The additive three-action friendly-versus-antagonistic discrepancy criterion

Let $S=\{s_3,s_2,s_1\}$ with $s_3<s_2<s_1$, and suppose the largest subtraction is additive:

$$s_1=s_3+s_2.$$

Let a friendly/antagonistic discrepancy mean a difference between the FvF and AvA self-interest games. Additive discrepancy conjecture. There exists a heap with a friendly/antagonistic discrepancy if and only if

$$\frac{s_2}{s_3}\neq k$$

for every $k\in\mathbb{N}$. When this holds, the first such heap is

$$x=(i+2)s_2+is_1=is_3+(2i+2)s_2,$$

where $i\in\mathbb{N}$ satisfies

$$i+1>\frac{s_2}{s_3}>i.$$

The statement is based on the paper's computational investigation and is presented without a proof.