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So, you like many before you have spawned yet another theory linking Egypt with America, and this time it was the Mayans (again). Cocaine mummies, pyramids of stone, polytheism. That being said, let’s expand on your hypothetical scenario. 

For starters, all of these indicators listed everywhere should have linked them more to Egypt and not the Middle East. And it did a little, even the origins of Inca civilization stemming from a ruler who’s name phonetically resembled someone who was a ruler in ancient Egypt. Cyrus never made it to Egypt. But you’re talking about Mayans anyway, and the rise of the Mayans is around the same time Cyrus conquered the Middle East. Other than that, it’s very complicated to create a link to anywhere who’s ever used a cubit. 

To know of the land beyond the sea was an implausibility because the chances of return would be very slim and a very tedious process in itself. There would have to be several large vessels storing food and grain for resustinance upon arrival, and that alone would take years of premeditated cultivation along with well equipped men and women of exceptional learnings and with society structure.  Then you’d have to survive the long trip while maintaining a course often against the winds: you could actually sail that boat and wind up anywhere twice. You’re talking about a secret cubit guild who fled Cyrus: they would actually have to sail through Gibraltar, around the whole of Africa or somewhat Southeast through the Indian and Pacific Oceans bypassing pretty much everywhere ideal just to reach this land that somehow only the cubit guild had come to know about.   It’s not an impossibility but it’s highly unlikely there. Moving on....

Now we’re looking at their languages. Mayan did have a syllabic written pictogram representation, indicating a cultural revolution from a previous society with a syllabic representation. It’s not like Chinese or Linear A, but a similar concept to Egypt yet entirely different in every way imaginable. If there were linguistic similarities then someone would’ve said something but the truth is either nobody knows how Mayan actually sounds, or the similarity wasn’t at all evident.  This indicates the written and syllabic alphabet concept is older than the oldest ruin ever found and not a copycat unless this cubit guild intentionally changed their belief system and language while traveling years to hide from someone who couldn’t pursue them, leading me to believe they would’ve had no intention of preserving their former way of life in the cubit guild thereby making this endeavor pointless to begin with.  Plus these Mayan , eh...refugees made no mention that they were refugees from Cyrus or anywhere distant at all. 

Now we look at Mayan DNA. Very variant but somehow (somehow) traced to a crossing at Beringia long long ago. I can accept the fact that Eskimo/Inuit populations DNA are similar to those further south, but nobody entertains that these people might’ve just travelled North after their arrival from some other point on the continent, and nobody liked the Northwest Territory so breeding after that was a little incestuous thereby creating less variance. I won’t completely deny that even the Mayan culture was just as easily influenced by a seafaring immigration of refugees. However these indicative genetic markers found in Europe or Asia Minor were found in the Paracas, who preceded the Nazcas and Incans etc. This actually led me to believe that the Mayans were actually a product of indigenous culture evolution and not a Sumerian descendance. This fleeing band you thought of would have therefore been vastly outnumbered by an already present earlier group of societies and would’ve had no choice but to adopt the language and culture of these peoples. But there’s one fatal flaw in this scenario:

The wheel was not in the Americas, ever. They used a pampoose to drag things over the very rough landscape. This highly advanced warlike sacrificial cubit guild who crossed several oceans successfully multiple times just couldn’t manage to remember how to build a wagon let alone a chariot.  You’re actually better off entertaining the Book of Mormon explanation or alien colonization and their extraterrestrial knowledge of the cosmos. 

You’re talking about Masonic foundations made manifest in Mayan cultures.  All I can say is if all these indigenous bands were influenced by outsiders, why are there so many tribes and bands with countless dissimilar languages. They were very isolated, they had to be to become so different. Their gods were entirely different, their games were different, their system of calculations were different, their artistic expression was different except in the fact they managed to build step pyramids, but even their technology was different. However: theirs were much smaller and were used entirely different. These builders often hauled stone from nearby places, it’s apparent to me these Mayans actually used stones that were already present.  

I won’t deny that all this knowledge and society structure came from like sources, but it was long before Cyrus gave the Jews freedom of religion

Let’s say there was a ruling class in the earliest of days. Kings of Sumer, King Sargon of Akkad, Hammurabi, Kings of Persia, protoLatium, King Minos of the Cretans, pharaohs. Let’s say the earliest ruling class spread far and wide to establish their dominion over lands not yet inhabited by anyone. This would be evident in genetics and linguistics, and it is. Over time Egyptians spoke Khemite, French spoke Gaulish, and the whole time this ruling class continued to expand and eventually didn’t realize their neighbor kings were their kinsmen. So they were practicing political weddings and treaties, campaigns and wars,  establishing themselves as god kings with some divine right and birthright, and yet nobody knew they were already somehow related. So every war fought was generally not commonfolk seizing crowns and kingdoms but princes and kings and emporers overthrowing each other. Let’s say this dominion spread long before the R1b gene made manifest in Europe, this ruling class crossed into America and began to establish common dominion there too, multiple times (multiple crossings). This became Mayans jealous of Incan territory, Paracas slaughtered, Nazca assimilation into neighboring organizations. This all became god king rule all identifying some king as their being of idolizations. Such was evident in China and Australia, Easter Islanders and other Polynesians. Which would explain why all these kings even of recent times share a common genetic marker (R1b and a couple others). Wars and  revolutions destroyed lineages and birthright claims, all knowledges lost and constantly reestablished since before Dynasty One of Egypt. Let’s say this was brought upon by the end of the Ice Age, every land more accessible by crossing ice sheets and land bridges instead of by the later need for a boat. There’s an interesting theory: oceanic vessels weren’t invented because they weren’t necessary. All these lands were known because people had more easily been there, obviously before  glacial melts 11,000 years ago caused all these access points and regions to disappear, isolating everyone and destroying their records. 

It might be a bit far fetched, but when you read the bible and other ancient accounts you’ll find it’s more likely a blood related ruling class evolved to become Mayans with similar knowledge and society structure than fleeing from Cyrus the Great