what does very small 4v relay is
Since the documents only say that a HIGH output is 4V and don't specify a current limit I would look for a relay that will activate on 4V at 20 mA.
in comparison to the magnet I described, 0,81wire, 40m 1v 1Amp
Look for a relay whose contacts can handle 1A r more at 1V or more.
is it all, the battery, the IR detector, and the relay, on 4v or adding a transistor if on 5v. no controller.
Correct. No controller needed if you only want to close a relay contact. The sensor will run on 5V. The output of the sensor is enough to activate a transistor. If you have 5V for the sensor you also have 5V for the relay. The transistor can be chosen to provide as much current as the relay needs so the relay coil current is not critical. That will allow you to pick a less expensive relay.
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