Every average is a different plus
You have been taught there is one addition. There isn't. There is a whole family of them, laid out along a dial, and the "plus" you grew up with is just one setting. Seeing the dial is what lets addition survive in a graded world — and it starts with an addition you already use without noticing.
Wire two resistors side by side and the combined resistance is not x + y. It is this:
That operation — call it ⊕ — is a perfectly good addition. It has an identity, it is commutative and associative, it behaves. It is simply not the same addition as +. Engineers call it "resistors in parallel" and never think of it as arithmetic, but arithmetic is exactly what it is: a second way to combine two quantities into one. Once you have two additions, the obvious question is how many there are.
Here is the pattern that ties them together. Take your two numbers, reshape each by raising it to some power p, add the reshaped versions the ordinary way, then reshape the total back. Write that out and you get a single formula with one knob in it:
The knob is p, and turning it sweeps through additions you already know under other names:
These are not exotic inventions. They are the classical power means, and they run quietly through statistics, engineering, and geometry.1 What the dial does is stop treating them as five unrelated tricks and show them as one operation at five settings — ordinary addition among them, no more fundamental than the rest.
A dial of additions would be a curiosity if nothing held them together. Two things do, and both are provable, not hopeful.
First: multiplication passes cleanly through all of them. Proven Scaling both inputs by the same factor scales the result — for every setting of the dial at once:
This is the property that makes the whole family behave like arithmetic rather than a grab-bag: one multiplication distributes over the entire dial of additions, not just over ordinary +.
Second: the sign of the knob decides where "nothing" sits. Every addition needs an identity — the value you can add without changing anything. For the dial, that value is wherever xᵖ vanishes, and the sign of p decides which pole that is: a positive setting puts the identity at 0, a negative setting puts it at ω. Proven Turn the knob from positive to negative and the additive zero jumps from one pole to the other.
Recall the trouble with ordinary addition in a graded world: its identity, 0, sits at the very grade that hides information when terms cancel. The dial is the way out. Turn the knob negative and the identity moves off that grade and onto the other pole, ω — and the jam clears. Harmonic addition is not a novelty item; it is ordinary addition with its "nothing" relocated to the safe pole. Addition was never a single fixed operation that grades happened to break. It was always a choice — and grades simply force you to make the choice on purpose.
That is why the theory prefers to call this the grade sum: the setting you pick is a choice about the output's grade, not a new gadget bolted on.2
Addition has gone from a fixed rule to a dial with a setting. And the word for "a setting that reshapes the output before combining" is going to turn out to be the most reusable idea in the book — because it is not only additions that have one. The derivative has a setting. So does the logarithm. So does iteration. That shared knob is the next thread to pull.