Can someone explain some of the steps in this paper clearly? I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$

Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, April 1997, IFTUWr 911/97 (Citeseer^x)

In particular, line 6 where it relates his result on the partial Brun's sums to first occurrence prime gaps. And also the following quadratic. 
moreover it seems to be inconsistent with his previous paper in line (4) as it ignores some terms in line 28 of this one.

Marek Wolf, Generalized Brun's constants, preprint, March 1997, IFTUWr 910/97
  (Citeseer^x)

 A: Marek Wolf's preprint First occurrence of a given gap between consecutive primes 
(Preprint IFTUWr 911//97, 1997) was written in Wrocław University's Institute of Theoretical Physics (IFTUWr). The preprint employs "physical" notation, where $f\sim g$ means
$f$ and $g$ have the same order of magnitude.
(See e.g. Mathworld on the usage of the tilde symbol.)
In number-theoretic asymptotic notation, the formula should be read as
$$
p_f(d) \asymp \sqrt{d} e^{\sqrt{d}} \tag{1}
$$
(for infinitely many first-occurrence gaps of size $d$). 
Wolf's recent preprint arXiv:2002.02115 
clarifies this notation ambiguity (see footnote on page 12). It also generalizes the formula to first occurrences of gaps between primes in an arithmetic progression (P): $r+nq$, $n\in{\mathbb N}$.
Namely, the first occurrence of a gap $d$ between primes $p\equiv r$ (mod $q$) 
is conjectured to have end-of-gap prime
$$
p \asymp \sqrt{d} e^{\sqrt{d/\varphi(q)}} \tag{2}
$$
infinitely often.
(Note that, in the derivation of this formula, omitting a small factor is okay, as long as we are not changing the order of magnitude of the result. Formulas $(1)$ and $(2)$ do not assert asymptotic equivalence of the left- and right-hand sides.)
