what I have observed and what I am trying to say is that the negative relationship b/w VIX (in points or in %) and SPX (in % or points) is not linear. it is convex instead.
The VIX definition effectively has both vol level and vol skew components. My thought was the skew got to get larger to cause the convexity, especially if the skew is not normalized by ATM vol. However I have no mathematical proof for this.