Exact certificates for three open cases of the Heo-Kiem range-criterion problem: (m,n,k,l) = (5,5,3,5), (6,6,3,7) and (7,7,5,7)

For integers m,n,k,l with k+l=m+n-2 and an integral zero Pᵏ⁺ˡₘ₋₁(k)=0 of a binary Krawtchouk polynomial, Heo and Kiem (Linear Algebra Appl. 567 (2019) 167-179, Question 1.2) asked whether there are subspaces D, E of C^m (x) C^n of codimensions k, l such that no nonzero x, y satisfy x(x)y in D and conj(x)(x)y in E. We give explicit pairs (D, E) for the open cases (5,5,3,5), (6,6,3,7) and (7,7,5,7) (hence also (5,5,5,3), (6,6,7,3) and (7,7,7,5)), each with a matrix-inequality certificate (as in arXiv:2611.08518, Sec. 5) verified in exact rational arithmetic. The release contains the note (PDF + LaTeX), the data, and a verification script (python scripts/verify_all.py; numpy + sympy; about 3-15 minutes). Work done with AI assistance (Anthropic Claude); reviewed by separate AI referees (reports included), not yet by a human expert.