RefacteA useful change of coordinates can remove an interaction term. Sometimes it works through the first few orders and then encounters a term that no further adjustment can remove. Calculating that boundary is itself a successful refactor: impossibility acquires a visible mechanism.
1. Ask for coordinates in which multiplication adds
Write a quantity near one as 1+x. Multiplying two such quantities gives
Thus the corresponding operation on the coordinates is F(x,y) = x+y+xy. The interaction is the term xy. Seek a formal power series
such that L(F(x,y)) = L(x)+L(y). Its linear coefficient is one, so it would be an invertible formal change of coordinate. Everything here concerns coefficients of series; no convergence or choice of logarithm branch is involved.
2. Determine the correction from the defect
Call L(F(x,y))−L(x)−L(y) the defect. Expanding through total degree three gives
The first coefficient tells us what to try. Where two is invertible, choose c = −1/2. Where three is also invertible, the next coefficient asks for d = 1/3. These are the first terms of the familiar series for log(1+x). The coefficients are forced by the interaction we want to remove.
Now work over a field of characteristic three, so 3 = 0 and 2 = −1. The quadratic equation becomes 1−c = 0, giving c = 1. The cubic coefficient then equals −1, whatever d is. We can repair the first defect, but the next one is beyond this correction.
3. See exactly why the next adjustment cannot help
Take the repaired coordinate L₂(x) = x+x². Its full defect is
The leading remainder is the nonzero cubic polynomial −xy(x+y). Add a candidate correction dx³. In characteristic three, cubing a sum introduces no mixed terms:
The cubic correction affects degree six, not degree three. Every correction of degree at least four can affect only terms of degree at least four. None can touch the cubic remainder. Therefore no series with the prescribed linear coefficient turns this multiplication law into addition.
This argument separates two questions: the quadratic repair exists, and the complete coordinate change does not. Calling the denominator three “undefined” would have identified a problem with one formula. Comparing the defect with every available correction proves the stronger impossibility.
4. Check the obstruction through repeated multiplication
Three successive copies of x under the law F give x³, because (1+x)³ = 1+x³. Under addition, three copies of anything give zero. A coordinate respecting the two laws would therefore satisfy L(x³) = 3L(x) = 0.
But the leading term of L(x³) is x³. It cannot be zero. This independently exposes the same incompatibility between the laws.
5. A failed repair becomes a method
We have replaced the vague search for a clever substitution with a sequence of concrete questions: compute the defect, determine what the next coefficient can change, and compare the remaining terms. Both a successful correction and an obstruction are explicit outputs.
The full study of the correction organises these calculations by polynomial weight. It then explains the spectral-sequence differential that records the obstruction, and how the same cubic term contributes to a new addition law. Those constructions extend the coefficient calculation; they are not prerequisites for seeing why this coordinate repair stops.