Proof that a US citizen can return to US with a passport expiring in 6 months

There are two parts to this question:

Will US border guards deny entry to a US citizen without a valid passport

No.

International law† says that a country may not deny entry to a returning citizen. US law says the same. The person only has to prove they are a US citizen, they don't need a passport. This doesn't mean admittance to the US will be hassle-free.

Will airlines transport a US citizen to the US from abroad without a valid US passport?

They might not, if the airline doubts the passenger has documents normally required to be allowed entry, the airline may decline to carry the passenger in order to avoid the costs to the airline of returning the refused passenger.

†Universal Declaration of Human Rights - Article 13 paragraph 2.

From US Customs and Border Protection - INFORMATION CENTER:

All U.S. citizens traveling internationally by air are required to present a valid U.S. passport to board a flight to the Unites States.

Note the to board a flight portion of that statement.

In theory, if you (as a US Citizen) without a valid passport or any other proof of citizenship, arrive at a port of entry cannot be denied entry.

Assume that they will not accept your word of honour, but start a (possibly) long and costly varification procedure that you are a US Citizen.

What are the passport requirements for visitors to the U.S.?

As a general rule, passports must be valid for six months beyond the date the traveler will exit the U.S., however, the U.S. has signed agreements with a number of countries to waive this requirement. When such an agreement is in place, the passport must be valid for the entire period of the visitor's intended stay, but the additional six month validity period is not required.

Sources:

• U.S. Citizens - Documents needed to enter the United States and/or to travel Internationally.
• Technical Requirements for Passports (machine readable)
  • What are the passport requirements for visitors to the U.S.?

Check this discussion Why would airlines not let a US citizen with an expired passport board a plane back to the US?

In short:

• A US citizen cannot be denied entry (passport or not) if they manage to get to a US border
• The US government can easily force airlines to implement any passport rules they like.

These two things do not need to be related. Don't try to apply logic to government rules and behavior, it typically doesn't work this way.

Is a US citizen living abroad considered to be like a visitor, who needs 6 months (since there's a plan to return?) or is the only determinate that the passport is valid.

No. A US citizen is a US citizen regardless of where they live an even if they have never set foot in the US ever. They cannot be denied entry when arriving at a US border. However, they can certainly be denied boarding if they don't meet the passport rules set out by US government here https://www.usa.gov/enter-us

or is the only determinate that the passport is valid.

Yes. You NEED a valid passport to board a plane. You DO NOT need a passport to enter the country (although it certainly makes the process a lot simpler).

So the answer for you friend is:

If you manage to get to the US border by a means that doesn't check passports you can enter the country (after some scolding from CBP, that is). If you can drive up from Canada or Mexico, you are good. However, no airline will let you board a plane without a passport that meets requirements set out by the US

Timatic, the database used by airlines to determine what documentation is needed, states:

• For a US citizen going to the US:

• Passports and other documents accepted for entry issued to nationals of USA must be valid on arrival.

(emphasis mine).

• For comparison, for a French citizen going to the US:

• Passports issued to nationals of France must be valid for the period of intended stay.

• And for various categories of Chinese citizens:

• Passports and other documents accepted for entry must be valid for a minimum of 6 months beyond the period of intended stay.
• Hong Kong (SAR China) passports must be valid for the period of intended stay.
• Macao (SAR China) passports must be valid for the period of intended stay.

So the rules depend on what passport you hold. For some, there is a 6 month validity beyond the period of intended stay. For others, just the period of intended stay. For US citizens, it just needs to be valid on arrival.

Other countries have similar variable rules depending on the nationality, type of visa, etc.