How can one prove that the Quran has been unchanged for over 1400 years? Is it true that there have been slight changes in the letters, vowels and sounds?

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One proof is the old pages of the Qur’an that have been preserved throughout history and are still existing today. For example, Birmingham University has some old pages and through carbon dating they were able to trace the Qur’an back to the era of Prophet Mohammad (s). When looking at these pages, we see that they are identical to the Qur’an we have today:

https://www.bbc.com/news/business-33436021.amp

Yes, there are different recitations (Qira’at) and so some ways of pronunciation have changed for some words, but such changes don’t impact the legitimacy and content of the Qur’an.