An empirical subleading correction to Wolf's formula for consecutive prime gaps

Wolf's 1998 conjecture gives Ng (N) Cg Li₂ (N) (-g Cg / N) for the count of consecutive prime gaps of size g up to N. (We adopt the variant with Cg in the suppression exponent; the original preprint has (-g/ N), see Sec. ~1. ) On a dataset of n = 140 points covering N 10⁵, 10^13, the residual R (g, N): = (Ng^emp (N) /W (g, N) ) is captured by a two-parameter linear model without intercept of the form R (g, N) a\, ( (g) ) - b\, N, where (g) is the number of distinct prime divisors of g and is any strictly monotonic transform. On the primary-filtered support (g) \1, 2\, so \, , , -1, -1\ give identical fits (see: altforms) ; the relevant arithmetic feature is (g), the functional form is not pinned down by the data. For a numerical quote we take the linear representative, a = 0. 349, b = 0. 202, on the heuristic of: heuristic; the relaxed-filter ranking (at _ = 4) lies outside Wolf's regime of validity (loss L 87 vs. \ 1. 11) and is not evidence about the correction inside it. The originally fitted square-root form gives a = 0. 855, b = 0. 202. The coefficients depend on the filter: tightening to 0. 33, 0. 95 shrinks |a| and |b| by about 20\% (Tab. ~tab: filter). We have no theoretical derivation for either value. A likelihood-ratio test does not require a free intercept (p = 0. 19) ; BIC prefers the no-intercept form. Hold-out-one-N cross-validation gives median R²₂ₕ = 0. 92 (range 0. 69, 0. 94), with a plateau at R² 0. 94 for N 10⁹ and R² = 0. 874 at the largest held-out value N = 10^13. Features that break the two-value bijection (Cg, ₀ (g) ) are rejected by 47. Under a relaxed filter (_ = 4, n = 553, outside Wolf's regime of validity) linear is preferred over by 10, in line with the heuristic of: heuristic. A five-parameter model that jointly reparametrises the Wolf exponent fits better (nominal = -110. 8; rescaled to n₄₅₅ 25, ₄₅₅ -19. 8). Because the histograms are not independent, the nominal n = 140 values overstate evidence; we report the rescaled values alongside. The two-parameter form is the minimally parametrised descriptor, not the best fit.