Cartesian-Carrier Symmetric Monoidal Structures on Games

## Abstract We classify symmetric monoidal structures on the category of finitely branching, finite-time impartial games whose tensor carrier is cartesian product and whose coherence maps lie over the standard cartesian coherence bijections. The datum is an admissible rooted transition law on pairs of finite rooted games, satisfying finite naturality and coherence equations. This presents the cartesian-carrier moduli as an effectively closed profinite space with continuum many concrete points. We construct uniform internal homs using the cofree game, identify Conway and selective sum as the two coordinate-local points, and give associativity criteria for height-drop and path-length families. The genuinely lax unital problem is explicitly left open. This AI-generated manuscript is a private review draft reconstructed after the earlier chat-generated files became unavailable.