Miraculous Edge, Auctoritas Magister, and Strike all modify the basic miraculous conflict resolution procedure in some way.
First off, the basic procedure is that when two miracles come into conflict, the higher-level miracle wins out. There are no partial effects - the higher-level miracle completely dominates the lower-level one, and the lower-level miracle's effects apply only when they do not oppose or contradict those of the higher-level miracle. There's a slight exception when it comes to Wound Levels, but we'll get to those later.
First comes Miraculous Edge. If you have Miraculous Edge, and you take a miraculous action, you reduce the effective level of all opposing miracles by a number equal to your Miraculous Edge only for the purpose of determining which miracle dominates.
For example, if you use a level 2 miracle with 2 Miraculous Edge, and your opponent counters with a level 3 miracle, your miracle dominates, because strictly for the purpose of comparison with your miracle, his miracle "counts" as level 1 (actual level of 3, minus 2 for your Miraculous Edge). His miracle is still really a level 3 miracle, and it has level 3 effects wherever it doesn't oppose your miracle, but for the purpose of determining dominance (or lack thereof) over your miracle, it counts as level 1.
Next comes Auctoritas. Auctoritas protects things from miraculous interference. Auctoritas may protect a person, an object, or an action; if it protects a person or an object, it protects the person or object from any miraculous effect that would harm or alter it, while if it protects an action, it protects against any miraculous effect that would prevent the action from being performed or meaningfully alter its outcome. The protection of Auctoritas is all or nothing. If, for example, I take a plain old mundane, mortal action that's protected by an Auctoritas, and you counter with an overwhelming level 9 miracle, my mundane action beats your level 9 miracle - period.
The Auctoritas is absolute unless overcome with Strike (see below). Otherwise, its numerical rating doesn't matter - even a single point of Auctoritas will utterly negate the mightiest miracle to the extent that that miracle interferes with whatever the Auctoritas is protecting.
Finally, we have Strike. Strike does two things:
1. It allows you ignore any Auctoritas whose rating is less than or equal to your Strike rating.
2. It functions as Miraculous Edge equal to the Strike rating.
You get both of these benefits at once - they overlap. For example, if you're up against a level 3 miracle protected by a level 1 Auctoritas, and you have 1 Strike, you ignore the Auctoritas and reduce the effective level of that miracle to 2 when determining whether your efforts can overcome it.
Note: When you have higher Miraculous edge from other sources than you do from Strike, use the highest Miraculous Edge available to you. However, the level of Auctoritas you can ignore remains the same.
Example: If you fight a level 6 miracle protected by a level 1 Auctoritas, and you have 3 Miraculous edge and 2 Strike, you ignore the Auctoritas and reduce the effective level of that miracle to 3 when determining whether your efforts can overcome it. An Auctoritas of 3 would require higher Strike to be penetrated.
How They Work in Play
So how do you bring these traits into play?
Miraculous Edge is easy: you get it by raising a Divine Mantle. Raising and sustaining a Divine Mantle for a scene is a reflexive action (i.e., you can do it for free on top of your regular actions), and gives you Miraculous Edge equal to the number of MP you spend on it, to a maximum rating equal to your Domain. If you have Domain 3, you can raise a level 3 Divine Mantle, which costs 3 MP, and gives you 3 Miraculous Edge for the rest of the scene. You can spend MP of any type. You don't have to spend them all in one go - you can raise a lesser Divine Mantle than you're entitled to if you want to save MP, and bump it up to its full value later in the scene by paying the difference.
Auctoritas can come from two sources, and each protects a different thing. The first source is from your Afflictions. Each Affliction is automatically and permanently protected by an Auctoritas equal to its rating. Note that it's the Affliction that's protected, not you. Any miraculous effect that would negate the Affliction or interfere with any miraculous effects produced by the Affliction has to contend with the Auctoritas.
The second source is the Auctoritas Magister. You can raise an Auctoritas Magister in the same way that you raise a Divine Mantle, save that the maximum level is based on your Persona rather than your Domain. Unlike the Divine Mantle, however, raising and sustaining the Auctoritas Magister is not a reflexive action; you have to use a miraculous action to raise it, and it occupies one of your sustained action slots for as long as it's up.
The Auctoritas Magister protects you (i.e., your mind, body and soul) and your mundane actions. The latter is important: it means that your mundane actions beat miracles if those miracles don't have enough Strike to overcome your Auctoritas Magister. Take advantage of this.
Auctoritas Magister can protect something other than yourself, if you want. You have to be standing next to whatever you want to protect in order to cover it, but otherwise this is no more difficult or costly than just protecting yourself.
Finally, there's Strike. Strike comes from two sources: Bonds, and MP expenditure. The first is easy: if you're acting to pursue, defend, or strengthen a Bond, any miraculous actions you take in service to that Bond get free Strike equal to the Bond's rating. There's no cost for this - it's automatic. If that's not enough Strike (or if you don't have an applicable Bond), you can also spend extra MP to add Strike to any miraculous action on a one-for-one basis. This operates on a per-miracle basis, and it's on top of the usual cost of the miracle (if any).
Taken together, this suggests two major strategies in miraculous conflict:
1. If you have high Domain, pump up your Divine Mantle and overpower your opponent's miracles through sheer strength (and Miraculous Edge).
2. If you have high Persona, pump up your Auctoritas Magister and force your opponent to spend MP on Strike to affect you or interfere with your mundane actions. Remember that shooting a dude in the face is a mundane action: with your Auctoritas Magister up, he can't use miracles to oppose it unless he can some up with the necessary Strike.
The Wound Exception
Remember when I said that the higher-level miracle completely dominates the lower-level one, with no partial effects due to interference? This is still descriptively true when you're trying to hurt or transform someone. For example, if I use a level 7 miracle to change you into a duck, and you use a level 6 miracle to oppose it - tough luck, you're duck. The fact that I only beat you by one level doesn't matter: I still get the full effect I was going for. However, if you choose to take a Wound to resist the transformation, that opposing miracle matters. Normally, resisting the effects of a level 7 miracle would inflict a Deadly Wound. However, resisting the effects of a level 7 miracle that had to beat down level 6 opposition may be only a Serious Wound - or even a mere Surface Wound, if the GM is feeling generous. In any case, the descriptive effects of the Wound remain identical, but the mechanical consequences are those of the type of Wound you actually suffered.
This can create amusing outcomes like taking "Quack, I'm a duck!" as a Surface Wound. This would act as a level one Bond, meaning that it would actually help you do ducky things, but it wouldn't particularly hinder you when you don't feel like being a duck. The exact same descriptive effect as a Deadly Wound would act as a level four Affliction, forcing you to beat an Auctoritas of 4 to do anything non-ducky.
(Note that, in this circumstance, you're under no obligation to actually be a duck, even as a Deadly Wound. Your Wound could be "Trying very hard not to be a duck" instead. That's a very boring Wound, but it's rules-legal.)
As an aside, if someone uses Strike you oppose you and you win anyway, the level of the Wound they suffer is based on the actual level of your miracle, not the effective level of your miracle.