The Word-Search Constants C₂(ABBB) = 8/5 and C₃(ABB) = 6: On Two Questions of Schildkraut and of Halberstam and Schildkraut

For a word \(w\) and a toroidal grid of letters in dimension \(d\), an appearance of \(w\) is a cell together with one of the \(3^d-1\) directions in \(\{-1,0,1\}^d\)\({}\setminus\{0\}\) along which the letters spell \(w\), and \(C_d(w)\) is the supremum, over all grids, of the number of appearances per cell. We prove that \(C_2(\mathsf{ABBB})\)\({}=8\)\(/5\). This answers Question 9.3 of Halberstam and Schildkraut (Words with repeated letters in a grid, 2025) and the last part of Question 11 of Schildkraut in the problem session of the Oberwolfach workshop Combinatorics (Oberwolfach Rep. 23 (2026)). We also prove that \(C_3(\mathsf{ABB})=6\), which is the case \(d=3\) of the first part of that Question 11 and of their Question 9.2. Both values also hold for \(n\times\dots\times n\) arrays without wraparound, as in the report. The lower bounds are known constructions. The upper bounds are computer-assisted: averaging lemmas, proved in the text, reduce them to \(65{,}536\) and \(2^{27}\) integer inequalities, one for each pattern of a \(4\times4\) or a \(3\times3\times3\) window; these are given by exact certificates and were checked in integer arithmetic by independent programs. In both cases we classify the extremal grids. For \(\mathsf{ABBB}\) they are the two five-queens lattices, on tori with both sides divisible by \(5\); for \(\mathsf{ABB}\) in dimension \(3\) they are the grids in which the letters \(\mathsf{A}\) form a family of parallel planes modulo \(3\). Every window that is not of this kind costs at least \(3/70\), respectively \(47137/294912\). So for these two pairs \((d,w)\), and only for these, we show that the supremum is attained by a periodic grid; Question 10 of the report (Question 2.5 of Halberstam and Schildkraut) asks this for all pairs. For \(\mathsf{ABBB}\) we also determine the possible pairs (density of \(\mathsf{A}\), concentration): their closure is the quadrilateral with vertices \((0,0)\), \((\frac15,\frac85)\), \((\frac38,\frac32)\), \((1,0)\). It follows that Question 9.6 of Halberstam and Schildkraut, a stability statement with constant \(1\) for the letter distribution of near-extremal grids, has a negative answer for the pair \((\mathsf{ABBB},2)\) as stated; it holds with the optimal constant \(\frac{14}5\) for grids on two letters. Finally \(18\)\({}\le C_4(\mathsf{ABB})\)\({}\le\frac{62}3\), \(54\)\({}\le C_5(\mathsf{ABB})\)\({}\le64\) and \(\frac{24}5\)\({}\le C_3(\mathsf{ABBB})\)\({}\le\frac{11}2\). Question 10 in general and the values of \(C_d(\mathsf{ABB})\) for \(d\ge4\) remain open. This is an unrefereed note.