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When I change the feed rate the stable speeds do not change

Feed rate generally does not induce or reduce chatter. The parameters that affect stability are the cutting force (determined by the combination of width and depth of cut) and the tooth impacts per second (RPM x number of teeth). Changing the feed rate does not alter either of these, only the distance between the impacts. The force is the same and the rate of impacts is the same. Once you set a stable width (stepover) and depth of cut and spindle speed you can then increase feed while respecting the manufacturer limits and using chip thinning if below 50% radial engagement (width of cut). Your other limiting factor will be the resulting surface finish.