How can historians and archaeologists verify which ones are fake and which ones are not?

tl;dr: We can't.

Thanks for asking!

This r/AskHistorians answer was brought to you by u/JimeDorje a lazy sonnofabitch who should actually answer the question...

Ok, let me try again.

The Buddhist Canon is collectively, or more accurately "generally," referred to as the Tripitaka, which translates to the "Three Baskets of Scripture." Those three baskets are

• The Sutras

• The Vinaya

• The Abhidharma

Working backwards: the Abhidharma (Tibetan: mngon pa) we can (pretty much) dismiss right away because it deals with metaphysics. The Buddha (generally speaking) did not like metaphysics and refused to answer metaphysical questions (metaphysical by his day's standards, not ours) saying that they were basically irrelevant to enlightenment. However, his philosophy left a lot to be desired and under the doctrine of upaya (Tibetan: thabs), also known as "skillful means" later Buddhist thinkers set down their conclusions based on the Tathagata's doctrine into what has become known as the Abhidharma. The Meta-Dharma, if you will. In the Abhidharma, you'll read all sorts of texts about the mind, how it functions, how we perceive the past present and future, how karma (the law of action and reaction) works, etc. More than once it has been summed up to me that the Abhidharma is the (Tibetan: phung po lnga) the Five Aggregates and its logical extension.

The Vinaya (Tib: 'dul ba) is literally "Discipline." These are the monastic rules and codes of conduct. There are separate Vinaya for monks and for nuns and the Vinaya is roughly codified as things the Buddha's followers did, that lay people of ancient India saw, went to the Buddha and criticized, the Buddha meditated, and then banned such and such an action. Examples include handling money, having sex, having sex with animals, having sex with women, men, getting handjobs, wearing commoners clothes and not mendicant robes, eating meat thath as been slaughtered for them, etc. The general idea is that renunciate life is supposed to be, you know, renouncing the world. So participating directly in the world like any other Tshering, Dorje, or Haribhadra was tantamount to giving up the religious life. There are many many different perspectives on the Vinaya that different schools have and many have tried to either correct for or avoided talkign about. For example, the Vinaya pretty much bans all sorts of sexual arrangements, but missed out on outercourse, which has become a well-known problem in (especially Bhutanese) monasteries as elder monks have taken advantage of this loophole. Other times monastic establishments have simply gotten around things like "not handling money" by hiring lay followers to literally handle their money. For the most part, however, these are still established as the things the Buddha said, unlike the Abhidharma.

Finally we get to the Sutras, which are literally defined as the words of the Buddha. In Tibetan Sutra is "mdo" but more specifically referred to as the "bka' 'gyur," which translates quite literally "happened [to have] said" but can probably be interpreted like, "What was said [by the Buddha.]"

Now, you mention different Buddhis sects. Generally speaking, there are three divisions of Buddhism by the Buddhist Canon they derive their teachings from: the Pali Canon (holds sway in Sri Lanka and southeast Asia), the Chinese Canon (authoritative in Chinese, Korean, Japanese, and some Vietnamese traditions), and the Tibetan Canon (authoritative in traditions originating in Tibet, Bhutan, Nepal, Mongolia, and northern India). Pali being the only Indian language among the three canons, it's tempting (and often promoted) that the Pali Canon is the "original" Buddhism and thus, Theravada Buddhism the "original" variety, but reality, history, and religion, are of course more complex. That said, we can rule out the possibility that the Buddha spoke Tibetan or Chinese considering that it wasn't until the mid-first millennium C.E. that Sanskrit and Pali sources were translated into Chinese, while in Tibet two waves of translation occurred, mostly from Sanskrit in the 8th and 12-13th Centuries C.E.

These Canons do not all align and are based on what various translators were able to carry with them from India to their homelands (i.e. China and Tibet). The Chinese Canon, for example, has volumes that were translated twice, once by Fa Xian and once by I Ching, and thus will tell the same story from the same source but different translation options made (similar to the Gospels of Mark and Matthew. Similar, not the same). Some translations from Chinese to Tibetan did occur, but these were relatively few and Chinese derived Tibetan Buddhism was stamped out in a series of waves, first from Tsenpo Trisong Detsen's (alleged) mass conversions to Indian Vajrayana Buddhism over Chinese Mahayana Buddhism, and later from the Tangut Rebellion against Chinggis Khaan which... well, ended poorly for everybody.

Anyway, that leaves us the Pali Canon.

Now, disclaimer, I am not a Pali specialist. And going over a whole history of the Pali Canon (or just the Buddhist Canon in general) is unwieldy and to cover it with any sense of thoroughness is far beyond my capability. I direct you to Andrew Skilton's A Concise History of Buddhism which has a whole chapter dedicated to the Tripitaka and a whole separate one to the Abhidharma and then subsequent chapters dedicated to the later developments of Buddhist philosophy, all through their primary textual development.

What I can tell you is that the Canon was theoretically developed at the First Buddhist Council after the Buddha's Parinirvana (i.e. death) c. 543 B.C.E. Ananda, the Buddha's distant cousin and attendant was so busy taking care of the Buddha that he was unable to attend all of the Buddha's sermons, so he asked that the sermons be repeated to him at length. After his master attained Nirvana, (i.e. died) Ananda recited the entire teachings of the Buddha which were then theoretically passed on verbatim until they were eventually written down and later translated.

How Indian ascetics memorize and pass along texts is... well, a topic of intense study and unlike other traditions of religion or story telling, Indian philosophy places a lot of emphasis on language and precise memorization technique. It's remains a matter of intense academic debate how historians are supposed to consider these traditions of memorization and repitition from an etic perspective (those emic to that tradition, of course, have no problem with it).

That said, we in western traditions place a lot of emphasis on the written tradition and that said, the first scriptures that we can attest in the Buddhist tradition do not occur until centuries after the Buddha's Parinirvana. So we can no more confirm that Siddhartha Gautama said them than we can that, say, Jesus said anything attested in the Gospels, which were also written down centuries after his alleged Ascension.

This then leads to "Who was the Buddha anyway?" and "How does history and tradition interact in Buddhist and Indian tradition?" and more general ideas of the Buddha as a literary figure and "Does the Dharma matter if it all proves to be ahistorical?" On both side of the East/West divide of Buddhism there are plenty of bullshit, misattributed, or misunderstood quotes attributed to the Shakyamuni but hold little water in Buddhist tradition, or misunderstand what exactly he was saying.

One of my favorite quotes, for example, "Anger is like grasping a hot coal to hurl at someone else: you are the one who gets burnt." Tracing this quote to its source leads to an actual Sutric Pali source, specifically the in the Majjhima and Digha Nikayas,

Householder, suppose a man took a blazing grass torch and went against the wind. What do you think, householder? If that man does not quickly let go of that blazing grass torch, wouldn’t that blazing grass torch burn his hand or his arm or some other part of his body, so that he might incur death or deadly suffering because of that?

Except in the metaphor, the blazing torch isn't anger, but sensuality. A vastly different meaning.

I can highly recommend the blog Fake Buddha Quotes run by a scholar and monk who traces quotes attributed to the Buddha and Buddhist Masters across history and space to the weirdest places (often times to Alcoholics Anonymous literature, weirdly enough). If you're looking for sources that are confirmed Buddhist translations of the Tripitaka, they're not hard to find.

• In the Buddha’s Words: An Anthology of Discourses from the Pali Canon, trans. Bhikkhu Bodhi

• The Dhammapada, trans. Narada Thera

• The Sutta Nipata, trans. John Ireland

• The Udana and Itivuttaka, trans. John Ireland.

• The Middle Length Discourses of the Buddha: A Translation of the Majjhima Nikaya, trans. Bhikkhu Bodhi

• The Long Discourses of the Buddha, trans. Maurice Walshe

• The Numerical Discourses of the Buddha: A Complete Translation of the Anguttara Nikaya, trans. Bhikkhu Bodhi

• The Connected Discourses of the Buddha: A Translation of the Samyutta Nikaya, trans. Bhikkhu Bodhi

Many of these (and much more, see the r/Buddhism reading list in the sidebar) are available online and can be traced back to sources in the Pali (and others) Canon.

As for how much of it can be confirmed to have roots in an Indian Prince turned ascetic some 500 years before Christ is still under academic debate, however.