Hi rws,
Welcome to Seattle. The Seattle Public School District has one of the best gifted programs because it was designed by Dr. Nancy Robinson of the Robinson Center for Young Scholars at the UW. The program is being updated and you can find the latest update on here, not sure when it'll be put in place:
http://www.seattleschools.org/UserFiles/Servers/Server_543/File/District/Departments/advancedlearning/2190SP%20FINAL%20DRAFT%208.4.16%20(2)%20(2).sbm.WJ.pdf

Briefly put, there are 2 tiers to the program: Highly Capable and Advanced Learner(formerly known as Spectrum). Both programs go from K-8. Highly capable is in self-contained classrooms offered in a number of elementary and middle schools. Advanced Learner is the lower tier where students sit in regular classrooms but get differentiated instructions. All middle schools offer Advanced Learner but only some elementary schools offer it. The eligibility for each program is as follows:

Highly Capable: 98th or 99th percentile in at least 2 categories of CogAT. 95% and up for ELA and math in the district achievement test (MAP).

Advanced Learner: 87th percentile and up in at least 2 categories in CogAT, 87th percentile and up in ELA and Math in the district achievement test.

Those who move in from another gifted program OOS can submit their cogAT scores and latest achievement test scores for consideration on a one-on-one basis, or do the summer testing.

The Highly Capable cohort self-contained program ends in 8th grade. From there they go on to an accelerated APX program at Garfield HS or accelerated IB program (IBX) at Ingraham HS.