The Sprague–Grundy conjecture for near-square rectairs

The near-square rectair conjecture fails in both parity branches, already at the first two values in its stated range $r\ge7$.

Write $\lambda'$ for the conjugate partition. The exact normal-play recursion is

$$G(\varnothing)=0,\qquad G(\lambda) = \operatorname{mex}\!\left( \{G(\lambda\text{ with one row deleted})\} \cup \{G((\lambda'\text{ with one row deleted})')\} \right).$$

Every recursive call strictly decreases the number of boxes.

For $r=7$ and $k=5$, the source's rectair definition gives

$$R^5_{7,6}=(6,6,5,4,3,2,1).$$

Its distinct followers and exact recursively computed Grundy values are

$$\begin{array}{c|c} \text{follower}&G\\ \hline (6,6,5,4,3,2)&0\\ (6,6,5,4,3,1)&2\\ (6,6,5,4,2,1)&0\\ (6,6,5,3,2,1)&4\\ (6,6,4,3,2,1)&0\\ (6,5,4,3,2,1)&0\\ (5,5,5,4,3,2,1)&2\\ (5,5,4,4,3,2,1)&4\\ (5,5,4,3,3,2,1)&0\\ (5,5,4,3,2,2,1)&4\\ (5,5,4,3,2,1,1)&0\\ (5,5,4,3,2,1)&5 \end{array}$$

Consequently

$$G(R^5_{7,6}) = \operatorname{mex}\{0,2,4,5\} = 1,$$

whereas the conjecture predicts $3$, because $k=r-2$ and $r$ is odd.

Conversely, take $r=8$ and $k=6$. For

$$R^6_{8,7}=(7,7,6,5,4,3,2,1),$$

the fourteen distinct followers have Grundy values

$$0,0,0,0,0,0,0,1,2,2,2,2,2,4.$$

Therefore

$$G(R^6_{8,7}) = \operatorname{mex}\{0,1,2,4\} = 3,$$

whereas the conjecture predicts $1$.

The parity error is also exposed by the source's own subsequent conjecture. Its padded staircase satisfies

$$\mathrm{PS}_n=R^{n-1}_{n+1,n}.$$

Thus Conjecture 3 predicts $G(\mathrm{PS}_n)=3$ for even $n\ge6$ and $1$ for odd $n\ge7$, while Conjecture 5 predicts exactly the opposite on both infinite families. The exact cases $n=6,7$ identify Conjecture 3 as false. The suggested parity repair for all larger $r$ remains a separate open claim.

Source: I. Bašić, E. Gottlieb, and M. Krnc, “CRIM: A Natural Game on Integer Partitions,” arXiv:2606.16828, Conjectures 3 and 5.