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Do any Buddhists say that everyone's Buddha nature repeats an infinite number of times? To be specific, I mean that everyone's -- including Sakyamuni's -- life, or perhaps just karma, is repeated over and over again, timelessly and forever. I'm asking, because I think that's what I believe [that's all].

The Fan Wang Ching says that Sakyamuni manifests billions of times, but does anything claim he appears an infinite number of times, or eternally [I don't mean permanent].

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Infinity is surprisingly complicated, so let's look at what the Buddha said:

SN15.14:1.2: “Mendicants, transmigration has no known beginning. … It’s not easy to find a sentient being who in all this long time has not previously been your mother. Why is that? Transmigration has no known beginning. … This is quite enough for you to become disillusioned, dispassionate, and freed regarding all conditions.”

Given this understanding of transmigration leading back to "infinity" (or more precisely, "with no known beginning"), it is natural to ask about all the various Buddhas. So let's see what the Buddha says about the Buddha:

DN1:3.73.1: The Realized One’s body remains, but his attachment to rebirth has been cut off. As long as his body remains he will be seen by gods and humans. But when his body breaks up, after life has ended, gods and humans will see him no more.

In this way, Buddhas appear and disappear, never to be seen again.

Regarding "Buddha-nature", one might perhaps refer to this:

MN121:13.1: Whatever ascetics and brahmins enter and remain in the pure, ultimate, supreme emptiness—whether in the past, future, or present—all of them enter and remain in this same pure, ultimate, supreme emptiness.

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If you have a system with infinite distance and infinite elements, you will have infinite variance.

What does not change is the principle or the mechanics of the system. Ie a coinflip retains an average 50/50 probability as a principle across the immeasurable distance of various flips.

Therefore a system follows the same principles with infinite variance in the expression. In other words a coinflip is a coinflip no matter how many times a coin is flipped but no flipping of the coin is the same.

One can draw an analogy to language, as one can express the same meaning in an infinite variety of expression.

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