The Multiplication Table Near Perfect Squares: Concentration, Typicality, and the Slow Approach to Ford's Exponent

We study the density ρ(N,c) of distinct entries of the N×N multiplication table in the square-gap ((cN)², (cN+1)²], equivalently the density of integers near (cN)² possessing a divisor within a factor 1/c of their square root. Exact computation is carried to N = 10^8, an exact-integer Monte Carlo to N = 2^1024, and a continuum model of factorizations, anchored to the exact tiers at seven seam heights, to N = 2^{10^18}. All predictions were frozen before their data and every failure is logged. Three results emerge. First, square-centered windows are statistically indistinguishable from generic windows of the same magnitude, and the gap-to-gap coefficient of variation fits N^β with β = -0.440 ± 0.031, so the unproven localization of Ford's theorem to √x-scale windows is empirically a concentration statement with a square-root exponent. Second, the local decay of ρ attains Ford's shape -δ - (3/2)t^{-1} in t = log log x only from t ≈ 13–16, and the exponent δ = 0.086071 itself is never attained. At t = 39.5, the model height of quintillion-bit integers, the effective exponent still reads 0.077. The deficit follows a 1/t law with coefficient b = 0.333 ± 0.032 in the mature regime, a constant floor being rejected at Δχ² = 56.8. Third, an exactly computable ballot skeleton, Ford's order-statistics mechanism stripped of all divisor structure, reproduces δ, the t^{-3/2}, and the 1/t coefficient to within 1–2 standard errors. The lossy steps in the reduction from divisors to the ballot event therefore carry nothing at subleading order, and the divisor-specific contribution is localized to a transient dying near t ≈ 25. The subleading law is a measurement in the model, anchored to exact integers at the seams. It is not a theorem, and we state the theorem-shaped objects it calls for.