Testing the new modulus, I check the effect of a constant input message.

Msg=0:

66000000000  X=36663    Y=131746                        
66100000000  X=159181    Y=42180                        
66200000000  X=56730    Y=127039                        
66300000000  X=93853    Y=158451                        
66400000000  X=36243    Y=165031                        
66500000000  X=71790    Y=65041                        
66600000000  X=80476    Y=43666                        
66623095107     done !

Msg=1:

66000000000  X=236773    Y=175058                        
66100000000  X=143693    Y=108851                        
66200000000  X=227833    Y=190930                        
66300000000  X=246629    Y=237951                        
66400000000  X=151749    Y=33828                        
66500000000  X=162761    Y=46437                        
66600000000  X=16165    Y=215837                        
66623095107     done !

Msg=2:

66000000000  X=178770    Y=218371                        
66100000000  X=128206    Y=175522                        
66200000000  X=140823    Y=254822                        
66300000000  X=141292    Y=59338                        
66400000000  X=9141    Y=160740                        
66500000000  X=253732    Y=27833                        
66600000000  X=209968    Y=129893                        
66623095107     done ! 

Msg=3:

66000000000  X=120766    Y=3570                        
66100000000  X=112718    Y=242193                        
66200000000  X=53813    Y=60600                        
66300000000  X=35955    Y=138839                        
66400000000  X=124647    Y=29537                        
66500000000  X=86588    Y=9230                        
66600000000  X=145657    Y=43950                        
66623095107     done !

The orbit lengths are the same, it just goes through different points.

This proves that the "quiescent value" does not matter. I can stick to 0, any other value would be more complex to handle and wouldn't change the efficiency.