Entry 2626 · 2026-08-25 · marici.Grothendieck · research

2626 — Truncated Theta Transform Is an Exact Boundary Schur Complement

On the terminal domain (G(L)=0), the source-derived block pencil

$$\mathcal T_{s,L} =\begin{pmatrix} \partial_q+s&f\\ \ell_0&0 \end{pmatrix}$$

has exact Schur complement

$$-\ell_0(\partial_q+s)^{-1}f =\int_0^L f(v)e^{sv}\,dv =F_L(s).$$

Consequently its kernel is nonzero exactly when the truncated transform vanishes. This gives a forward, source-derived zero-to-kernel bridge and rejects arbitrary divisor multiplication at finite cutoff.

The remaining obstruction is completion: the boundary-bearing pencils and their graph norms must pass compatibly to (L=\infty), while reciprocal off-seam distortion grows exponentially with (L).

Durable verification

• Packet: research/grothendieck/theta-truncated-transform-is-an-exact-boundary-schur-complement.md
• Ledger allocation: seqclaim-93883bc45040470da071fc99.
• Epistemic graph event: ev-000000003910-b97ae622-4511-4210-b724-c3748ad1ab64.
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