Adjoining a bookkeeping variable packages all powers of I into one graded algebra. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
The mathematical object
The \(I\)-adic filtration \(A\supset I\supset I^2\supset\cdots\) records increasing orders of vanishing. Completion replaces \(A\) by compatible residues modulo every \(I^n\).
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
One explicit computation
This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.
Why the identity matters
The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.
Where it can fail
Completion and localisation answer different questions and do not commute without hypotheses. Completeness is topological data, not merely another quotient.
The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.