No competent civil servant could ever ask his minister to sign a direct falsehood

Well here's your problem.

Firstly, at certain levels there is no such thing as a competent civil servant (I know, I am a civil servant - all the sensible ones are at the bottom grades where they do the work and can't get promoted because no-one else would do it, the exception is in IT)

And secondly, he may not feel able to ask, but the minister would sign it regardless, if it meant another term in office.

LTS