Excuse me? I didn't define "T" as a complex pattern. T could actually be any pattern or shape. And I didn't say that they were looking for T. I said they were looking of an inference of IP by the existence of T+R+O, which is what they would have if they found such a signal.
So you're saying that "any pattern or shape" plus a "receiver" plus an observer would mean "intelligent purpose" on the part of the sender or originator of the pattern or shape?
That is clearly not the case. Surely you can think of examples.
I'm not saying that the function can not occur apart from intelligence. Again my wrist watch runs quite well with out a single intelligent sole around. It is the detection of intelligence that requires an intelligent observer. However I can detect intelligence in the transmitter receiver relationship of the parts within my watch. I am saying that this type of design function requires an original intelligence to create the process.
I understand what you're claiming, but I'm pointing out that the claim isn't supported by observation. Because there are transmitter-receiver relationships in human designs is no reason to infer that all transmitter-receiver relationships are intelligently designed. SETI does not infer this, otherwise they would already be claiming to have identified intelligence elsewhere in the universe.
Unintelligent things can certainly originate signals. SETI are looking for a signal that they think shows the symptoms of artificiality. Your watch certainly could be said to contain signs of artificiality, and a timepiece made by intelligent alien life forms might well have such signs. But that does not mean that all phenomena by which we can measure time (the sun, the moon, tides etc.) are intelligently designed.
Let's reshape your formula. AT (artificial transmission) + R (receiver) + O (observer) = IP (intelligent purpose). Now, that might be (tautologically) true, but it's useless, because it leaves us with the problem of establishing the "A" in AT. Whether or not SETI is a worthwhile project depends on their evidence and reasoning in relation to the artificiality of a persistent, simple narrow band signal. It's a technical question, and a number of lines of evidence are involved.
I would argue that there's no simple universal formula for SETI (or archaeologists) to determine "A" (artificiality) in objects. So, even looking for signs of biological intelligence isn't straightforward, although it can be done.
As for what non-living intelligence might do if it existed, we haven't got a single example, and we have no idea, let alone a formula.