Here are a couple of rules that you can use for your case.
Rule 110c- A league cannot adopt a rule that would have the effect of not permitting any prebowling/postponements. The league must grant a prebowl/postponement when the team is unable to field a legal lineup for the following reasons: 1. some bowlers are participating in tournaments..., 2. There is sufficient cause, and 3. An emergency situation.
This one is out of the What is the answer booklet from 2003-2004
#117 When teams pre or post bowl unopposed, can the opposing team see the scores bowled by the team bowling first? There is no WIBC/ABC rule to prevent the opposing team from seeing the scores. When a league allows teams to bowl unopposed, rules can be adopted to govern this situation.
So basically it says that you do not have to leave the building, unless the league says that you cannot see the scores until the games have been made up. I would seriously suggest calling up your league president and or secretary and ask him/her if they can produce a copy of the league rules so this situation can be cleared up.
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It does not matter what type of shot you throw, it's the results on the scoresheet that counts.