We introduce two families of infinite sums, SA (d, K) and SB (d, K), obtained by applying a floor-function block-alternating reciprocal transform to two sequences of quadratic rational factors. The construction is parallel to the infinite-product families A (K, d) and B (K, d) of the companion paper, and several results run in exact analogy. For K=1 and d=1, 2, 3, 4, we obtain closed-form evaluations of SA (d, 1) in terms of pi², Catalan's constant G, log 2, and algebraic numbers, and of SB (d, 1) in terms of the hyperbolic cotangent coth (pi/2) and related functions. A notable structural feature is the reflection identity SA (d, 1) + SB (d, 1) = (sum over positive blocks only), which holds for every d and in which the negative-block contributions cancel completely. For general integer K >= 2 with d=1, the sum SA (1, K) takes rational values expressible in terms of odd harmonic numbers. We also establish the exact formula SA (inf, K) = (2K-1) *H₂₊/ (2K) as the sum-family analogue of A (K, inf) =2K, with SA (inf, K) ~ log A (K, inf) + gamma as K tends to infinity. Finally, we prove the pi/2 unification principle: lim₊->₈₍₅ SB (inf, K) = -pi/2 = log B (inf, inf), showing that the limit of the sum family equals the logarithm of the corresponding infinite-product value. We also remark that, since sigma (n, d) is periodic with period 2d and induces a decomposition modulo 2d, the constants at each level d are identifiable with values of Dirichlet L-functions of period 2d; this arithmetic connection is noted but not developed here.
Masanori Fujii (Fri,) studied this question.