Weak Vishik Equivalence and Linear $2$-Basis Blindness for Totally Singular Quadratic Forms

Let K be a field of characteristic two, put k=K², and let U⊂ K be the value space of an anisotropic totally singular quadratic form. For a∈ K k, the anisotropic dimension after the one-step extension K(√ a)/K is U(a)=ₖ₍ₐ₎k(a)U=ₖU-12ₖ(U∩ aU). Thus weak Vishik equivalence becomes an equality of multiplicative-overlap profiles. We derive direct value-space reconstructions of the norm field and the similarity-factor field, and we show that orthogonal complements for a nondegenerate associative pairing have identical one-step profiles. Over K=₂(s,t,u,v), we construct explicit nonsimilar eight-dimensional value spaces $U,V$ whose one-step profiles agree for every a∈ K. In addition, their complete coordinate profiles agree for every $2$-basis obtained from $(s,t,u,v)$ by a matrix in ₄(₂). Nevertheless, the degree-four intermediate field $k(t,su)$ separates them, with profile values $4$ and $3$; it is coordinate for the nonlinear $2$-basis $(t,su,s,v)$ but for no linear change of the standard basis. Thus the unrestricted implication ``weak Vishik equivalence implies similarity'' fails for anisotropic totally singular quadratic forms in characteristic two. This does not answer Zemkov\'a's Question~Q, which concerns full Vishik equivalence over all field extensions. To the best of our knowledge, our construction gives the first explicit nonsimilar weakly Vishik-equivalent pair in the totally singular characteristic-two setting, as well as the first example showing that all linearly changed coordinate profiles can miss a nonlinear intermediate-field distinction. The only computer-assisted step is an exact symbolic verification over ₂(s²,t²,u²,v²) for the $35$ linear degree-four coordinate fields. The results also translate into minimum-rank profiles of symmetric matrices with prescribed diagonal.