Differential structures of games

Question

Is there one categorical structure explaining both the Leibniz-type rule for moves under Conway addition and the Rota–Baxter-type rule for recursive game values? This is Question 5.1 of Games as recursive coalgebras.

The target is an explanation of the Nim-sum calculation, not just a new name for an identity already obtained using the Sprague–Grundy theorem.

Established equations

The move map satisfies

When a Bouton monoid is induced by a finite-powerset algebra $\beta$, Proposition 5.2 proves

These are the starting data. General differential-category terminology alone does not establish a deriving transformation or show that it recovers these formulas.

Next step

State the candidate structure and derive both displayed equations from its axioms. Then give the Nim example and identify precisely which facts about Nim-sum are inputs and which are consequences. If the desired identification is assumed through Sprague–Grundy theory, the argument has not yet supplied an independent explanation.

This question is distinct from the internal hom for Conway addition, which concerns an adjunction rather than a differential structure.

Comparison source: Differential Categories Revisited.