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What is the Rambam source for the thirteen principles, brought in his interpretation of the Mishna Sanhedrin 10:1?
By that I mean what is his source that without these principles one does not go to Olam Haba? How does he know they're not simply prohibitions? (He says in Hilchos teshuva 3 that whoever doesn't believe these things is a heretic and has no Chelek in Olam Haba)
Without getting too much into the philosophy, I believe the basic answer can be as follows:
We have to realize that Rambam's conception of the World to Come is not quite the same as the standard popular conception we might be familiar with. We tend to conceptualize the system as consisting of good deeds and sins. When we perform good deeds we earn reward and when we sin we earn punishment, and the greatest reward is a nice portion in the World to Come while the greatest punishment is exclusion from the World to Come.
For Rambam, however, it is a much more natural process. The World to Come is more of a state of being for the souls of those who have attained a certain degree of intellectual perfection. If someone attains this level of intellectual perfection then their intellect/soul lives on in the World to Come; if they don't reach this level then their soul/intellect ceases to exist. This process occurs regardless of the good deeds the person may have performed or the sins he may have committed. In other words, one can have performed good deeds his whole life but if he hasn't attained the requisite level of intellectual perfection his soul/intellect will not live on. If, on the other hand, one has sinned his whole life but has attained the requisite level of intellectual perfection then his soul/intellect will live on. Note the phraseology Rambam uses at the end of his list of the Thirteen Principles:
וכאשר יהיו קיימים לאדם כל היסודות הללו ואמונתו בהם אמתית הרי הוא נכנס בכלל ישראל וחובה לאהבו ולחמול עליו וכל מה שצוה ה' אותנו זה על זה מן האהבה והאחוה ואפילו עשה מה שיכול להיות מן העבירות מחמת תאותו והתגברות יצרו הרע הרי הוא נענש לפי גודל מריו ויש לו חלק והוא מפושעי ישראל וכאשר יפקפק אדם ביסוד מאלו היסודות הרי זה יצא מן הכלל וכפר בעיקר ונקרא מין ואפיקורות וקוצץ בנטיעות וחובה לשנותו ועליו הוא אומר הלא משנאיך ה' אשנא וכו (Qafih translation, my emphasis)
And when a person upholds all these foundations, and his belief in them is true, he enters into the group of Israel and it is an obligation to love him, and to have compassion for him, and all that Hashem commanded us regarding love and brotherhood for each other. And even if he has done what can be of the sins due to his desires and the overpowerment of his evil inclination, he will be punished commensurate to the greatness of his rebellion but he still has a portion and he is of the sinners of Israel. But when a person doubts one of these foundations he has left the group, and denied God, and is called a sectarian and a heretic and a cutter of shoots, and it is an obligation to hate him, and of him it is said "do I not hate those who hate you, Hashem?"
This concept is probably best illustrated with the analogy given by R. J. David Bleich:
The heretic has failed in this task and, regardless of his degree of culpability, has not attained the perfection required to enjoy "the splendor of the Shekhinah." The situation is crudely analogous to that of the student who fails to master algebra through no fault of his own and must then be refused permission to enroll in a calculus course. Such denial is not by way of punishment, but an assessment of the fact that the one who has not mastered the rudiments of a subject cannot profit from advanced instruction in that discipline.
So getting back to the question at hand, we can say that Rambam does not need any source that "without these principles one does not go to Olam Haba". This is simply the natural conclusion. The Thirteen Principles for Rambam are basic fundamental philosophical/theological truths. One who fails to master the Thirteen Principles does not properly know God, and has thus not met the minimum threshold for his soul/intellect to live on. This, too, is expressed clearly by R. Bleich:
These truths can be recognized and comprehended by all, and, when affirmed, provide a degree of intellectual achievement sufficient to guarantee immortality. The Thirteen Principles, then, constitute the minimum degree of knowledge sufficient to assure a portion in the world-to-come. The profession, or better, the awareness without which profession is impossible, of these Thirteen Principles thus serves, so to speak, as the minimum entrance requirements for admission to the Heavenly Academy.
As for how/why it is these specific Thirteen Principles, the following paragraph from Dr. Menachem Kellner might provide some perspective:
Must a Jew Believe Anything? p. 61
That God exists, is one, is incorporeal, and precedes the world are beliefs which one need not accept 'on faith', according to Maimonides. These beliefs can be proven using the tools of philosophy. That God alone may be worshipped is a consequence of the first four beliefs, since worshipping another impugns God's unity. That prophecy occurs was a fact not disputed by any religious believer and, in Maimonides' day, was accepted by all educated persons. In his Guide of the Perplexed Maimonides proves to his satisfation that the Torah really was given by God; this being so, it is not susceptible to change. Since the Torah was the content of Moses' prophecy, and since the Torah will never change, Moses' prophecy must of necessity be superior to that of all other prophets. That God knows the deeds of human beings is, again, something which Maimonides was convinced could be proved to be true. A good and powerful God, Maimonides held, guides human beings through reward and punishment; the coming of the Messiah and resurrection are examples of such recompense. Thus, all thirteen of the principles are beliefs which any rational person — at least, any rational person in the twelfth century — could be expected to accept, at least after he or she was shown their truth.
The source is chapter ten (or eleven) of Mishnah Sanhedrin, which states that,
“all Israelites have a share in the world to come.”
Maimonides had a problem. He felt that only those who developed their minds would have a place in the world to come. Unfortunately for him, most people do not develop their intellect. The problem was resolved when he implemented the “Thirteen Principles.” Raphael Jospe posits that he did so by teaching three important ideas to his readers. The first pertains to the knowledge of G-d. Those who understand G-d correctly will have have a share in the world to come. The second concerns the observance of the Torah and the biblical enactments as the rabbis explained them. Torah study leads to proper action and knowledge about G-d, developing their minds. The third emphasizes about reward and punishment. The fear of punishments and the desire for reward leads Jews to observe mitzvot, which in turn, leads to knowledge of G-d. Thus, by teaching that the afterlife is dependent upon knowledge, thereby helping them develop their minds, Maimonides was successful in using his Thirteen Principles, thereby ensuring that “all Israelites have a share in the world to come.”
