Ferrers-Cell Formulas for Chapoton's q-Zeta Numerators: Ballot Words, Interval Structure, and Trapezoidal Spectra

This record contains a mathematical study of Chapoton's q-Zeta numerators for Ferrers-cell lattices and their root-poset specializations. For every nonincreasing boundary sequence, we prove an explicit positive descent enumerator over boundary-admissible ballot words, obtained from a constructive signed EL-labelling, and recover the full multivariate descent-set polynomial by a finite transfer-matrix recursion. For the trapezoidal family, we derive Gaussian-binomial formulas for intervals and t-slices. At q=1, the global profiles admit a Jacobi-polynomial transform with simple negative zeros, strict interlacing along fixed-offset rank sequences, algebraic generating functions, and an explicit arcsine push-forward limit. The balanced boundary recovers the standard type-B and type-C positive-root posets under Chapoton's fixed denominator normalization, yielding an all-rank reversed-ballot formula, coefficientwise q-positivity, sharp slice degrees, and the specialization [t^k]H(1,t)=binom(r-1,k)^2. Type A appears as the rectangular specialization. The results provide a uniform Ferrers-cell mechanism linking q-Zeta numerators with ballot-word enumeration, interval structure, and spectral consequences.