The authority map is the useful thing to know, because the Bureau's reach is a list rather than a subject. 12 USC 5481(12) defines "enumerated consumer laws" as eighteen named statutes, and the Bureau's rulemaking and enforcement authority attaches to those plus title X itself. In the statute's order they are: the Alternative Mortgage Transaction Parity Act; the Consumer Leasing Act; the Electronic Fund Transfer Act, except its interchange section; the Equal Credit Opportunity Act; the Fair Credit Billing Act; the Fair Credit Reporting Act, except two sections; the Homeowners Protection Act; the Fair Debt Collection Practices Act; subsections (b) through (f) of section 43 of the Federal Deposit Insurance Act; sections 502 through 509 of the Gramm-Leach-Bliley Act, with one carve-out; the Home Mortgage Disclosure Act; the Home Ownership and Equity Protection Act; the Real Estate Settlement Procedures Act; the S.A.F.E. Mortgage Licensing Act; the Truth in Lending Act; the Truth in Savings Act; section 626 of the Omnibus Appropriations Act, 2009; and the Interstate Land Sales Full Disclosure Act.
The carve-outs are what prove this is a map and not a shorthand. Paragraph (F) brings in the Fair Credit Reporting Act "except with respect to sections 615(e) and 628 of that Act (15 USC 1681m(e), 1681w)", which are the identity-theft red flags rules and the records disposal rule. Those two stayed with the agencies that had them. Anyone reasoning from "the CFPB administers the FCRA" to a conclusion about either of those sections gets the wrong answer.
The $10 billion line splits who examines and who sues, not whose rules apply. This is the single most misstated thing about the agency. A Bureau regulation, such as Regulation Z or Regulation F, binds a covered institution whatever its size. What the asset line divides is supervision and enforcement. Under 12 USC 5515(a) the Bureau's supervisory and enforcement authority reaches an insured depository institution or insured credit union "with total assets of more than $10,000,000,000 and any affiliate thereof", and 5515(b)(1) gives it "exclusive authority to require reports and conduct examinations on a periodic basis" of those institutions. Under 12 USC 5516(a), for an institution with total assets of "$10,000,000,000 or less", 5516(d)(1) provides that "the prudential regulator is authorized to enforce the requirements of Federal consumer financial laws and, with respect to a covered person described in subsection (a), shall have exclusive authority (relative to the Bureau) to enforce such laws." The Bureau keeps two narrow footholds below the line: it may include its examiners "on a sampling basis" of the prudential regulator's examinations (5516(c)(1)), and where it believes there has been a material violation it notifies the prudential regulator in writing, which must respond in writing within 60 days (5516(d)(2)). So a complaint about a small community bank is a matter for that bank's prudential regulator, even though the rule the bank broke was written by the Bureau.
The boundary against the Federal Trade Commission runs the opposite way from the shorthand. The two agencies are not divided by institution size. The FTC Act's own enforcement provision, 15 USC 45(a)(2), excepts "banks, savings and loan institutions ..., Federal credit unions ..., common carriers ..., air carriers and foreign air carriers", so the FTC's unfair-or-deceptive-practices authority does not reach depository institutions at all, of any size. The Bureau reaches them, subject to the supervision line above. In the other direction, 12 USC 5481(14) is explicit that "Federal consumer financial law" as the Bureau administers it "does not include the Federal Trade Commission Act", so a practice the FTC would attack under its own statute is not automatically within the Bureau's remit either.
The complaint function is a routing mechanism rather than an adjudication. The Bureau's own description of the process at consumerfinance.gov/complaint is that it forwards the complaint to the named company and asks the company for a response, which is why a workable mailing address is a requirement rather than a formality. A complaint is not a claim, and the process produces a company answer rather than a determination or an award.