If highSyzygy(M) chooses the p-th syzygy, then highSyzygy(M,Optimism=>r) chooses the (p-r)-th syzygy. (Positive Optimism chooses a lower "high" syzygy, negative Optimism a higher "high" syzygy.
Functions with optional argument named Optimism :
highSyzygy(..., Optimism => ...), see highSyzygy -- Returns a syzygy module one beyond the regularity of Ext(M,k)
twoMonomials(..., Optimism => ...), see twoMonomials -- tally the sequences of BRanks for certain examples
Caveat
Are there cases when positive Optimism is justified?
See also
mfBound -- determines how high a syzygy to take for "matrixFactorization"
highSyzygy -- Returns a syzygy module one beyond the regularity of Ext(M,k)