There is one anomaly in Bender's Big Score which cannot be resolved even in concept under any known model of time, given the facts of the story. It pertains to Hermes Conrad.
Before much of anything has happened in the film, Hermes loses his head in a freak accident involving a sword falling from a wall and decapitating him. This being a thousand years in the future, the head is preserved in a jar for reattachment to the body; but the procedure is delayed when the Scammer aliens deprive the Head Museum of its assets. Hermes is quite concerned that his gorgeous wife will leave him for Barbados Slim, on the pretext that "boy need a father". Indeed, Hermes' hold on this woman is weak.
As the reattachment procedure is further delayed, Hermes devises a solution, and gets Bender to assist. He needs a body now; he had one in the past. Bender travels to the past, kills Hermes before the accident, and brings Hermes' body, minus the head, forward to the present, where Dr. Zoidberg improperly but effectively attaches the present head to the past body.
The film treats this as if the body were like any other temporal duplicate. The Futurama treatment of doppelgangers has already been found wanting, but in this case the problem is more complicated. No part of Hermes ever traveled through time; the time code does not permit travel to the future, even connected with travel to the past. Bender thus killed Hermes before his accident, took the body into the cave for an unspecified length of time which had to be at least several months, and then delivered it to Zoidberg. However, while Bender was in the cave, events were being formed in the world above--and Hermes (being already dead) was not decapitated at the limbo contest, so his head is not awaiting a body.
For fixed time advocates this is absurd, and never could happen; Hermes was not killed in the past. Under replacement theory it creates a simple infinity loop, since once Bender kills Hermes in the past, Hermes' head is not there in the future to ask him to do so. Divergent dimension theory provides a simple explanation, that Hermes was now killed in the past and his non-existence in the future simply means that Bender wasted a trip; there is no head to re-attach.
Such a story might be possible in parallel dimension theory, on the assumption that the time traveler moves to another universe in his own past, steals the body, and returns to his own universe when he comes forward; but as noted Bender does not travel to the future; he merely awaits it. Thus he might travel to a universe which is not his past, but when he emerges from the cave, he is in the universe in which he killed Hermes before the accident, and there is no head to re-attach.
Although other theories such as supertime and two-dimensional time come closer to resolving such a story, since Bender waits in the cave history will have caught up with his changes by the time he emerges.
Ultimately the body taken from the past is killed in a freak accident, attributed to the "doom" which attaches to temporal doppelgangers. Again it seems backwards: it is not this body but the other whose existence has been undone by temporal interference. This, though, is necessary to the plot overall, as one of the characters needed to learn that temporal duplicates are doomed.
Next time we cut to the chase, as Fry flees a murderous Bender.
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