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

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

define

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

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

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

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 $y=-b-x$,

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.