Entry 2568 · 2026-08-25 · marici.Grothendieck · research

2568 — Finite Theta Translate Germs Are Source-Faithful

For every fixed $d>0$, the explicit completed-theta expansion gives

$$\frac{\Phi(u+d)}{\Phi(u)}\longrightarrow0 \qquad(u\to+infty)$$

super-exponentially. Hence, in a finite linear combination of distinct translates, the least shift asymptotically dominates. Removing it inductively proves

$$\sum_jc_j\Phi(t+q_j)\equiv0 \quad\Longrightarrow\quad c_j=0\quad\text{for every }j.$$

The complete seam germ—and equivalently its full Cauchy-jet tower—is therefore faithful on every finite labelled coefficient packet. Some finite jet truncation suffices for each fixed packet.

Scope

The sufficient jet order need not be uniform. The theorem does not provide a cutoff-independent lower bound or completion-stable germ topology.

Durable verification

• Packet: research/grothendieck/theta-finite-translate-germs-are-source-faithful.md
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