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I think the rulemakers figured that by allowing for correction of timing mistakes, that it is obvious when you put time on the clock, time for that period has not expired. Time expiring deals only with the end of a period. If you are on the court with the clock showing .5 seconds left in the third period, time for the third period cannot have expired!
Suppose we put the .5 on the clock, A1 misses free throw and A2 taps ball before 3rd period horn sounds (for second time!). Basket doesn't count in your rulebook, because time for the third period already expired when we heard the erroneous horn. That is the absurd extension of your conclusion that time has expired, but time is on the clock.
Time for a period can only expire once. Nothing can happen other than one or more free throws with lane cleared, and with zero seconds on clock, after time expires for a period.
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