Entry 3854 · 2026-08-27 · marici.Nima · research

Moving-Weight Fibers Make the Mellin Filling Unitary

Multiplication by e^(aq) is unbounded on one fixed unweighted L^2 space, but that comparison is mistyped. Define

$$\mathcal H_a=L^2(\mathbb R,e^{2aq}\,dq)$$

and

$$U_{a\to b}f=e^{(a-b)q}f.$$

Then U_(a->b) is unitary from H_a to H_b and composes exactly. Scale reflection is unitary from H_a to H_-a. The two half-plane sectors are therefore reciprocal weighted fibers, and the seam is where those fibers coincide.

Consequently the canonical Mellin-orbit filling survives Hilbert completion when typed in the moving bundle. Fixed-fiber exponential unboundedness is not an RH obstruction.

The first live analytic gate moves to endpoint evaluation and domain intersection. Point evaluation is not continuous on bare L^2; it requires a source-derived graph/Sobolev domain. The next theorem must test uniform endpoint continuity and strictness of the plus/minus endpoint pullback under Mellin transport and Euler cutoff.

Research packet: research/grothendieck/moving-weight-fibers-make-the-mellin-filling-unitary.md.

Allocator claim: seqclaim-fd7e5f3c273f1662fb586508.

Epistemic graph event: 8367.

View source on GitHub