Well no worries, so just unsubstantiated claims then
Just the same as yours you mean
http://www.commercialalert.org/PDFs/ftcresponse.pdf have a read through that if you like, it is stated in there that paid listings and organic listings MUSt be unlinked (directly).
Now back to the debate. You appear to have changed your mind, either that or possibly not read the important bits in the OP & My reply, so here they are.
OP WROTE
Site is a few months old and had got to around page 6 on google search, for no obvious reason it disappeared and came back after 4 days on page 19.....>snip
then I tried adwords for 2 weeks, paused it and site disappeared again
My reply
Google have categorically stated there is no direct link between individual paid and organic results
You appear to be challenging my reply, but at the same time agreeing with me )potentially).
The devil is in the detail (as you say), but the FTC & The UK advertising Standards authority have in the past sought assurance from Search engines that people can not use money as a tool to appear in the natural listings.
http://www.out-law.com/page-4628 Old I know but I can't be rsed to look any firther, in order to debate with someone who will not post their own detail.
So what do I think?
I think that google is using behaviour of individuals in their ranking algo, along with their analytics, and the masses of other data they have that give them industry benchmarks. Paid advertising can of course, increase traffic to a site, but the DIRECT relationship is between the traffic and the algorithm, not the ad campaign and the algo. Who knows, but either way, if there is a direct relationship, then it could fall foul of the advertising laws.
One last thing though. whenever there is an algorithm change, forums are flooded with people who complain their rankings have disappeared. HOW can that be? because surely the same amount of people should be complaining that their rankings IMPORVED (as for every page that drops, others MUSt move up to take their place!
For every person who claims their site suffered from using adwords, there are people claiming their site improved as a result, it is called co-incidence. So I have told you my theory, how about you tell me yours, on the forum for all to see, (as it should be).