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

Wolf's 1998 conjecture gives Ng(N) ~ Cg · Li₂(N) · exp(-g Cg / ln 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 exp(-g/ln N), see Sec.~1.) On a dataset of $n = 140$ points covering N ∈ [10⁵, 10¹³], the residual R(g, N) := log(Ngᵉᵐᵖ(N)/W(g, N)) is captured by a two-parameter linear model without intercept of the form R(g, N) ≈ a\,φ(ω(g)) - b\,loglog N, where ω(g) is the number of distinct prime divisors of g and φ is any strictly monotonic transform. On the primary-filtered support ω(g) ∈ \1, 2\, so φ ∈ \√ω, ω, logω, ω-1, √ω-1\ give identical fits (see {sec:altforms}); the relevant arithmetic feature is ω(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 {sec:heuristic}; the relaxed-filter ranking (ω √ω at ρₘₐₓ = 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 ρ ∈ [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 R²CV = 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 = 10¹³. Features that break the two-value bijection (log Cg, σ₀(g)) are rejected by ≥ 47. Under a relaxed filter (ρₘₐₓ = 4, $n = 553$, outside Wolf's regime of validity) linear ω is preferred over √ω by ≈ 10, in line with the heuristic of {sec:heuristic}. A five-parameter model that jointly reparametrises the Wolf exponent fits better (nominal = -110.8; rescaled to neff ≈ 25, eff ≈ -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.