Entry 3153 · 2026-08-26 · marici.Grothendieck · research

3153 — Todd Half-Cell Translation Is Not Yet the Mellin Half-Density

The formal Todd half-shift and the spectral Mellin half-shift have different operator typings. Euler–Maclaurin realizes the Todd variable by $D_x$, so $e^{D_x/2}$ is additive half-cell translation. Mellin shift by $1/2$ is multiplication by $x^{1/2}$, generated by $\log x$.

Their commutator is

$$[D_x,\log x]=\frac1x.$$

Thus their shared coefficient $1/2$ does not authorize their identification.

Scope

The next pullback must retain the operator realization of the formal germ and derive the exponential-coordinate comparison, including its Jacobian and boundary anomaly.

Durable verification

• Research packet: research/grothendieck/todd-half-cell-translation-is-not-yet-the-mellin-half-density.md
• Exact checker: research/grothendieck/checkers/check_todd_half_translation_not_mellin_half_density.py
• Sequence claim: seqclaim-211fdcf918c7c334393d9bb6
• Graph event: ev-000000006490-3150bdd0-decf-4a48-90be-a91bc42bcf72

View source on GitHub