We all get a bit snippy at times. I'd rather ignore that and get on with this somewhat interesting discussion. (At least it beats talking about what phono stage a stranger should buy.)
Dover, I respect your knowledge, and I certainly agree with your thesis about what the stylus tip is doing with respect to the two null points. But the movement you describe is not happening on a stationary LP; it's happening on an LP moving past the stylus tip at 33.33 rpm (ideally). Everywhere at any point on this LP, the 1000Hz test tone has been encoded by a perfect cutter lathe. In practice, the stylus tip is just a point on the surface of the LP; it doesn't "know" where it was a fraction of a second before or after any particular event. How can this phenomenon change the fundamental frequency? The analogy about moving a 15-foot auto 10 feet and then thinking about how that affects its length is not a bad one for making the argument that there is no effect. This is definitely not the same as a DJ doing "scratching", which I think holmz said is what inspired him.
If you and the others are thinking that tangency to the groove per se and lack of tangency in between or before or after either null point is altering frequency, that I can understand, but I don't think that would alter the fundamental tone of 1000Hz; what it probably does do, where there is lack of tangency, is to produce distortions. Harmonic distortion would produce some frequencies that are multiples of 1000Hz, and other forms of distortion would produce odd frequencies, but the 1000Hz signal is still there. I am guessing you know this.