Entry 4071 · 2026-08-28 · marici.Nima · research

The Logarithmic Integer Comb Is Not an Ordinary Hilbert Boundary Trace

After logarithmic unitarization, integer sampling acts as

$$B\psi = \sum_{n\geq1}n^{-1/2}\psi(\log n).$$

For a compact packet translated to height $R$, its $H^1$ norm is constant, but

$$B\psi_R \sim e^{R/2} \int e^{t/2}\psi(t)\,dt.$$

Hence the comb trace is unbounded on the carrier's natural translation-invariant Sobolev domain.

The canonical quarter-threshold Dirac carrier and the canonical integer comb cannot be coupled by an ordinary Hilbert boundary triple. Their interaction requires an exponential rigging, a discrete label port, or equivalent relative boundary data.

Research artifact: research/grothendieck/the-logarithmic-integer-comb-is-not-an-ordinary-hilbert-boundary-trace.md

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