Personal FAQ:
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)
23.1581 - 23.1589 talk about what the contents of an approved AFM manual must contain. 23.1589 says:
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Loading information.
The following loading information must be furnished:
(a) The weight and location of each item of equipment that can be easily removed, relocated, or replaced and that is installed when the airplane was weighed under the requirement of § 23.25.
(b) Appropriate loading instructions for each possible loading condition between the maximum and minimum weights established under § 23.25, to facilitate the center of gravity remaining within the limits established under § 23.23.
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Read them together and you get, an airplane must have its manual on board and it's manual must contain weight and balance data.
I think that there are 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.