Existence and Construction of Reel Strips from Mathematical Specifications

A constructive theory for designing slot machine reel strips from target specifications (RTP, hit rate, volatility). Introduces the co-location matrix as the central design object, proves three-coordinate sequential decoupling via the rearrangement invariant, and establishes reachability of the achievable set by exact splice linearity. Constructibility is decidable from the co-location matrix before any strip is built. A worked example constructs a complete 5×3 ways-pay base game with compound multiplier wilds, with verified reel strips provided.