Mine is

LENS:
1-788-858-
L0002996

only 3 0's between the L and the last 4 digits of the SN

They might both be correct. The lens with my camera does have what looks like a physical serial number on it, there is a sticker near the manual/servo switch.
There is also a small QR code.

But the menu shows 00000000.

My 300 reports 00000000 as the lens serial number with firmware is V1.12. I believe it is the same body and lens as before. I did own an EX3 and I am sure that it had a serial number for the lens. in the menu But I have not had one for over a year so I could be misremembering. But when I pushed for ...

I have just received my fixed 300 from Sony. The CA problem seem so be cured.

Interesting, I first was told by Vistek I was getting a new camera. Then that Sony Canada have performed a major firmware update on my camera that an end user could not apply by themselves, and that they would ship my original camera back to me. I too would not be happy with an non serialled lens.

No it is 00000000

In my case Sony say that it is problem with the camera, possibly the optical block. They are replacing the camera and lens

Vistek are being very good about this. They could not reproduce the same problems with their demo 300. I guess I just got a rogue lens. Camera is great otherwise. It felt bit more front heavy than the Ex3. I am shipping it back tomorrow. Hopefully Sony are going to replace the lens or the whole kit....

I just noticed there is no serial number for the lens on the settings menu. Electronics rather than optics but another concern
abut the 14x zoom. Please let me know if the 300 you receive reports a lens serial number.

I have the 14x lens. I am going to shoot some more tests t but I don't want to take the camera outside as it is -17c here today. I did have the camera ground shipped 1500km from Vistek in Toronto, I wonder if the bumpy roads shook something loose.