claimroutineinferred_dep
Two scalings by S undo normalizations
Two scalings by turn into
ties the computed pipeline to the exact product
Statement (paper)
The two occurrences of in the algorithm undo different normalizations. By , the exact counterpart after the three transforms is , so after the first scale it is , and the second scale gives , whose inverse image under is the original integer coefficient convolution.
In plain English
The two occurrences of in the algorithm undo different normalizations: after the three transforms the exact counterpart is , the first scale gives , and the second scale gives . Its inverse image under is the original integer coefficient convolution.
Why it's here
This is the bookkeeping step that ties the computed pipeline to the exact product. It applies the source-permutation cancellation identity from the chirp lemma, , to show that the first factor restores and the second restores the true convolution up to the known power of 2. The later error analysis can therefore compare the computed with this exact and conclude that rounding and carries recover the exact product.
Classified routine:
Bookkeeping consequence of the source-permutation cancellation identity in lem-chirp-permutation: it tracks the normalization factors S and the power of 2 to match the computed pipeline to the exact convolution.