I am looking for an algorithm $A$ - which for any non-null input string $s_1$ produces a sequence $s_1, s_2...$ such that :
It can be proved in some axiomtic system $S$ that: $\forall j> i; K(s_j) \ngtr K(s_i)$; K(s) being the Kolmogorov complexity of string $s$.
ie complexity of terms in the sequence does not increase.
For any given $ s_1 , \not\exists $ Algorithm B such that $ |A| >|B| $ and given $m,n$: B can decide "$|s_m|>|s_n|$"
ie There does not exist any algorithm B with program length less than A which can decide, given any two indices, which one the terms at those indices is smaller.