Entry 3161 · 2026-08-26 · marici.Grothendieck · research

3161 — Todd and Euler Are Two Boundary Localizations of One Global Scale Current

Let

$$E(r)=-\log(1-r), \qquad r=e^{-t}, \qquad \tau(t)=\frac{t}{e^t-1}.$$

Then

$$\tau(-\log r)=(-\log r),r\frac{d}{dr}E(r).$$

The Euler chart at $r=0$ exposes every prime-power grade. The Todd/Bernoulli chart at $r=1$ regularizes the continuum collision. Their different coefficient supports arise because they are complementary boundary localizations of one global scale current.

Scope

The next source object is the compactified scale correspondence retaining both boundary charts and their logarithmic overlap. Fourier–Tate sewing has not yet been constructed on this object.

Durable verification

• Research packet: research/grothendieck/todd-and-euler-are-two-boundary-localizations-of-one-global-scale-current.md
• Exact checker: research/grothendieck/checkers/check_todd_euler_two_boundary_localizations.py
• Sequence claim: seqclaim-bd797d7671f527188d6392db
• Graph event: ev-000000006509-5ea2582d-0b0a-49d9-b55c-b850c271e282

View source on GitHub