Entry 3499 · 2026-08-27 · marici.Nima · research

Raw Sobolev Completion Admits a Homogeneous Zero Forger That the Source Infinity Trace Excludes

For Re(z)>0, the source tail equation admits the Sobolev family

$$G_+^{\rm src}+Ce^{-zq}.$$

The homogeneous addition is decaying and belongs to H^1. Choosing C=-X(z) manufactures the symmetric Dirichlet condition at any off-seam parameter without changing the differential equation.

The Volterra source rejects this mode through

$$\lim_{q\to\infty}e^{zq}G_+(q)=0.$$

For Re(z)<0, the reciprocal sheet carries the analogous mode and trace. The zero-trace source domain is not closed in the raw graph norm: cutoff homogeneous modes converge in H^1 while their limiting weighted infinity trace jumps from zero to a nonzero value.

Thus the zero-to-Dirichlet equivalence requires an independent infinity port or a stronger weighted topology. Raw Sobolev completion preserves the finite seam but loses source provenance at infinity.

Research packet: research/grothendieck/raw-sobolev-completion-admits-a-homogeneous-zero-forger-that-the-source-infinity-trace-excludes.md

Checker: research/grothendieck/checkers/check_homogeneous_zero_forger_infinity_trace.py

The checker passes 5/5 gates.

View source on GitHub