The Colomban Cri-Cri is a small twin engine plane with an empty weight of ~200 lbs and a MTOW of ~400 lbs.
Can I fly this in controlled airspace? What are the requirements for aircraft registration?
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If there were any minimum weight requirement for manned aircraft in controlled airspace, the requirement would probably be in 14 CFR Part 91, Section 126 through Section 135. These are the sections which describe the operating rules for various types of airspace. I looked through those sections, and there are no weight requirements.
I also can't imagine why there would be a minimum weight requirement for any class of airspace.
Like most aircraft, the Colomban Cri-Cri must have an airworthiness certificate and a registration certificate (14 CFR 91.203). The regulations describing getting an airworthiness certificate are in 14 CFR 21, Subpart H.
Some aircraft weighing less than about 254 pounds are "Part 103 ultralights" and do not need to be registered. However, this doesn't apply to the Colomban Cri-Cri, because Wikipedia says that it has a cruise speed of 100 knots; Part 103 ultralights must be "not capable of more than 55 knots calibrated airspeed at full power in level flight".
It's not a weight thing, it's an equipment thing. Need a transponder, ADS-B-Out (starting Jan 1 2020) and 2 way radio. OR, calling ahead and making arrangements if not so equipped.
I don't imagine flying something that light around big jets would be very much fun, altho some of them have smaller runways for smaller GA aircraft.
Boston, for example: https://img.airnav.com/aptdiag/w240/00409.gif?v=MZPOGC