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56/57mm is a slightly awkward number as I think it sits right on the limit of a lot of calipers' range. They quite often seem to be something-57mm.
Please just leave that bike exactly as it is. What will actually be left over from that in a fixed gear? Frame, forks, bars, stem, post, saddle and possibly brake levers and front wheel? I think it's just not worth it. I'm not much of a bike purist and I've put modern bits on my own early-80s bike, but mine is a fairly bog-standard light tourer, not a top-of-the-line racing bike. I'm sure you can find another frame to build a fixed gear on.
OK, so to the best of measuring ability, the reach is 56/57 mm (centre of hole to centre of rim).
The bike is a Peugeot PX10. Yeh I do suspect that doing a ss conversion on this bike might exasperate the purists! (especially as it is very well maintained components wise). I have been after the bike in my size, this colour for a few years, always with the intention of converting it and was finally able to pick it up on ebay. Since getting it I have been considering keeping it as it, as it certainly does ride very nicely - but overall am still feeling inclined towards conversion.