#### A Guide to Known Counterabstractional Phase-Transitions in Number Theory

##### or: The Set of All Numbers Which Might Be Bears

You are likely already aware of the existence of SCP-1313, but in case not, here is an overview: a set of logical operations, definable in terms of mathematics, which when completed will produce a fully grown bear. The act of completing the calculations, which are notably simple enough to complete by hand, causes what we call a counterabstractional phase transition. During such a phase transition, an object which exists in possibility space

^{1}will gain additional defining attributes (position, momentum, etc) which can only be possessed by things which exist in a physical location. The abstract object will therefore lose its abstract nature and manifest in the appropriate physical location (notably, an identical abstract object will continue to exit in possibility space. It cannot be decided with our current set of extended axioms whether this is the same object or a copy.)

SCP-1313 is, itself, a moderately pathological concept at best. However, it has always been the firm opinion of the HPCD that where there is a bear, there may possibly be lions and/or tigers in addition. For this reason, and due to its various attributes which are extremely annoying to mathematicians^{2} SCP-1313 has been the subject of intense scrutiny by HPCD number theorists.

SCP-1313 (and the similarly annoying SCP-033) are what we call "mirage numbers" in anomalous number theory.^{3} A mirage number is, under normal circumstances, unreal^{4}. We did not "miss an integer", and the square of a prime number cannot be "neither even nor odd". The key to a mirage number's existence is in the calculations used to determine it (this is why said calculations tend to be unusually round-about, and more direct methods often lead to inconsistent results). The calculation process itself is essorhetorically active and inserts the mirage number into local reality at the appropriate point in the number line. Essentially, when you find a mirage number, it is because you have convinced the universe it must exist in the process of finding it. Most mirage numbers decay very quickly, existing just long enough to, say, manifest an adult bear. SCP-033 is extremely unusual in its high level of stability and self-propagating nature; it is the prevailing opinion of the HPCD that it was originally produced intentionally as a weapon of some sort.^{5}

The HPCD considers these counterabstractional phase transitions to be a form of nomenclative hazard. The calculations which lead to the relevant number, in conjunction with the number itself, amount to an elaborate way of identifying the abstract object. The act of naming imparts additional features to the object and it manifests as mentioned above.

The principle goal of SCP-1313 research has been finding a way to identify similar abstract objects and determine their properties *without* manifesting them. The universe contains a very large number of objects which, were they to suddenly manifest anywhere in the vicinity of earth, would bring an abrupt and unceremonious end to human life. As modern computing technology has vastly expanding the scope of calculations which can be performed, assessing the risk of accidentally manifesting such an entity was considered a priority.

While the research was arduous and involved the manifestation and subsequent demise of an inordinate number of bears, Site-89's HPC team was eventually able to devise a method of searching for bear numbers.^{6} "Bear numbers" are numbers in the "bear set", which is short form of "the set of all numbers that might be bears". The construction of the bear set cannot be described without going through most of the calculations involved in SCP-1313 and it is quite easy to accidentally complete the calculations from that point. As I'm sure neither of us wishes for a sudden ursine encounter, I will therefore refrain from further describing the construction.[[/footnote]] Fortunately, from this point, determining the effects of each object in the bear set proved trivial, as the numerical identity of each object is the same as its identity when manifested. This sounds nonsensical, of course, but such things are the norm when dealing with mirage numbers. Within the distorted reality produced by the bear set's construction, it becomes obvious that, for example, SCP-1313 is both a number and a literal adult bear.

With this brief introduction out of the way, the following is a representative list of entities within the set of all numbers that might be bears.^{7} The point at which calculations are performed is described as the origin. Momentum is relative to the origin. Adjacency refers to how likely the calculations are to be encountered while doing various types of math.

Identity | Risk | Adjacency | Notes |
---|---|---|---|

An adult male giraffe. Manifests three light years away from the origin, travelling in a random direction at 33% lightspeed. | Notice | Very High. Easy to accidentally manifest while preforming quartic algebra. | Many of these have likely been manifested over the course of human history. |

SCP-1313 | Caution | Medium | None. |

A squid. Manifests ~20 meters above the origin (orientation determined by gravitation field) with no momentum. | Caution | Low. | Recommend not doing too much calculus outdoors. |

A single proton. Manifests in a random location within π light-seconds of the origin, moving directly toward the origin at a random velocity between 99 and 100% of the speed of light.^{8} |
Notice | Medium-high. | While it's technically possible, this is statistically unlikely to account the Oh-My-God particle or any similar phenomena. |

A small potato composed mainly of isotopes of element 121. Manifests within five meters of the origin. | Critical | Negligible. | Theoretical only. Manifestation of the base object would be extremely destructive. Attempts to manifest a much much smaller derivative in a controlled environment to study decay chains of element 121 are under consideration. |

An adult male sun bear. Manifests within about a kilometer of the origin, moving in a random direction at 3% lightspeed. | Danger | Low. | This one is on the Reimann Hypothesis watch-list, although the calculations include several cognitohazards, so it's unlikely a civilian would be able to complete them. |

Approximately one trillion adult lions, Manifesting in random locations which do not intersect solid objects, including other lions, within about half a light second of the origin and with no momentum. | Critical | Very low. | I told you there'd be lions. |