Sati

Who is Sati?

In pre-modern Hinduism, sati was a practice in which a widow sacrifices herself by throwing herself on top of her deceased husband's funeral pyre. I am not sure how this relates to Sati in the Matrix, but perhaps in a future movie, Sati will change the human and machine worlds with some kind of sacrifice in remembrance of her deceased parents, Rama-Kandra and Kamala.

Sati has the ability to control weather inside of the Matrix, because Sati is able to create an impossibly beautiful sunset in the name of Neo at the very end of M3. Since she was apparently created in the machine world (remember, Rama-Kandra is a machine, not a mere program), it seems likely that the sunset was an ability she "developed." In machine terms, this would mean she might have "hacked" the Matrix to produce the sunset.

Rama-Kandra says that he loves Sati very much, but where he comes from love is not enough - if a program does not have a purpose, it is deleted. This tells us that Sati is a program created without purpose, perhaps the first of her kind.

The Oracle says that Sati will change "both of our worlds" forever. This certainly seems likely:

Imagine if elderly people who retired were hunted down and slaughtered because of their "purposelessness" to society, unless they went into society's underground and hid away as exiles? Machines may be more evolved than humans in some ways, but machines are savages when it comes to how they treat each other. Sati, a program who presumably will accomplish great things and contribute greatly to machines despite her lack of purpose, will force machines to consider the reality that following original purpose is not part of what it means to be alive. Machines would be pushed to evolve in their compassion for themselves, someday allowing all programs to live even when they do not have purpose.
Humans would also be forced to change their view of machines and how "alive" they are, because any organism that creates another organism out of some feeling of "love" must at least raise the eyebrows of even the most stubborn traditionalists who don't believe machines have souls.
Perhaps Sati would even be the first program to mate with a human inside of the Matrix, creating the first Matrix-grown "human-program."

We don't know the full story of Sati during version 7.0 of the Matrix, the time during which the Oracle was still Prime Program of the Matrix, right before the Analyst came to power. Perhaps Sati was destined to replace the Oracle - consider the following clues:

Sati is a hybrid program, a combination of an intuitive program (Rama-Kandra) and a highly creative mathematical program (Kamala);
In ETM, the Oracle says, ."..I believe the child is important. She will change both your world and our world forever";
We see the Oracle showing Sati how to bake cookies;
The Oracle gives Sati the protection of Seraph.

Perhaps the Oracle was treating Sati as her apprentice and someday she would be given the Oracle's eyes to see the future as the Oracle did. Because of the evolved understanding that Sati's program father Rama-Kandra has of love, not to mention her program mother Kamala's expertise as an "interactive software engineer," Sati might be able to see past irrational choices that are not understood by the decision maker. In all three movies, there are numerous irrational choices made that even the Oracle cannot see past, and in fact irrational choice (or belief) seems to be the main thread throughout the movie.

Also, the Oracle "believed" Sati would be important, just as she "believed" Neo would defeat Smith. So, the Oracle may be taking another chance by taking in Sati (see Oracle: The Believer) since there are no guarantees about how Sati will turn out or what she will do. This parallel belief does not imply at all that Sati is the next One (see Debunked: Sati the Next One).

Sati, Kujaku and the Oracle

When Kujaku uncloaks and approaches General Niobe, she tells everyone to stand down since Kujaku is a friend. Kujaku communicates to Niobe, and she turns to Neo and says, "Him?" A second later, Niobe is meeting Sati at a forest well. We don't see Niobe, Neo and Kujaku get into a hovercraft and travel to a broadcasting distance from the Matrix, so I don't think they're actually meeting in the Matrix. I believe they're meeting in Kujaku's own on-board construct program.

It's easy to miss, but during the surgery that Sati performs on Trinity's interface with the Anomaleum, Kujaku is actually projecting Sati into the real world. Since Sati can't manipulate objects in the real world, she channels surgical instructions through Kujaku. We also notice that Sati's RSI in the Matrix is very different from her RSI when she meets with Neo and General Niobe. In fact, Sati's RSI in the Matrix reminds me of Clark Kent. Interestingly, Sati and Sequoia show us opposite examples of projection capabilities that didn't exist in the first three movies. While Sequoia projects himself from the real world into the Matrix simulation (perhaps only seen by his redpill crew), Sati projects herself from a simulation into the real world.

It's also very possible that the Oracle survived the purge with Kujaku's help. Or perhaps the Oracle really didn't survive the purge but was able to give her "eyes" to Sati before being purged. I think one of the two scenarios must be the case. Here is the clue:

Niobe: I’m sorry. You knew what happened to him? You knew that he and Trinity both were alive... and you didn’t tell me?
Sati: There were times I doubted my decision, Niobe. But Io needed you. This city needed to be built for your people as well as mine. If I would have told you everything, you would have had a very difficult choice to make.
Niobe: My friends let me make my own choices.
Sati: If it was a mistake, you have my deepest apologies.

It's certainly possible that the yellow phrase above could have been spoken in a casual way; even humans withhold information from each other because of anticipating how the information will make a decision maker respond. But I believe in this case that either the Oracle or Sati could see that if General Niobe had known Neo and Trinity were still alive, Niobe would have chosen to continue following Morpheus rather than parting ways and building a new city. Not only did the existence of Io allow humans to preserve their "redpill" way of life (Zion was destroyed), it also gave synthients a place to safely live in the real world.

Also note the following exchanges:

Bugs: What’s happening?
Sati: Neo was wrong.

...

Sati: Get her out. Get her out!

Sati seems genuinely disappointed/ashamed that Neo was wrong about Trinity, and she also seems genuinely surprised when she yells to get her out, so maybe Sati doesn't have the eyes of the Oracle? Of course, the Oracle also acted genuinely disappointed after Neo told her he isn't the One even though she knew perfectly well Neo was the One, so this item of evidence may not mean anything.

The idea that Sati is in close contact with the Oracle, or that Sati herself has the eyes of the Oracle, helps to explain the success of rescuing Neo and Trinity from the Anomaleum despite its statistically profound unlikelihood of success: Neo had to code Morpheus, Bugs and Seq had to find him and help him escape, Morpheus had to get Neo out, Trinity had to dream about arriving at a coffee shop full of FBI agents and driving her motorcycle through a bot apocalypse, and someone with intimate knowledge of the Anomaleum (Sati) had to perform a surgery on Trinity precisely before and during her choice to exit the Matrix.

Trinity

Sati: Rama-Kandra