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I write sci-fi, and I don't love fudging important things. I have a character that was a victim of genetic experiments (I know how original). I want to find a semi-reasonable way to explain this, because her developing pyrokinesis isn't accidental.

The condition Hypothyroidism, an under active thyroid gland, can cause Hypothermia or low body temperature. In an attempt to treat the symptom of low body temperature, scientist used genetic engineering to alter your character. They gave her the ability to speed up the atoms in her body to produce heat on demand to warm herself. However, your character eventually discovers that she can speed up the atoms of anything that she can see and focus on, that is located in her general vicinity. This would give her the ability to set things on fire from a distance.

You specify your character underwent genetic experiments. How does that happen in the real world? Perhaps your character was experimented on in the context of a clinical trial.

Your character might have been born with a genetic disease. There are many. Ornithine transcarbamylase deficiency is an example, and there are clinical trials looking at genetic modification of people born this way in hopes of overcoming or sidestepping the disease. https://clinicaltrials.gov/ct2/show/NCT00004498?cond=ornithine&draw=2&rank=5

Or perhaps she has a childhood cancer like a brain tumor and undergoes experimental gene therapy or infusion of her own genetically altered T cells to treat the cancer.

As opposed to "experiments" done out of malice or some perversion, you can have the experimental procedure done in hopes of helping her, and it actually worked - she was cured of her illness.

But later there are consequences. And she was not the only participant in this clinical trial...

Any scientific explanation of pyrokinesis should start with a definition. As usual, wikipedia provides...

Pyrokinesis is the purported psychic ability allowing a person to create and control fire with the mind

...from that article, a useful initial premise arises...

Parapsychologists describe pyrokinesis as the ability to excite the atoms within an object until they generate enough energy to burst into flame. Science fiction works define pyrokinesis as speeding up the movement of molecules in order to increase temperature.

So we want to create a character whose mind can excite atoms at a distance to the point of combustion. Atoms usually get excited when thermal energy, such as heat from an adjacent fire, is applied to them. Depending upon the substance being ignited, the amount of thermal energy does not need to be very high. Gasoline fumes for example, only need a spark.

So let's make the job easy for our character, beginning our explanation with some real world possible sensory enhancements which allow her to sense the presence of easily combustible compounds at a distance. Since sharks can reportedly smell a single drop of blood at many miles distance, I don't think anyone will complain if our little fire starter can smell and precisely locate her favorite volatile substances at 20-30 feet.

From there, it is just a small step to light the proverbial match. If your fictional world already has telekinesis, then the mechanism for that match already exists. Your fire starter is just a telekinetic who moves energy instead of matter. Stealing enough heat from the surrounding matter and focusing it on the volatile target is all it takes to get the process started. If your world doesn't have telekinesis, you can explain her ability away by saying that it is still telekinesis, which remains impossible at anything above the energy level.

None of this will give your character fireball throwing abilities. For that, you probably need to focus her telekinesis down below the atomic level, allowing her to split a few otherwise stable atoms to release the required heat. Giving her that much power however jumps the shark in a big way, leaving you with every little that can threaten or challenge your character later in the story. Fireballs also don't make a lot of sense strategically. Why throw it at a target when you can just create it inside their skull, bypassing their defenses.

Without the subatomic energy boost, your character will still have a considerable range of abilities, depending mainly on what combustible compounds are nearby. She will also have very useful limitations which can help in your story creation.

If you can be satisfied with a super hero who can set an enemy's clothes on fire rather than incinerate them, then these ideas should make it look like you are at least trying to play within the laws of physics. It is still not science because at its' core, it is still dependent on telekinesis which has not yet been scientifically proven. But beyond that fundamental flaw, a pyrokinetic character, created using these ideas, could be a physical-law-abiding member of your otherwise real world.

There is a scientific rationale for a person developing pyrokinesis because of a genetic experiment.

Assume that in this fictional world effectively everybody, human beings that is, have the genetic capacity (or at the very least the genetic basis for those powers) for them to have pyrokinetic powers. However, this capacity is not expressed, because of a genetic blocking factor. The genetic experiment unblocks the innate biological mechanism repressing her pyrokinetic powers.

Viola! She now develops her pyrokinesis.