False. The obligation to check your schedule implicit at 0000 of the start of your on-call block breaks rest. That's why 1000 is the earliest possible (non-FRMS) assignment on Day 1--you have zero obligation to check your schedule prior to starting on-call, so you have to check at 0000.

Contractually, if it's posted by 1500 (prev) / 1200 (new) the last day, it's allowed to be less than the 12 (prev) / 18 (new) hours report time. Both of those times exceed the minimum 10 per 117. Since you are never required to check your schedule unless you are on call (even on your last non-fly day), you have to break rest at 0000 (i.e., when you do have an obligation to check your schedule) to determine if you have a hypothetical 1000 report. The exception to this is if you've been notified & acknowledged the assignment ahead of time--such as the posted-by-1200 trip. That gets you off the hook for the 0000 phone call to notify you (since you've already ack'd).

Same story for FRMS rest. If you ack the rest + associated trip (because CS went fishing & caught you), then it's yours, and 0000 of your first on-call day goes by uneventfully. If you have not been legally notified, though, your rest cannot begin earlier than 0000.