Where on earth in the FAR's is the requirement to have the weight and balance (AROW) information in the airplane?
There is no blanket requirement in the regs that this information be carried aboard the airplane. However, under the section "AIRPLANE FLIGHT MANUAL AND APPROVED MANUAL MATERIAL"
§ 23.1589 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.
Combine the above with 14 CFR 21.5, which says
21.5 Airplane or Rotorcraft Flight Manual.
(a) With each airplane or rotorcraft that was not type certificated with an
Airplane or Rotorcraft Flight Manual and that has had no flight time prior to
March 1, 1979, the holder of a Type Certificate (including a Supplemental
Type Certificate) or the licensee of a Type Certificate
shall make available to the owner at the time of delivery of the aircraft a current approved Airplane or Rotorcraft Flight Manual.
and combine that with 14 CFR 91.1 which says
§ 91.9 Civil aircraft flight manual, marking, and placard requirements.
(a) Except as provided in paragraph (d) of this section, no person may operate a civil aircraft without complying with the operating limitations specified in the approved Airplane or Rotorcraft Flight Manual, markings, and placards, or as otherwise prescribed by the certificating authority of the country of registry.
(b) No person may operate a U.S.-registered civil aircraft—
(1) For which an Airplane or Rotorcraft Flight Manual is required by § 21.5
of this chapter
unless there is available in the aircraft a current, approved Airplane or Rotorcraft Flight Manual or the manual provided for in § 21.141(b);
All of the above applies to aircraft with no flight time prior to March 1, 1979. Prior to that, some aircraft had as part of their type certificate that current weight and balance information be carried aboard the aircraft, or had a requirement for an AFM.