I finally got a reply from the FAA.

I am pasting it in here. Even though they say this and at least it is coming from an FAA person, i am still not comfortable until I can read it in the regulations themselves. Why can't they just add that one sentence it to the regulations" Any good lawyer could do that in about three pages.

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Mr. Culbertson,
Thank you for contacting the FAA's Unmanned Aircraft Systems (UAS) Support Center.
Control line models do not require registration.
Sincerely,
Paul
Paul Hamilton
PLH
Please follow up with any further inquiries at UASHelp@faa.gov. Additional information is also available at
https://www.faa.gov/uas/.
We appreciate your feedback. Please select: UAS Safety and Integration Division AUS-400.

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Are control line model airplanes required to be registered? There is great confusion in our community. The AMA (our Community Based Organization) says control line, free flight and indoor model airplanes are exempt from registration but we can find nothing in anything coming from the FAA that confirms this.