Block-Alternating Reciprocal Sums, Special Values, and a Companion to Infinite-Product Theory

We introduce two families of infinite sums, S_A(d,K) and S_B(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 S_A(d,1) in terms of pi^2, Catalan's constant G, log 2, and algebraic numbers, and of S_B(d,1) in terms of the hyperbolic cotangent coth(pi/2) and related functions. A notable structural feature is the reflection identity S_A(d,1) + S_B(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 S_A(1,K) takes rational values expressible in terms of odd harmonic numbers. We also establish the exact formula S_A(inf,K) = (2K-1)*H2K/(2K) as the sum-family analogue of A(K,inf)=2K, with S_A(inf,K) ~ log A(K,inf) + gamma as K tends to infinity. Finally, we prove the pi/2 unification principle: limK->inf S_B(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.