Saturation Defects and Price Dependence in Synchronized Optimal Farey Parent Words

The paper grows out of our study of optimal-completion words and rank-three Farey parent–child constructions. Its central question is whether the cyclic difference data of two genuinely optimized parent words contains enough integral and numerical information to reconstruct the optimal daughter. The paper gives an exact quotient-based description of the lattice generated by synchronized parent differences and determines its determinant ideal and saturation defect without expanding the full joint period. Genuine positive unimodular infinite families show that the source-difference lattice can have unbounded index, or even rank one, while an essential new daughter optimum exists. A further family exhibits an index-29 defect that one absolute source letter repairs, together with an exact three-way price tie at a parameter wall: the optimal ray changes while the source alphabet and lattice stay fixed. Its main novelty is demonstrating, within authentic optimized Farey data, both arithmetic information loss under cyclic differencing and the impossibility of certain price-blind selection rules. A general transfer algorithm using complete ordered parent words and numerical prices remains open.