We know that Stargates only reintegrate matter at the destination once the entire object has gone through the gate on the departing side. This behavior is made explicit in the Atlantis episode 38 minutes (Season 1, Episode 4). The episode makes clear that a whole person is not a discrete object to be reintegrated, but rather the entire system that person is a part of. (In 38 minutes half of a puddle jumper enters the gate, but then gets stuck so two crew members and half the jumper have dematerialized, while the several other Atlantis members and the back half of the jumper remain outside the gate. The crux of the episode is that if the back half of the jumper does not enter the gate, then the front half + 2 crew members will never rematerialize.)
However, this behavior does not seem to apply to gates in the Milky Way. A few notable times this behavior did not apply and partial objects got rematerialized:
Ernest Little field in the episode Torment of Tantalus from Season 1. We can clearly see in the archive footage that Ernest is attached to a breathing apparatus on the Earth side of the gate (as he is in some sort of diving suit) and that the breathing tube and apparatus remain on the earth side (there is close up footage of the severed breathing tubes in the archive Daniel is watching). So, how could he have reintegrated at Heliopolis? Should not the gate have been waiting for the rest of the breathing apparatus to come through?
In episode two (“The Enemy Within”), when Kawalsky is taken over by a Goa'uld symbiote, O'Neill kills him holding his head half-way in the gate and calling for it to be closed, thus cutting off half of Kawalsky’s head.
Teal'c, in his efforts to recover O'Neill from Edora in "A Hundred Days," shoots a projectile attached to a rope through the gate. The gate on Edora is buried, and the projectile lodges itself in the rock on the other side, allowing Teal'c to climb up the rope on the Earth side, through the gate, and be suspended above the event horizon on the Edora side. How can the projectile and rope rematerialize on the Edora side while some part of the rope is still on the Earth side (so that Teal'c may climb it)?
In 200 the puppets manage to rematerialize without their puppet strings going through the gate!
Is this a difference in behavior between Milky Way gates and Pegasus gates or is something else going on here? Does the gate technology make some sort of distinction between ropes and non-ropes? (It is not between animate and inanimate objects as a partial puddle jumper is unable to rematerialize.)