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Law 16B:
1. {a} After a player makes available to his partner extraneous information that may suggest a call or play, as for example by a remark, a question, a reply to a question, an unexpected* alert or failure to alert, or by unmistakable hesitation, unwonted speed, special emphasis, tone, gesture, movement or mannerism, the partner may not choose from among logical alternatives one that could demonstrably have been suggested over another by the extraneous information.
* i.e., unexpected in relation to the basis of his action.
{b} A logical alternative action is one that, among the class of players in question and using the methods of the partnership, would be given serious consideration by a significant proportion of such players, of whom it is judged some might select it.
2. When a player considers that an opponent has made such information available and that damage could well result, he may announce, unless prohibited by the regulating Authority (which may require that the director be called), that he reserves the right to summon the director later. The opponents should summon the director immediately if they dispute the fact that unauthorized information might have been conveyed.
3. When a player has substantial reason to believe that an opponent who had a logical alternative has chosen an action that could have been suggested by such information, he should summon the director when play ends*. The director shall assign an adjusted score (see Law 12C) if he considers that an infraction of law has resulted in an advantage for the offender.
* It is not an infraction to call the Director earlier or later.
What all this means, in practice, is that when you believe that something has happened which might convey UI (see the list in Law 16B1{a}), you should ask the opponents if they agree with you. If they do not, they are supposed to call the TD. Of course, what usually happens is that they don't agree, they get upset,
and they don't call the director. So you may have to call him yourself. If things go well, they agree something has happened which might convey UI, and there's no need to call the director (yet). At the end of play, if you believe an opponent who had UI chose an action that could have been suggested by it, when he had a logical alternative, you should call the director. People who don't believe they were damaged frequently decline to call the TD, on the theory that it doesn't matter, but in fact even if there was no damage, if a player has illegally chosen a suggested alternative, the TD should take the opportunity to educate the player about his obligations. That can't happen if the TD isn't called.
Side note: The footnote to Law 16B3 contradicts the established usage mentioned in the Introduction to the Laws, which states that when a player "should" do something, failure to do it is an infraction. I suspect a better wording for the intent (or at least, what I believe to be the intent) of this footnote is "he should summon the director. He does so when play ends." IAC, I would not, as director, fault a player who called me "earlier or later".
Heh. I see I still haven't directly answered your question. The purpose of calling the director when there is a disagreement as to whether UI may be present is so that the TD can establish, at the time, whether there was. He cannot (yet) rule on the question of whether UI was illegally used, whether it caused damage, and so on, so that's why he says "call me back at the end of the hand if you feel you've been damaged." As I said above, it's probably better, from the point of view of education, for the TD to say "call me back if you feel the player with UI has chosen an action suggested by it, when he had a logical alternative", but that's likely to confuse people. Note also that if the TD determines there was no damage, you're not going to get a score adjustment, and most players are more interested in getting that than in educating their opponents (or seeing that the TD does so).
In the case you cite, it appears your opponents weren't damaged, so they would probably, in practice, not have needed to call the TD (save for the education thing, but he should have already done that). But it sounds like the TD should have ruled that there was no break in tempo, so he should have told the table that, and that no further call would be necessary. Did he rule there was a break?