For multiple cyclic actions, like your analysis case (thermal + wave induced movements), the single relevant result, with obvious significance, is the so-called "Usage Ratio", meaning the Accumulated Damage ratio, which is:

[ (Applied Cycles No)/(Allowable No of Cycles) ]_Load#1 +
[ (Applied Cycles No)/(Allowable No of Cycles) ]_Load#2 +
[ (Applied Cycles No)/(Allowable No of Cycles) ]_Load#3 =

(N_applied/N_Allow)1 + (N_applied/N_Allow)2 + (N_applied/N_Allow)3 < 1.00,

and which needs not to exceed 1.00.

Personally, I am always interested in that resultant/overall "Usage Ratio", which is the relevant fatigue analysis result.

I am not aware about that "Minimum Allowable Cycles" number. Have a look in caesar II documentation, but it's unlikely you'll find the clarification there...
Maybe somebody from COADE/Intergraph/Hexagon... might clarify this issue, which is anyway, in my opinion, less relevant for a fatigue analysis result interpretation.