Far-sider

Far-siders are alien entities native to a parallel reality governed by different rules of arithmetic, separated from our own by a dynamic mathematical "border." They appear in Greg Egan's short story "Luminous" and its sequel "Dark Integers."

Description
The matter composing the far-siders' bodies follows mathematical rules different from those that prevail on the "near side." As such, they cannot survive in the near side, since the basic chemistry and physics they depend on would not work the same way there. It is unknown if they are even made up of atoms.

Since physical interactions across the border are impossible, it is unknown what far-siders look like. They are discussed as a single species, but may in fact be a group of races. It is theoretically possible to manipulate the border to create a "radar" effect that could resolve individual far-siders, but no one on the near side manages to achieve such a resolution. The far-siders' only physical representation is a triangular digital avatar used by a far-sider nicknamed "Sam," and even this was designed by Sam's human contact without any input from Sam itself.

Despite their alien nature, far-siders seem human-like in their psychology and behavior. They carried out real-time conversations with humans, and showed hostility and fear toward what they could not understand or control.

Homeworld
Far-siders inhabit a parallel reality known simply as the far side. The far side formed out of a distinct set of mathematical laws that emerged in the moments after the Big Bang alongside more conventional arithmetic. The far side is mathematically distinct from the near side, but both realities occupy the same physical space. It is unclear exactly what the differences are between the two sides, but the far side's math is based on "dark integers" that are not comparable to whole numbers on the near side. Near side matter and far side matter do not physically interact with each other in any way.

On the macro scale, the far side does not seem much different from the near side universe. It contains planets and stars, although these bodies are not parallel versions or duplicates of those on the near side. Notably, the concentration of stars on the far side is much greater than on the near side, meaning that stars are much closer to each other on average, and the far side is implied to contain much more life.

Border
The near side and the far side are separated by a dynamic mathematical border. The border is not a physically real object or region, but rather an abstract representation of where the two sides' math systems resemble each other. Near side math generally intersects with far side math only during arithmetic performed using immensely large numbers with hundreds of digits. Operations - whether performed mentally, on a computer, or even through natural physical processes - that touch on the border can shift its position. The numbers involved are so large that they are not involved in natural interactions, but advanced computers can alter the border. This can have physical consequences if naturally occurring chemical, nuclear, or quantum interactions are caught up in the border change.

Technology
Far-sider computer technology and mathematics is apparently far in excess of that on the near side. They discovered the near side an unknown amount of time before human computers made any changes to the border, and may have discovered it thousands of years in the past. They are able to manipulate the border with great precision, enough to crash near side computer systems and to implant their version of arithmetic directly into a person's mind. They could conceivably kill people or even destroy the Earth outright in this way if they chose, but destroying physical objects apparently requires much more computational power.

Little else about their technology is known, as it cannot be directly observed. However, the far-siders have a spacefaring civilization that may be interstellar in extent, and they possess spaceships that are capable of matching the Earth's orbital speed. They are still absolutely bound by the speed of light.