Sharp Total Apéry Factorization Multiplicities and Tie Proliferation along Genuine Farey Parent–Child Steps

The paper connects our rank-three Farey optimal-word program to factorization theory in three-generated numerical semigroups. For a primitive positive signed source with one positive and two negative letters, it counts every additional minimum-length original-labelled Parikh factorization over the fundamental residual classes and identifies the total exactly with excess factorization multiplicity in an Apéry set. It proves a sharp maximum at fixed semigroup multiplicity and characterizes equality, in the minimally three-generated case, by the coprime-reduced nontrivial generator pair (2,3). More strikingly, for every sufficiently large eligible multiplicity it realizes this extremum on an authentic strictly positive unimodular rank-three Farey parent–child step, although both independently optimized parent words have no ties. A restricted bounded-extension reblocking also recovers every daughter optimum in that constructed family. Its main novelty is the joint extremal counting theorem and its realization in genuine Farey optimization, connecting numerical-semigroup multiplicities to new optimal ties that are invisible in the separate parent solutions. Unrestricted original-parent-word transfer remains open.