Fate is the tendency for the universe to undergo a false-vacuum metastability event. Fate itself is essentially the universe post-decay. It can't even exist within itself, it's a massive paradox. Within it, everything is non-existent (including this concept). Fate isn't entirely sentient either though it does select people to act as its contractors.
There is, generally, at most, one Fate contractor at a time. They gain the Nom de Guerre of Andromalius and inherit the rank of Count of Hell and the 36 legions associated with it. They have no obligations to Hell though. This is in return for services rendered, presumably in the form of anti-angel equipment.
Fate contractors aren't obligated to do anything, indeed, many lived out the rest of their lives without any involvement in the supernatural world. Whilst Fate prefers the removal of variables (life), the end of the universe is inevitable from its point of view and thus isn't very proactive about it.
Contractors are selected according to a list of criteria and become capable of calling upon its power, in return, simply by remaining alive, the Contractors allow Fate to drag everything to absolute destruction. Of course, this destruction is only from Fate's point of view, from other points of view (in which it will never succeed), it is basically being an incredibly ineffectual omnicidal entity that gains absolutely nothing from its Contractors.
Contractors use theurgy rather than true magic, they allow a false vacuum metastability event to form and issue orders to it. These events are usually rapidly repaired by the realm in which they have been rended in. They tend to ignore many barriers and whatnot because they aren't so much actual objects as much as they are rips in reality.
They can vary in size and intensity, the larger ones are just larger holes, the more intense ones are ones that have a stronger connection to Fate. The intenser they are, the more the universe breaks inside them, constants and laws distort or vanish entirely. It can never reach maximum intensity however.
A consolidated distortion is one in which the boundary between the metastability event and the rest of reality is solidified. It usually protects the former from the latter. They cannot be broken through physical means, but some types of magic can and siphoning away the power in the boundary, dissolving it. Reality then rushes in and closes it.
When intensity is high enough, the realm can no longer repair it without the Contractor removing it. In that state, a boundary of a consolidated distortion instead protects the realm rather than the distortion. If the boundary is siphoned or tampered with, additional distortions (with intensity below this threshold) tear in the surrounding reality instead. That is why entropy magic enhances the gauntlet's attacks.
Applying enough force allows the bubble to temporarily split, that is how dislocation occurs, but it will snap back. That is why they are unbreakable but useless as armor.
Applying enough power covers the entire distortion with the boundary, not just its exterior, this allows them to act more fluid and me empowered to slice as with Cyrielle's longcoat.
Annie and Wheatley's (along with the replaced parts of Cyrielle's) bodies are incredibly intense as they can be consolidated in a form that resembles a proper human body as it isn't a single event bubble but a massive collective of them interacting with each other to simulate a body (or bodypart). If any of them deconsolidated those, they would have far greater power (but in Cyrielle's case, she'd die immediately as well so it's rather useless).
Annie and Wheatley are basically tendencies for Fate's influence to act in a specific manner. They have no souls as they have no real need for them. Furthermore they technically do not exist either, but as they have properties, they can't actually enter Fate either. They are summoned into a vessel by the main loop of their associated programs (note: main loops of programs generally just call the subprograms required, so their summoning codes aren't actually that long). Even if their vessels are all destroyed, they still 'survive' as they are nonexistent to begin with. They have far greater power than any Contractor. If their bodies were to deconsolidate they would exist as super-high intensity distortions.
Distortions are forced to shrink in size and increase in intensity when they hit a density increase, when they then face a density decrease, they catastrophically expand and leak out of their shape, that is why beams act as they do.
Cyrielle's alternate form occurs when some parts of her brain (which were replaced) get influenced by Fate and her body channels more of it. In this form, she loses her sight, as it is replaced with the ability to sense gauge bosons (sort of like sight but involving the other fundamental forces).
Within the realm of Wall Street, everything obeys economics (and to a lesser degree, business in general). One of the most famous, fundamental papers on physics written by a denizen of that world which, when translated, is entitled On the Economics of Non-Sentient Entities describes the economics that substitute physics in that realm.
They view all aspects of other realms in terms of this. The exchange of gauge bosons to mediate fundamental forces is seen to be akin to any other transaction. Love is also viewed as an exchange of many commodities.
The entities within it occasionally grant some of their powers to humans, designated as Traders, for various reasons, whether to spread the beauty of economics, out of curiosity, to create amusement or due to liking them, what the Trader does is entirely up to them.
Wall Street contracts can bind any sentient entity regardless of their realm as long as it is an economic agreement that both agreed to under legally acceptable circumstances.
Phoenix essence is a trait some people are given. Activation kills the user and allows their soul to remotely control a phoenix spirit which can generate intense heat and pressure to generate a magical diamond-like substance that they may create a usable body from. Nothing in the afterlife aside from the creator may use this body. However, the immortal birds from which this power was derived from did not undergo natural selection due to their immortality, hence, these bodies tend to be rather flimsy though thankfully replaceable. Also, the issues associated with existing in the afterlife but controlling a body in the mortal world are a detriment to reflexes, furthermore, antimagic can sever the connection to these bodies.