You won't find a specific regulation that says that the weight and balance numbers must be on board. I've seen the issue bandied about some on various forums. I'll offer the collected wisdom (?) of those discussions.
There are two primary sources of the requirement.
1. If you remember your 91.213 airworthiness analysis, ultimately, whether a piece of equipment is required or not is based on the airplane's Type Certificate Data Sheet . In some cases, not all, the TCDS contains language like:
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Current weight and balance report including list of equipment included in certificated weight empty, and loading instructions when necessary, must be in each aircraft at the time of original certification and at all time thereafter...
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So, the presence of the weight and balance data is as much an airworthiness requirement as operating fuel gauges.
But not all aircraft have this requirement on the TCDS.
2. 91.9(b) contains the requirement for the presence of systems information on board. (The reference to 21.5 is a reference to the requirement for an approved manual (AFM) after 1979). Notice that the information, in some form, must be available on board, whether or not an AFM is required.
23.1581 - 23.1589 talk about what the contents of an approved AFM manual must contain. Without going into detail, the information includes loading information (23.1589) and references to equipment are strewn throughout the rest of the group.
Read them together and you get that "manual material" includes weight and balance and equipment data. If the airplane has an approved flight manual, the manual, which contains the "manual material" must be on board. And if the airplane doesn't have an approved flight manual, you need the information on board anyway.
I think that there may be some airplanes that still slip through the cracks but, like a lot of shorthand expressions for learning things, AROW covers the general rule. There are always exceptions.
