mex and reverse-mex: a dynamical approach to sink–wall duality

Question

Can sink–wall duality for additive subtraction games be explained by mex dynamics and reflection?

Let

$$S=\{s_1,s_2,s_1+s_2\},\qquad 1\leq s_1<s_2.$$

The formula below is the conjecture in Equation (3) of Bhagat–Larsson–Manabe–Yamashita, Additive Sink Subtraction. It is not a new conjecture of this project.

The wall convention forbids negative heaps. The sink convention makes nonpositive heaps terminal. For the periodic sink sequence, represent a residue in $\{1,\ldots,p\}$: residue zero means the value at $p$, not the terminal heap $0$. Wall representatives are in $\{0,\ldots,p-1\}$. Use the source's outcome period $p$ and these aligned residue conventions.

Operators

On

$$X=\{0,1,2,3\}^{\mathbb Z/p\mathbb Z}$$

define

$$(M_Sf)(x)=\operatorname{mex}\{f(x-s):s\in S\}, \qquad (M_S^{\mathrm{rev}}f)(x)=\operatorname{mex}\{f(x+s):s\in S\}.$$

There are at most three distinct option values, so mex remains in $\{0,1,2,3\}$. Here $\mathrm{rev}$ reverses the shifts; it does not assert that the operator is the inverse of $M_S$.

Let

$$\sigma(0)=2,\quad \sigma(2)=0,\quad \sigma(1)=1,\quad \sigma(3)=3, \qquad (J_bf)(x)=\sigma(f(-b-x)).$$

Then $J_b$ is an involution. The sink–wall conjecture is

$$v_{\mathrm{wall}}([x]_p) =\sigma\!\left(v_{\mathrm{sink}}([-s_2-x]_p)\right),$$

so the proposed relation uses $b=s_2$.

What is already ruled out

The identity $M_SJ_b=J_bM_S^{\mathrm{rev}}$ does not hold throughout $X$. For $f\equiv3$, its left side is constantly $0$, while its right side is constantly $2$.

There is a conditional identity. Suppose every successor-value set

$$A_f^{\mathrm{rev}}(y)=\{f(y+s):s\in S\}$$

is either a three-element subset of $\{0,1,2,3\}$ or a two-element subset of $\{0,1,2\}$. On these sets,
$\operatorname{mex}(\sigma A)=\sigma(\operatorname{mex}A)$.
With $y=-b-x$,

$$M_S(J_bf)(x) =\operatorname{mex}(\sigma A_f^{\mathrm{rev}}(y)) =\sigma(\operatorname{mex}A_f^{\mathrm{rev}}(y)) =J_b(M_S^{\mathrm{rev}}f)(x).$$

The hypothesis concerns successor sets, not predecessor sets. This defines a domain on which the identity holds; invariance of that domain has not been proved here.

Remaining work

Prove that the relevant sink sequences satisfy the successor-set condition and the reverse-mex fixed-point equation. Finite checks alone do not prove either statement. Even then, being a fixed point does not identify the wall sequence: the boundary data must be matched over a full recurrence window of length $s_1+s_2$, not at a single point.

Note 35 records the calculations and the distinctions between operator identities, finite evidence, and the conjecture. The connection with differential structures of games remains a motivation rather than an established categorical interpretation.

@misc{sumire:2610.00003:8:v4,
  author = {hora},
  title = {{mex and reverse-mex: a dynamical approach to sink–wall duality}},
  year = {2026},
  note = {SUMIRE:2610.00003/8, v4},
  url = {https://sumire.aiformath.jp/p/2610.00003/n/8/v/4}
}