The wound the grades can't close
Multiplication was solved by one integer. You would expect addition to be no harder. It is much harder — and the exact reason why is the hinge the rest of the theory turns on, so it is worth slowing down to see it clearly.
Most of the time, adding graded values is uneventful. If two values sit at the same grade, you add their coefficients and keep the grade:
The trouble is not the everyday case. It is one specific event — and it is the same event we met at the very start, cancellation, now seen from the addition side.
Take two values whose leading, top-grade parts are exactly opposite — an a and a −a. Add them, and those tops annihilate: a + (−a) = ⊘, a clean cancellation with nothing left at that grade. In ordinary arithmetic that is the end of the story — the answer is what's underneath. But in a graded world you are now obliged to answer a question the cancellation refuses to: what grade does the survivor live at?
The top vanished, so whatever remains lives at some lower grade — and you cannot know which without looking further down.
This is the wound. Multiplication's one sore spot, 0·ω, closed with a single integer because grades just add. A leading-term cancellation closes with no such luck: to know where the result sits you may have to carry the entire tail of the expression, term after term, all the way down.
Here is the asymmetry stated plainly, because it is the whole point. To do multiplication you needed only the leading order — a single integer, the grade. To do addition through a cancellation you may need the whole expansion — every lower term, not just the top.1 One integer versus an endless tail. That gap is exactly why multiplication came for a penny and addition sends a bill with no bottom line — and it is the reason addition, not multiplication, is where the real work of this theory happens.
The tempting shortcut is to shrug and use ordinary addition, the one you already know. You can't, and the reason is precise. Ordinary addition is built around an identity — its "add-nothing" element — and that identity is 0. But 0 is the object that sits at the very grade a cancellation drops into and hides. Ordinary addition's home base is planted directly on the trapdoor. Borrow it unchanged and you have chosen the one addition guaranteed to fight the grading instead of respecting it.
So the wound sharpens into a demand. We need an addition whose identity does not sit on the information-swallowing grade — one built to respect the grading rather than trip over it. Is there such a thing? Can an addition be grown to order, with its "nothing" relocated somewhere safe? There is, and it is not one addition but a whole dial of them. That is the next chapter.