On the Combinatorial Core of Second-Order Quantum Argument Shifts in $U{\mathfrak{g}\mathfrak{l}}_d$

We provide a complete, self-contained proof that reduces second-order generators of the quantum argument-shift algebra in the universal enveloping algebra $U{\mathfrak{g}\mathfrak{l}}_d$. We prove the necessary combinatorial identities – expressed as relations among polynomials with rational coefficients – by induction.