Large-scale computation and out-of-sample validation of an empirical correction to Wolf's prime-gap formula

We report a computational study of the consecutive prime-gap counting function N_g(N), the number of pairs of consecutive primes (p_n, pₙ₊₁) with pₙ₊₁ - p_n = g and pₙ₊₁ <= N, conducted up to N = 3 * 10^14. The starting object is Wolf's 1998 heuristic W(g, N) := C_g * Li_2(N) * exp(-g*C_g/ln N), where C_g is the Hardy-Littlewood singular series. Using a single-threaded C generator built on libprimesieve we produce 28 bit-reproducible histograms covering N in [10^5, 3 * 10^14] and study the residual R(g, N) := log(N_g^emp(N) / W(g, N)) on a precisely defined filter. On a dataset of n = 169 filtered points the residual is captured, with no free intercept, by a two-parameter linear model R ~ a*sqrt(omega(g)) - b*log log N, where omega(g) is the number of distinct prime divisors of g. The model selected on n = 140 points with N <= 10^13 predicts the n_test = 29 disjoint points obtained at N in {3*10^13, 10^14, 3*10^14} with weighted out-of-sample R^2 = 0.986, mean residual -0.005, and refit coefficients drift by 2-4%, inside the pre-extension block-bootstrap intervals. Features that break the bijection g -> omega(g) on the filter (log C_g, sigma_0(g)) are rejected by Delta-AIC >= 52. We contribute (i) a filter on the (g, N) plane that admits Wolf's regime of validity; (ii) a reproducibility pipeline allowing bit-identical regeneration of all 28 histograms; (iii) an out-of-sample validation design for empirical formulas in arithmetic, where no ground-truth generative model exists and the prime sequence is a single realisation. All code, data, and notebooks are released under MIT.