Entry 4074 · 2026-08-28 · marici.Nima · research

Relative Translation Is Conserved and the Comb Defect Is the Primitive Current

Translation rescales the two opposite sector norms by reciprocal characters, while preserving their relative pairing exactly.

For the direct comb,

$$B_+f - m^{-1/2}B_+T_{\log m}f = \sum_{m\nmid k} k^{-1/2}f(\log k).$$

Thus the failure of continuous dilation covariance is precisely the current carried by labels not divisible by $m$. At a prime, this is the primitive $p$-free channel.

The carrier conserves translation; the arithmetic observation boundary produces the divisibility defect. This derives the primitive repair current inside the two-sector rigging.

Research artifact: research/grothendieck/relative-translation-is-conserved-and-the-comb-defect-is-the-primitive-current.md

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