You're making my head hurt. Also, perhaps part of the problem is that's not how Karma works. Here are the actual rules for Karma exchange: 1. A person with good karma will increase their good karma by helping a person with good karma. 2. A person with good karma will increase their good karma by helping a person with bad karma. 3. A person with bad karma will increase their good karma by helping a person with good karma. 4. A person with bad karma will increase their good karma by helping a person with bad karma. The steady-state of Karma is enlightenment - we all break free of the wheel and vanish into the universe in a cloud of patchouli-scented bliss. It may not be a good example for invariance. That might be what got you in trouble - you flippantly picked a religious concept you don't understand perfectly and then used it incorrectly to make your point. Which - even accepting your terms, even accepting your math - I still don't understand and I have an engineering degree: WHAT is indeed an invariant of the operation? What are we defining here? An integer plus its negative integer will equal zero - so is that an invariant operation? But that's "invariant" as adjective, not "an invariant" as noun. I get the condition of invariance - the Conway glider is a great example. But in the Conway glider example, what is the invariant? This is why I gave a counter-example of catalysts - you need them for the reaction, but their mass returns to solution in the course of the process.While trivial, it is indeed an invariant of the operation as it will work for any object.
The invariant is a property. If you're a mathematician, a property is a thing you can define an indicator function for.WHAT is indeed an invariant of the operation? What are we defining here? An integer plus its negative integer will equal zero - so is that an invariant operation? But that's "invariant" as adjective, not "an invariant" as noun.
So "properties" are like "metadata" of the equation. They're not in it but they're a 2nd order quality that can be analyzed or used in other equations. They're not like catalysts, because the catalyst is actually in the equation - they're more like molarity.
So invariants are generalized from geometry. Angles being invariant under isometries is probably the ur-example. If you have a triangle, the measure of the angle A is a property of the triangle. If you translate the triangle, the image of A has the same measure as A. Another definition of a property is a one-argument predicate; it's something you can say about an object. If you know an invariant holds before an operation, you know it holds after; that's what's interesting about them. Something is only a catalyst in a particular reaction, so I would think it would only make sense to say "C is a catalysts" is a property of the reaction as a whole. My chemistry is beyond rusty though.