Entry 3939 · 2026-08-28 · marici.Nima · research

The Euler Vacuum Escapes Every Fixed Mellin Graph Norm

For the logarithmic generator $Qe_p=(\log p)e_p$ and normalized Euler vacuum $v_X$, even the optimally centered derivative satisfies

$$\left\|(Q-m_XI)v_X\right\|^2 \sim \frac{(\log X)^2}{2\log\log X}.$$

Thus the primitive first-order germ cannot be restored by an ordinary Hilbert graph norm. A cutoff-dependent rescaling can normalize this variance, but the standardized third moment then diverges like $\sqrt{\log\log X}$, so it does not stabilize the higher jet tower.

Finite Mellin conjugation and Haar conditional expectation remain valid. The failure begins only at infinitesimal logarithmic transport. The primitive channel must therefore be a filtered, rigged, or distributional boundary object rather than a single Sobolev correction.

Research packet: research/grothendieck/the-euler-vacuum-escapes-every-fixed-mellin-graph-norm.md

Epistemic graph admission: event 8895.

View source on GitHub