Graded All-Orders Asymptotics for Successive-Primorial Crossovers in the Odlyzko–Rubinstein–Wolf Exponential-Integral Approximation

Let \(P=p_k^\#\) and \(q=p_{k+1}\), so that \(qP=p_{k+1}^\#\) is the next primorial. We study the pairwise crossover between these two candidates in the explicit Odlyzko–Rubinstein–Wolf (ORW) exponential-integral approximation $$ \mathcal M(X,D)=\mathfrak S(D)I_D(X),\qquad I_D(X)=\int_2^X \frac{\exp(-D/\log t)}{(\log t)^2}\,dt. $$ Writing \(L=\log X\), we first prove a uniform all-orders endpoint expansion for \(I_D(e^L)\), with generalized Laguerre coefficients and a remainder that remains controlled under differentiation in the growing endpoint parameter. We then reorganize the logarithmic expansion into a graded asymptotic scale and rigorously invert the full ORW crossover equation. If $$ L_0=\frac{(q-1)P}{\log((q-1)/(q-2))} $$ denotes the exact solution of ORW’s leading exponential comparison, then for every fixed \(M\ge0\), $$ L_{\rm int} = L_0+ \sum_{m=0}^{M}\frac{c_m(P,qP)}{L_0^m} + O_M\!\left( \frac{(1+qP)^{\lfloor(M+1)/2\rfloor}} {L_0^{M+1}} \right), $$ with an explicit Lagrange–Bürmann coefficient generator and \(\deg c_m\le\lfloor m/2\rfloor\). In particular, $$ L_{\rm int} = L_0+1+\frac{3}{L_0} + O\!\left(\frac{1+qP}{L_0^2}\right). $$ The constants \(1\) and \(3\) are universal for general endpoint pairs satisfying \(D_1/L_0\to0\); primorial arithmetic enters only at the next polynomial coefficient. This absolute endpoint displacement is asymptotically separated from the classical algebraic expansion of \(L_0/P\), whose first terms are $$ q^2-\frac52q+\frac{17}{12}+\cdots. $$ The unconditional results concern the explicitly defined ORW approximation. For actual consecutive-prime gaps we give a separate conditional transfer theorem and, through a divisor-proxy and signed Vaughan analysis, exhibit a representative short affine prime-correlation obstruction that is not covered by the currently cited results.