Higher 2-adic Divisibility for Fixed-Hole Domino Tilings

Let Q_k be the free-boundary (2k+1) by (2k+1) square board, and let a_k(h) be the number of domino tilings after a fixed checkerboard-admissible cell h is removed. For every integer k at least 1 and every admissible noncentral hole on the central row or central column, the paper proves that 2^(k+1) divides a_k(h). More precisely, for h_(k,s)=(k+1,k+1+2s), with 1 <= s <= floor(k/2), either the count is zero or v_2(a_k(h_(k,s))) >= k+2s. The proof isolates a low-rank even-defect mechanism for integral matrices and applies it to a Kasteleyn-Percus matrix whose mod-2 channels and sign-defect ray control the Smith factors. This proves the fixed-hole hypothesis in Byun-Goddard Theorem 2.4 and yields its modulo-8 consequence for the total one-hole count; it does not prove Kong's stronger modulo-24 congruence, squarefreeness assertion, or the corresponding statement for arbitrary off-axis holes.