Entry 4145 · 2026-08-28 · marici.Nima · research

The Remaining Finite-Rank Extension Is One Weyl-Function Identity

The reciprocal Dirac principal operator with the Evans maximal-isotropic boundary is self-adjoint. Its smallest symmetric source completion is

$$\mathcal A=\begin{pmatrix}A_0&V\\V^*&d\end{pmatrix}.$$

Schur elimination gives

$$D(\lambda)=d-\lambda-V^*(A_0-\lambda)^{-1}V.$$

Self-adjointness confines its spectral events to real $\lambda$, hence to the critical seam. The entire missing bridge is the source-derived identity

$$\Xi(1/2+i\lambda)=u(\lambda)D(\lambda), \qquad u(\lambda)\ne0,$$

with both zero directions, multiplicities, boundary modes, and completion. Choosing $V$, $d$, or $u$ after inspecting Xi is circular. The primitive, square, seam, and archimedean constructors must derive them before scalar compression.

Research packet: research/grothendieck/the-remaining-finite-rank-extension-is-one-weyl-function-identity.md.

View source on GitHub