Erdős 68: factorial multiples of $\sum_{n\ge 2} 1/(n!-1)$ stay away from integers

Let

be the Erdős 68 constant. The claim is that for every integer $M \ge 3$ and every integer $m$,

that is, the factorial multiple $M!\,x$ never comes within $1/(M(M+1))$ of an integer.

Relation to Erdős Problem 68. This separation property strictly strengthens the irrationality of $x$ asked by Erdős Problem 68 (https://www.erdosproblems.com/68, open): if $x=p/q$ were rational, then for every $M\ge q$ the number $M!\,x$ would be an integer, contradicting the displayed bound at $m=M!\,x$. Conversely, mere irrationality does not imply the quantitative bound, so this lemma is a genuinely stronger statement.

Role in the decomposition. This is the hard core of the frontier leaf Erdos68.factorial_moment_partial_sum_bound (window avoidance for the fractional part of $A_M=\sum_{n=2}^{M}M!/(n!-1)$). Writing $A_M+t_M=M!\,x$ with $t_M=M!\sum_{n>M}\frac1{n!-1}$, the companion lemma Erdos68.factorial_tail_sharp_window places $t_M$ in $\bigl(\frac1{M+1},\frac1M\bigr)$; if $\{A_M\}$ fell in the window $\bigl(1-\frac1M,\,1-\frac1{M+1}\bigr)$, then $\delta:=\lfloor A_M\rfloor+1-A_M$ would also lie in $\bigl(\frac1{M+1},\frac1M\bigr)$, and hence

contradicting the stated separation at $m=\lfloor A_M\rfloor+1$.

Evidence. Exact rational arithmetic for $A_M$ combined with 60-digit evaluation of the tail verifies the displayed bound for every $3\le M\le 299$ (and the window avoidance of the leaf over the same range); no counterexample is known, and no proof for general $M$ is known.