Nine new replies? I must type quicker. Or less of it at a time...
edit: Typos corrected, e.g. "14 seconds" instead of "15 seconds", which I can't leave without looking like I'm innumerate
Yes, I see the confusion. And regardless of how far ahead of the lightspeed-signal you get (thus "information from the future") it won't ever make a round-trip in negative time, because it will always have spent a positive time amount travelling, and will reach its origin/destination after it was sent. Really, when I wrote up that travelling salesman example, I was being a bit flippant and didn't think that was what was meant, but was extending things out to perhaps slightly more ridiculous lengths than I should have allowed myself.
A signal going at 2c to a location half a light minute away will take (assuming, for now, the same inertial frame of reference for both ends) 15 seconds to get there. If the sender flashes a "sending" signal with a standard signalling light, it will have arrived 15 seconds before the flash is seen by the recipient, but it's still not instantaneous, just less delay. If the receiver flashes a "receiving" signal back (by standard means), the transmitters will see a response in 45 seconds (rather than 1 minute, if it had been a standard light-only signal sent out.) If the outward transceiver is bouncing the signal back, it will again take an additional 15 seconds of transit time. To the sender, that would be 15 seconds before the RCV-signal light, but it will still have been a 30 second round-trip time. Positive 30 seconds later.
Even sending a signal at ∞c will mean that the 0.5lmin distant receive will get the signal 30 seconds ahead of conventional traffic. If bounced, it will get back home 30 seconds ahead of the standard-type receipt acknowledgement. Woohoo, that's a full 60 seconds of travel time saved, surely putting us into "voice from the future" territory, compared to.... oh, 60 seconds of normal transit. What you're getting back is your signal, instantaneously (plus overhead). Which is no better than just using your information locally and not sending it outwards in the first place. No real time-travel here.
I am reminded of a story (I think a short story in a collection, I don't think it was the basis of a whole book, but if someone can recognise it it would interesting to reacquaint myself with those details) set in a future where there is galaxy-wide settlement, and communication is by an insta-transit coms-device that meant that we at least could talk to people hundreds of thousands of light-years away without having to wait (two times) hundreds of thousands of years. But part of the plot of this piece was about the annoying "beeeeep" or screech or somesuch that started each communication. It seems that somebody had worked out that this anomaly was actually a highly compressed signal consisting of the contents of absolutely every message that had been sent or would be. And so there was an organisation that (protecting the secret) had set about to extracting all the messages it could, in order that news of assassination attempts or natural disasters was (wherever possible) used to pre-empt events in whatever allowable way they could[1] to help avert the dire results that might have arisen without the foreknowledge.
However, while a good plot device (at least if wielded in the right hands) not relevant to our discussion, I believe. Just something that was dredged out of my memory from at least three decades back.
[1] I can't recall that they could change events, which meant that successful assassinations and actual casualty reports could not be done anything about, but I could be misremembering. As I've always been a believer in time-travelling information being of the self-fulfilling variety, I much prefer a clever story where "what was sent back and was done had already been back and done" to any number of plots where history gets overturned or (at the extreme silly end) you get "Fading Marty McFly Syndrome".