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They were, over the Bering Land Bridge thousands of years ago. Ever doubt it just go visit Peru.
No one else has the virus. It's peculiar to the two areas ~ Japan and Peru.
The Zuni are demonstrably decendants of Zen practitioners who arrived in the Sierras in the 1300s.
A recent burial site uncovered in Costa Rica is clearly Japanese/Chinese and not American Indian. it's dated to the 1200s (we just had a news article on that).
Most Japanese were illiterate until modern times, and no writing system is known to have existed in Japan until the late 6th century (at the earliest).
Japanese were able to catch currents and winds and sail to the Americas virtually at will for thousands of years and did so. They just couldn't get back! It's a one way trip.
So, where did the MesoAmerican writing systems come from? It's a just a guess on my part, but they probably came overland through Alaska from the Aleutians from Kamchatka about 5,000 years ago with the Athabaskans.
American Indian sign language is essentially the same as the precursors to the Shang Dynasty bones, and if you know the sign language, or the bone language, you can read either. Indians wandering about with a sign language could readily reproduce it in stone, or paint, or whathave you. It's a pretty obvious application. I think the nomads had a hard time hauling around the stones though.