Generator Calculus for Endpoint-Compression Words Free Structure, Typed Re-encoding, and the Role of Positivity

This paper develops the second step of the Generative Mechanism Atlas program: a calculus for endpoint-compression words. It studies typed words generated by M1(q)=1/(1+q)M_1(q)=1/(1+q)M1(q)=1/(1+q), M2(q)=2/(1+q)M_2(q)=2/(1+q)M2(q)=2/(1+q), and explicit re-encoding maps. The type-preserving core is shown to be a free monoid by a ping-pong argument on a positive invariant interval, while allowing the share-to-odds re-encoding produces formula-equivalent but typed-inequivalent mechanisms. The paper separates induced Möbius formulas, typed mechanism records, admissibility domains, and canonical traces.