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 SA (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 SA (1, 1) + A (1, 1) = 3/2, in which the irrational contributions ±pi²/8 cancel and the rational value 3/2 emerges without any free parameter. Second, the asymptotic expansion SA (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.