Gravity is Just Counting: The −3/2 log A Entropy Correction from Pure Arithmetic

The logarithmic correction ΔS = −(3/2) log A to black hole entropy, derived in loop quantum gravity only after fixing the Barbero-Immirzi parameter, arises here from pure arithmetic. We introduce a family of block-alternating infinite sums S_A(d,K) as additive companions to the infinite products A(K,d) and B(K,d). The main results are three. First, the exact identity S_A(1,1) + A(1,1) = 3/2, in which the irrational contributions ±pi^2/8 cancel and the rational value 3/2 emerges without any free parameter. Second, the asymptotic expansion S_A(inf,K) = log A + gamma − (log A)/A + (1−2*gamma)/(2A) + O(A⁻²), where A = 2K, which reproduces the full hierarchy of quantum entropy corrections (leading logarithm, universal constant, and power-suppressed terms). Third, Catalan's constant G and log 2 appear explicitly at level d=2, the same constants that govern spin foam volumes and holographic bit counting in loop quantum gravity. All results follow from the floor function sigma(n,d) = (−1)floor((n−1)/d) and elementary analysis. No geometric postulates, no kinematic assumptions, no free parameters.