Strict interval occupancy across block Lanczos iterations

Let $A\in\mathbb R^{n\times n}$ be symmetric and $V\in\mathbb R^{n\times p}$ of full column rank, and suppose the block Krylov space $\mathcal K_s$ has full dimension $sp$. For all $1\le k<j\le s$ and $1\le i\le(k-1)p$, with Ritz values listed with multiplicity, the open interval $(\theta_i^{(k)},\theta_{i+p}^{(k)})$ between Ritz values of iteration $k$ contains a Ritz value of iteration $j$. Suppose the compression $H_j$ had no eigenvalue there. Then $q(t)=(t-\theta_i^{(k)})(t-\theta_{i+p}^{(k)})$ gives $q(H_j)\succeq0$. Meanwhile the span of $p+1$ Ritz vectors of iteration $k$ meets $\mathcal K_{k-1}$ in a nonzero $x$ on which the $q$-form of $H_k$ is nonpositive. On $\mathcal K_{k-1}$ both compressions agree with $A$, which forces $q(A)x=0$. Expanding $x$ in the block Krylov basis, a monic quadratic cannot annihilate it while $[V,AV,\dots,A^kV]$ has full column rank. The argument also excludes coincident endpoints.