The classification conjecture for P-positions in shrinking circular Nim with six piles

Counterexample: a fourth exceptional losing position, with an exact exhaustive outcome certificate.

In Oginuma and Shinoda, Shrinking Circular Nim, Journal of Information Processing 33 (2025), 1064–1070, doi:10.2197/ipsjjip.33.1064, Conjecture 1 asserts that every losing six-heap position of $\mathrm{SCN}(6,4)$ belongs, up to rotation and reflection, either to the family

$$(a,b+q,c,a+q,b,c+q),\qquad q\ge0,$$

or to one of exactly three exceptional orbits:

$$(5,9,10,7,8,12),\qquad (5,10,11,7,9,13),\qquad (5,11,11,8,9,14).$$

There is a fourth exceptional losing position:

$$\boxed{x=(9,19,19,14,15,24).}$$

Its three opposite-heap absolute differences are

$$|9-14|=5,\qquad |19-15|=4,\qquad |19-24|=5.$$

Every member of the asserted infinite family has all three opposite-heap differences equal to $q$, in every cyclic orientation. Thus $x$ does not belong to that family, and its minimum heap $9$ distinguishes it from all three stated exceptional orbits.

Here is a complete finite exact certificate that $x$ is losing. The already established five-heap losing positions are precisely

$$(t,t,t,t,t),\qquad t\ge1.$$

Every nonempty position with at most four heaps is winning, since its entire circle can be deleted in one move.

A six-heap position $v$ has a legal move to a five-heap losing position if and only if there is some cyclic index $i$ such that

$$v_i=v_{i+1}=t,\qquad \#\{j:v_j<t\}\le1. \tag{1}$$

Indeed, the two unselected adjacent heaps must remain equal to $t$; exactly one of the other four heaps is deleted, and the three surviving selected heaps must be reducible to $t$. Conversely, delete the unique heap below $t$, if one exists, or any selected heap otherwise, then reduce the other selected heaps to $t$. In particular, the adjacent pair need not be a pair of global minima: allowing deletion of a unique smaller heap is essential.

A move between positive six-heap positions $v$ and $w$ exists exactly when

$$w_j\le v_j\quad(0\le j<6),\qquad w\ne v,\qquad (w_i,w_{i+1})=(v_i,v_{i+1}) \text{ for some cyclic }i. \tag{2}$$

The equal adjacent pair is precisely the untouched complement of the four consecutive selected heaps.

For any six-tuple $v$, let $\kappa(v)$ be the lexicographically least among its twelve rotations and reflections. Enumerate canonical positive six-tuples within a fixed heap bound in lexicographic order. Whenever $v=\kappa(v)$ and $w$ is a six-heap follower, (2) gives

$$\kappa(w)\le w<v=\kappa(v).$$

Therefore all possible followers of $v$ have already received their exact normal-play outcomes.

Maintain the earlier losing six-heap positions, including all their distinct dihedral orientations, indexed by adjacent pairs. For each canonical position $v$:

1. Declare $v$ winning if (1) gives a five-equal losing follower.
2. Otherwise declare $v$ winning if an earlier losing orientation satisfies (2).
3. Otherwise declare $v$ losing and insert all its orientations into the adjacent-pair index.

These cases exhaust every possible losing follower, including all legal shrinking moves. Lexicographic induction therefore proves that this procedure computes exact outcomes.

With all heaps bounded by $23$, exhaustive enumeration processes exactly

$$12{,}410{,}432$$

canonical positions and finds exactly the three previously published exceptional losing orbits. Raising the heap bound to $24$ produces the additional losing position $x$ after exactly

$$14{,}998{,}729$$

canonical positions; it is the $820$th losing position encountered. Consequently, maximum heap size $24$ is the first possible size of a previously unlisted exceptional orbit.

As an independent check requiring no rotational or reflection identification, direct componentwise induction on the full rectangular box below $x$ evaluates exactly

$$9\cdot19\cdot19\cdot14\cdot15\cdot24 =16{,}374{,}960$$

oriented positions. It finds $1{,}504$ losing positions, identifies $2{,}387{,}712$ positions with direct five-equal losing followers, and again classifies $x$ as losing.

Thus the three-exception classification in the published conjecture is false.