I had read that there were Roman census records that proved a Rabbi named Jesus did live at about the right time, but now I can't find a source to back that up, so that's probably bunk.
If you compare that with records we have for the likes of Alexander the Great though being 400 years later, it's not that implausible. And you'd be discounting the Christian Gospels and Paul's Epistles which were mere decades after Jesus
Interesting, what kind of records do we have from Alexander's time? And yeah, I agree, the early gospels and the later Roman references probably indicate Rabbi named Jesus was crucified, but I don't think that a secondary source or religious texts really meet OP's criteria for, "physical proof." (Although we probably don't have, "physical proof," for a lot of historical events we generally accept have happened).
Completely unlikely since no such census records are extant.
People who are jnfamilhar with the historiography are very much overestimating the amount of primary source material which exists from the Roman Empire, simply because historians have been very good at extracting information from the miniscule fraction (relative to the amount which was produced) of extant written sources we do have from the period.
There are no such records. Just having any extant census records from the Roman Empire would have pretty sensational, let alone some stemming from Judea at the supposed time of Jesus.
Yeah, that makes sense. I wish I could track down where I read this to figure out if it's a bad source or I'm misremembering it. I may be mistaking Tacitus' reference to Christ, but I don't think it's that. I distinctly remember reading about some sort of population record of a Rabbi named Jesus and thinking, "Wow, I'm surprised a record like that survived." The problem is this was 10+ years ago, and search engines suck now, so I'll never find it again.