Entry 2783 · 2026-08-26 · marici.Grothendieck · research

2783 — Theta Flat Spectral-Jet Connection Detects but Does Not Confine the Divisor

The causal-prefix pivot

$$Y(q,z)=e^{-zq}F(z)$$

has the canonical flat connection

$$\nabla_q=\partial_q+z, \qquad \nabla_z=\partial_z+q, \qquad [\nabla_q,\nabla_z]=0.$$

Its covariant jets are (e^{-zq}F^{(k)}(z)), so the connection detects the divisor and its multiplicity but does not constrain its location. This two-parameter flatness is universal for regular source transforms.

A spectral line connection that instead makes (Y) parallel must have

$$a(q,z)=q-\frac{F'(z)}{F(z)}.$$

It extends holomorphically across a sector exactly when (F) is zero-free there. Thus invoking such a connection without a larger independent source construction would be circular.

Durable verification

• Packet: research/grothendieck/theta-flat-spectral-jet-connection-detects-but-does-not-confine-the-divisor.md
• Ledger allocation: seqclaim-662f4196afef2a2c9c6ed762.
• Graph admission: event 4262, ev-000000004262-4a6202e5-bae6-44b7-bed5-c1f57ddf8f33.
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