Women have used the LJBF rejection for a hundred years because it serves an ego preservation function for her. To a greater or lesser degree, women require attention and the more they have of it the more affirmation they experience, both personally and socially. The LJBF rejection is a social convention that has classically ensured a woman can reject a man yet still maintain his previous attention. It also puts the responsibility for the rejection back on his shoulders since, should he decline the 'offer of friendship', he is then responsible for entertaining this 'friendship'.
This of course has the potential to backfire on women these days since the standard AFC will accept an LJBF rejection in the mistaken hope of 'proving' himself worthy of her intimacy by being the perfect 'surrogate boyfriend' - fulfiling all her attention and loyalty prerequisites with no expectation of reciprocating her own intimacy.
The LJBF rejection also serves as an ego preservation for her in that having offered the false olive branch of 'friendship' to him in her rejection she can also sleep that night knowing that she (and any of her peers) wont think any less of herself. Afterall, she offered to be friends, right? She is excused from any feelings of personal guilt or any responsibilities for his feelings if she still wants to remain amiable with him.
I think the default response should be to take the LJBF as what it is - as a rejection (and her loss) ergo, you remove the reinforcer - attention. Up until the point you made an approach for her intimacy she was enjoying the benefit of your attentions. After an LJBF response her latent intent is to keep that reinforcer of attention. Do not reward her for this disingenuous response, she will only use it on you again or with another guy in a similar situation since it was reinforced the last time this circumstance was experienced. And should the next fellow reinforce it further she will internalize this as her standard response.
Obviously the best way to enact this is to do what you did; use a takeaway and turn down her LJBF. An outright refusal of her psuedo-friendship offer would be ideal, but not always possible given social settings, however a takeaway is always warranted. The problem I see with doubting her intent with the LJBF is that, most women, whether serious or not in their LJBF rejection, will almost always follow up with some kind of communication when you do remove your previous attentions. This was a previous reinforcer to her and like most animals when faced with a behavioral extinction, she will attempt to re-establish that reward. This is why if you do end up cutting all contact with her she will have a tendency to pursue - and depending on the individual sometimes more actively pursue - your attention, even if she has no intent of becoming intimate with a guy. Women do this in an effort to maintain self-affirmation (i.e. she wants to verify everything is 'OK' between you and her in an email or IM) after a rejection.
The problem is when a guy begins to doubt her seriousness in her rejection when this occurs. We always read guys on this forum state that a fellow ought to "stick to his guns" in situations of rejection and this becomes more difficult when she confuses him with an unexpected burst of spontaneous attention. It's the stripper effect only more personal. Guys will spend small fortunes on lapdances at the strip club because it provides him with something he's not ordinarily accustomed to - spontaneous feminine attention. Similarly, when a woman does a follow up to a LJBF rejection after a takeaway the reaction is like that for men. Maybe she does actually like him afterall? Maybe he does have a shot with the stripper in his lap, she's giving him confusing signals in either instance.
