- 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.

The invariant is a property. If you're a mathematician, a property is a thing you can define an indicator function for.

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.