Apple - Can the 2013 13" Macbook Air drive an external 4k monitor?
No, it won't support it. The Thunderbolt 1 controller in the Air only supports DisplayPort 1.1 and you need DisplayPort 1.2 to get 4K support, as you mention in your question.
Check out http://www.anandtech.com/show/7049/intel-thunderbolt-2-everything-you-need-to-know and http://en.wikipedia.org/wiki/Thunderbolt_(interface)#Thunderbolt_2
No. Thunderbolt 2 is required.
What is Thunderbolt 2?
Thunderbolt 2 is an update to the original Thunderbolt specification and takes the original’s two 10 Gbps bi-directional channels and combines them into a single 20 Gbps bi-directional channel. The amount of data able to go through a Thunderbolt connection hasn’t increased, but the throughput of a single channel has been doubled.
What does Thunderbolt 2 have to do with 4K video?
4K video (a video format that has four times the resolution as 1080p and is gaining popularity with video professionals) requires more bandwidth than the original Thunderbolt can accommodate. Thunderbolt 2 will be able to stream that 4K video and write it to disk at the same time.
Excerpt from this MacWorld Article
It might be able to.
According to http://www.anandtech.com/show/7007/intels-haswell-an-htpc-perspective:
Haswell does have 4K display support and we will have a dedicated section to see how well it works.
The whole section that they're talking about:
http://www.anandtech.com/show/7007/intels-haswell-an-htpc-perspective/7
A quote:
The good news is that Haswell's 4K over HDMI works well, in a limited sort of way.
So, maybe it will, maybe it won't, since mid-2013 and early-2014 MacBook Air only has mini-DisplayPort, so, you're adding the uncertainty of a rubbish adapter into the already fragile mix.
It would be nice if someone actually tries it extensively, and reports with some more certain results.
UPDATE 0:
According to http://www.redsharknews.com/technology/item/1646-editor-hacks-2011-macbook-air-to-output-4k from 2014-04, it looks like even 2011 MacBook Air may be able to support 4K resolution, albeit only at 25Hz and with some reliability issues, and 2011 had HD Graphics 3000, as per http://en.wikipedia.org/wiki/MacBook_Air#Specifications, whereas mid-2013 and early-2014 have HD Graphics 5000 from a Haswell CPU.
UPDATE 1:
Additionally, there is a report of a 2012 MacBook Air, which has merely HD Graphics 4000, supporting a 4K Seiki -- http://forums.macrumors.com/showthread.php?t=1681560. So, I'd say chances are very good that a mid-2013 MacBook Air wouldn't have much problems with 3840x2160@30Hz, either.