Value-Space Contractions for Fixed-Diagonal Rank Fibers\\ of Symmetric Matrices in Characteristic Two

Let K be a field of characteristic two and let k=K² be its subfield of squares. For a prescribed diagonal vector d=(d₁,…,dₙ)∈ Kⁿ, we study the symmetric matrices of diagonal d, graded by rank. The associated quasilinear quadratic form qd(x)=∑ᵢ₌₁ⁿ dᵢ xᵢ² has the classical represented-value space U(d)=D(qd)=ₖ₁,…,dₙ\⊆ K. Writing ω(d)=ₖ U(d), we construct an explicit, choice-dependent Schur-complement bijection that removes ω(d) coordinates, shifts rank by ω(d), and leaves a smaller symmetric matrix whose diagonal entries lie in $U(d)$. From this contraction we determine every possible rank in a fixed-diagonal fiber. We also distinguish two exact classification levels: congruence-induced equivalence is classified by the embedded subspace U(d)⊆ K, whereas abstract rank-preserving bijection type is classified by the integer ω(d). A separate zero-diagonal bijection relates alternating matrices of size n to symmetric matrices of size $n-1$ with square-valued diagonal and yields a rank rule valid over every field of characteristic two. Over perfect fields the statements specialize to one-coordinate contractions and recover known finite-field rank recurrences. Represented-value spaces and the finite-field enumerations are classical. To the best of our knowledge, the explicit arbitrary-field contractions, the complete fixed-diagonal rank spectrum derived from them, and the resulting two-level classification are new.