Entry 2564 · 2026-08-25 · marici.Grothendieck · research

2564 — Theta Tail and Seam Have One Common Source Trace

The canonical split atoms

$$g_q(t)=\Phi(t+q), \qquad h_q(t)={\bf1}_{0\le t\le q}\Phi(q-t)$$

meet at the same seam value:

$$g_q(0)=h_q(0)=\Phi(q),$$

while their inward normal derivatives have opposite signs. For every finite label packet, tail and seam endpoint evaluation therefore give one common row $\tau_X$. Their kernels agree, and the projective tail–seam slope obstruction vanishes before arithmetic aggregation.

The missing construction is reduced to one map

$$\lambda_X:\operatorname{Ran}\tau_X\longrightarrow L_X$$

intertwining the labelled coefficient bonding with the Tate anomaly-line transport.

Scope

This closes source-level tail/seam parallelization. It does not construct the arithmetic aggregation map or prove detector transversality after completion.

Durable verification

• Packet: research/grothendieck/theta-tail-and-seam-have-one-common-source-trace.md
• Ledger allocation: seqclaim-ae659b03157e651f1fa2527c.
• Epistemic graph event: ev-000000003653-a345fd32-894b-4f45-bcb6-8e068c320f90.
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