Does imposing an odd initial condition reduce spurs in MASH Delta-Sigma Modulators that has second order stages?
This is on of those folklore things. There are very few synthesiser practitioners in the world who use fractional-N, very few of them (even those who write books) know all they need to know, and much of what has been done is shrouded in commercial secrecy, so there isn't a generally accepted answer to this in circulation.
It's quite easy to get MASH, or one of the many other algorithms, to generate a number stream suitable for driving a synthesiser. It's quite difficult to engineer a synthesiser so that the full performance of that number stream is realised.
Of the many things that can be done to make a synthesiser generate spurs are (1) failure of isolation between reference and output sections (2) use of a phase detector with non-linearities (3) use of a divider with data-dependent propagation delay. While most engineers will accept (1) and strive for isolation with buffer amplifiers, decoupling, separate screened pockets etc, a surprising number do not accept that (2) or (3) need attention too.
Any of these hardware problems can give rise to spurs that can look for all the world like defects in the number stream. Setting the LSB of the DSM input so its output is 'busier' can mask some of these effects.
In a well-engineered synthesiser, a MASH 1-1-1 can give results that are 'perfect' in the sense that the RF output is indistinguishable from that of another synthesiser using the same VCO frequencies and PLL bandwidth, but using integer multiplication from a similar reference frequency.
Since various patents in the field have expired, a lot of people are coming new to fractional-N synthesis, and blithely making errors number (2) and (3). Many will use integrated divider + PSD solutions from some major manufacturers, which integration makes it impossible to maintain adequate isolation between output and reference. This all helps fuel the folklore.