Your first article names its sources as exclusively wikipedia - this would not be accepted as a reliable resource in an academic context. I'm a little dubious about some of the setup of the website.

The second article appears more objective & analytical but still does not name its siurces.

The final one is a BBC article - I've observed that the BBC has made many errors & ommissions in it's documentry programs. An example of this would be the programs by Dr Ramirez about Dark ages artwork - a program she ommitted to mention any of the outstanding & unique pieces of rk on the Isle of Man. So I am dubious about it too.

Besides this if you read the articles objectively with 2 & 3 you will note they support several things I have already said & which you don't seem to be seeing. For example they support that the Lowlands were already Saxon speaking when the Scots arrived being part of the Kingdom of Northumbria. They also comment that the Lowland Scots percieved themselves as English living in the Kingdom of the Scots & the Scots dialect was as accepted as Gaelic as a means of national identity.

When it comments about the Lowland areas it talks about Galloway & Ayrshire & as I've already pointed out this was from the sea going Norse Gaelic influences & does not imply a spread across the lowlands. It also mentions some places directly abbuting the Highlands, which have a strong history predating the Scottish invasion.

It also acknowledges that Gaelic placenames can come as loan words rather than implying that there was definitely Gaelic spoken in those areas. What it does seem to brush over except where it suits the article is where words are common to both P & Q Celts.

So I'm afraid you haven't really supported your statement that Gaelic is central to Scotland & Scottish identity or that it was spoken across all of the Lowlands.

My point remains & is supported by all of the articles that Gaelic is like Saxon an invading language and if you believe that the history of Scotland goes deeper than just the influx from Dalriada then you need to recognise it is no more right to put Gaelic on welcome to Scotland signs in the borders thanit is to put it in Brythonnic (infact less right if it is supposed to represent the language of the people of the surrounding areas.