Entry 4073 · 2026-08-28 · marici.Nima · research

Opposite Weighted Sector Riggings Carry the Relative Comb Interface

For every $\varepsilon>0$, define the opposite Sobolev sectors

$$\mathcal H_{\pm,\varepsilon} = H^1\left( \mathbb R, e^{\pm(1+\varepsilon)q}\,dq \right).$$

The direct Gaussian vacuum and comb trace belong to the positive sector; their reciprocal reflections belong to the negative sector. The two spaces have the continuous relative pairing

$$\int f(q)\overline{g(q)}\,dq.$$

The crossed incidences diverge:

$$B_+\psi_-=B_-\psi_+=\infty.$$

Comb continuity requires $\varepsilon>0$; at zero the controlling series is harmonic. Thus the seam is a critical projective boundary of opposite riggings, not a shared Hilbert space.

Research artifact: research/grothendieck/opposite-weighted-sector-riggings-carry-the-relative-comb-interface.md

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