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@Rik_Van_Looy
I'd be surprised if the overall shape of the 1.125" alter is any different to the 1" alter, which would suggest that wall thickness/strength is good for 1.125" and over-engineered at 1".....
If that's correct, I'd take it as an invitation to ream the 1" alter or integralter to 1.125" if you want without too much concern for strength.
If you're not blessed with access to a toolmaker, this can be done pretty easily/accurately on a workbench/workmate with only 2 tools, an adjustable reamer and a tap wrench.
I couldn't find an adjustable reamer that goes from 1" - 1.125" on a quick glance, though I've no doubt such things exist.
The 2 below will take you from 15/16" (23.8mm) to 19/16" (30.2mm) so not only will they be good for this, but they'll also enable you to ream most common seat-tube sizes and remove seized posts too making you the envy of all your friends...
http://www.tracytools.com/straight-taper-reamers?product_id=561
http://www.tracytools.com/straight-taper-reamers?product_id=563
http://www.tracytools.com/index.php?route=product/category&path=42
basically you adjust the cut depth as you go and use the steerer tube as a size gauge.
Note: this is lightly educated conjecture rather than hard fact or experience, so don't come to me with your dentist bills if it fails