This isn't functionally that far off some existing answers, but I wanted to stick a little more tightly to what I can cite (EDIT: I have corrected the initial quotes from script PDFs to match film dialog--all changes are minor, but check edit log if you care to see). This may make it less satisfying :)
If we can trust the Architect (and, as Neo makes clear, he does), the most direct answer to this question is (this is near the end of scene 269 from the October 27 2001 script--the page is numbered 120, but it's logical page 127 in the PDF):
You are here because Zion is about
to be destroyed, its every living
inhabitant terminated. Its entire
Given the architect's general precision and candor, I think it's defensible to interpret his "entire existence eradicated" quite literally.
This doesn't really answer your root material question (whether it's the same cavern or a new location), but it does hint that their plans account for eradicating all evidence Zion existed.
A few pages before this (scene 266, numbered page 118, logical page 124), after the Architect tells Neo that there have been 5 previous iterations, they have this exchange:
There are only two possible
... either no one told me, or no
I guess there are a number of ways to interpret this depending on how we imagine both Neo and the Architect understand the nuances of who "no one" includes. The Architect hasn't quite told Neo about the "intuitive program" yet, so Neo might include the Oracle in that set, or just humans outside of the matrix.
In any case, I'm inclined to take the Architect's answer very literally again--his absolute certainty that no one outside the matrix knows is further evidence that this is a core consideration of the design.
It isn't explicit in your question, but there's also (as has come up here) the question of why Neo wouldn't just tell the next 23 (which would make the question of evidence moot...)
Here's one more quote from the Architect (scene 270, numbered page 121, logical page 128):
The function of the One is to now
return to the Source, allowing a
temporary dissemination of the
code you carry, reinserting the
prime program after which you will
be required to select from the
Matrix twenty three individuals,
sixteen female, seven male, to
plus what Morpheus tells Neo in the first film (scene 43, numbered pages 43-44 and logical 44-45 in the March 29, 1998 numbered shooting script):
When the Matrix was first built
there was a man born inside who
had the ability to change what he
wanted, to remake the Matrix as he
saw fit. It was he who
freed the first of us, taught
us the truth. As long as the
Matrix exists, the human race will
never be free.
After he died, the Oracle
prophesied his return and
that his coming would
hail the destruction of the
Matrix, end the war, bring
freedom to our people. That is
why there are those of us who
spent our entire lives
searching the Matrix, looking for
There are a few things here that never seemed to square to me.
I doubt what Morpheus tells us about the man who can remake the matrix is "true" (especially if that man is end-of-the-last-iteration Neo), at least in the literal sense Morpheus seems to believe it. It's hard to imagine the system remaining stable for very long with an individual possessing that sort of power on the loose (consider that this description suggests vastly more power than rogue Smith or Neo ever demonstrate) unless that individual is basically just serving/supporting the system.
It seems like previous-iteration Neo would tell the first 23 people freed unless:
- He finds or is given some compelling reason to lie to the initial 23.
- He has no contact with them. He just picks who the machines release.
- It's actually a new iteration of Neo's archetype with the function/directive to free the first 23 and then die (but still--they have to all be freed in a fairly short timeframe to have much chance at rebuilding...)
There's not much bedrock to reason from out this far, but my personal suspicion has been that returning to the source is a terminal action for previous-iterations-Neo, and that the "man" Morpheus tells us about (and probably also the Oracle's prophecy about him) is just part of the mythos/backstory the machines synthesize for the first people they (the machines) release. This seems to square well enough (not that other ideas don't...) with the Architect's certainty that the humans won't figure this out.