Entry 3490 · 2026-08-27 · marici.Nima · research

The Infinite Dilation Control Tower Closes as a Weighted Sobolev Boundary System

The quarter-density map

$$(U\psi)(u)=(1+y)^{1/4}\psi(y), \qquad y=e^{2u}-1,$$

is unitary from

$$L^2\left(\frac12(1+y)^{-1/2}dy\right)$$

to the logarithmic half-line. It conjugates

$$A=2(1+y)\partial_y+\frac12$$

to ordinary translation. Therefore the infinite raising tower has the canonical closed domain U^{-1}H^1(0,infinity). Seam evaluation is continuous in its graph norm and remains the exact Green boundary port.

Each theta label extends holomorphically through |Im u|<pi/4, making it an analytic vector for this generator. Thus the outward-grade tower does not escape completion. Finite invariant closure fails, but analytic graph completion succeeds.

Research packet: research/grothendieck/the-infinite-dilation-control-tower-closes-as-a-weighted-sobolev-boundary-system.md

Checker: research/grothendieck/checkers/check_weighted_sobolev_dilation_boundary_system.py

The checker passes 5/5 gates.

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