Rationality of fixed-weight sums of zeta-like multiple values

We study two families of zeta-like multiple series, called the multiple η-values and the multiple ρ-values, defined respectively by shifted powers and rising factorials. Although each individual η-value can be expressed as a rational linear combination of Riemann zeta values together with a rational constant, our main result shows that fixed-weight sums of η-values are always rational. More precisely, we prove that ∑|s|=q+r+1ℓ(s)=r+1η(s)=∑|α|=q+r+1ℓ(α)=q+1ρ(α), where the two summations are taken over admissible indices of equal weight and complementary depths. Since every ρ-value admits an explicit factorial evaluation, this identity immediately yields closed rational expressions for the corresponding fixed-weight sums of η-values. In addition, we derive explicit formulas for the ρ-values, weighted sum formulas, integral representations for η-values, and several combinatorial consequences arising from these identities.