It is not an easy problem and not always easy to fix, but here are some remarks and suggestions:
A simple solution would be to turn off anti aliasing, it works with the file you send if you turn off Smooth line art in Acrobat Preferences.
Illustrator is a vector application that creates device independent, scalable artwork.
Unfortunately, devices have their limitations, desktop printers and computer screens are relatively low resolution pixel based devices and anti-aliasing is used to smooth artwork. This can cause thin lines that may be visible at different zoom levels. Turning anti-aliasing off will get rid of these thin lines, but can have side effects like jaggy curves. Rasterizing artwork at resolutions that are a multiple of 72 ppi and using art optimized anti aliasing can help. You can do this in Illustrator with a Rasterize Effect that is very flexible, you can always change it or disable it, it does not permanently rasterize the artwork.
To avoid thin lines, it is best to work in Points or pixels and make sure that your pattern tile dimensions do not contain fractional pixels. Internally Illustrator works in points, 72 points per inch, and the same is used to define the size of a pixel, 72 ppi. Unfortunately, units like mm, cm and meters cannot be perfectly converted to pixels and will create fractional pixels. But in the real world, there is no such thing as a fractional pixel so rounding will occur.
I your example, your pattern tile is 528,4979 pt X 522,6871 pt.
If you ever want to export a single pattern tile, the fractions will be rounded and cause additional visible pixels.
It would have been better if you would have started with a 528 X 523 pixel tile.
That does not guarantee that there will be no anti-aliasing artifacts, but they will not be visible when Smooth line art is turned of in Acrobat, when printing to high resolution printers or when exporting at multiples of 72 ppi (144, 288…) and using Art Optimised for anti aliasing.
An additional remark: Your document is using snap to pixel, that is meant for web graphics and is causing trouble with exact placement, it is better to turn it off.
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