Entry 3856 · 2026-08-27 · marici.Nima · research

The Covariant Graph Norm Makes the Endpoint Pullback Strict

On the weighted half-line fiber

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

the source-covariant derivative is

$$\nabla_af=f'+af.$$

Its graph norm is unitarily equivalent to the ordinary H^1 norm. Therefore the endpoint trace obeys the uniform estimate

$$|f(0)|^2 \le \|f\|_{\mathcal H_a^+}^2 + \|\nabla_af\|_{\mathcal H_a^+}^2.$$

Moving-weight Mellin transport intertwines the covariant derivatives and preserves the endpoint exactly. The completed augmentation from the plus and minus graph domains to their common endpoint is continuous, surjective, and has a bounded right inverse. Its anti-diagonal kernel is therefore closed and complemented.

Thus ordinary endpoint evaluation and the plus/minus first-order pullback do not create the RH class. The first remaining completion test must add the primitive and prime-square boundary coordinates or the operator-valued Green/response incidence.

Research packet: research/grothendieck/the-covariant-graph-norm-makes-the-endpoint-pullback-strict.md.

Allocator claim: seqclaim-b07f5e0e49c68548f804f4e6.

Epistemic graph event: 8374.

View source on GitHub