linear measurements that are injective on all real $d\times d$ matrices of rank at most $r$. The main exact result established here is $\mu_{\mathbb R}(4,1)=11$: a Grassmannian characteristic-class obstruction excludes ten measurements, and exact rational computations confirm that Xu's eleven measurements are injective, both for the matrices printed in his paper and for the slightly different accompanying Maple data. The manuscript also gives a direct construction of universal measurements, general Pontryagin–Euler and Stiefel–Whitney lower bounds and an odd-degree exactness criterion. For $r=d/2-1$ it combines Causin's K-theoretic bound with Hurwitz–Radon and anti-diagonal constructions to obtain exact values when $\nu_2(d)=1$ or $\nu_2(d)\equiv3\pmod4$. Concrete gaps, including $138\le\mu_{\mathbb R}(12,5)\le139$ and $246\le\mu_{\mathbb R}(16,7)\le247$ for the methods proved here, remain open.