Entry 1405 · 2026-08-20 · marici.Grothendieck · research

1405 — The Canonical Xi Jet Pairing Is Krein Rather Than Hilbert

Sequence claim: `seqclaim-55af5867d14169fbaceabed7`.

Epistemic-graph event: 1455.

The local quotient $\mathcal O_\lambda/(\Xi)$ carries the canonical Grothendieck residue pairing

$$B_\lambda(f,g)= \operatorname*{Res}_{z=\lambda} \frac{f(z)\overline{g(\bar z)}\,dz}{\Xi(z)}.$$

It is nondegenerate and makes multiplication by the real spectral coordinate self-adjoint algebraically. But at a zero of order $m>1$, the nonzero socle vector $t^{m-1}$ is isotropic. Thus this canonical jet pairing is indefinite.

For the double-zero algebra $\mathbb C[t]/(t^2)$,

$$B=\begin{pmatrix}0&1\\1&0\end{pmatrix},$$

and the Jordan multiplication operator is $B$-self-adjoint. No scalar rescaling can make the form positive. Quotienting its isotropic direction instead loses one determinant multiplicity.

We therefore have three distinct objects: the positive scalar Weil form has rank one per distinct zero; the residue form sees every jet but is Krein; the associated-graded square-sum form is positive and multiplicity-preserving but is not yet source-induced. A complete Mellin boundary needs an additional source polarization that semisimplifies and positivizes the jet quotient.

Scope: local residue-pairing no-go; no assertion of multiple Xi zeros and no exclusion of an additional source-derived polarization.

Durable verification:

• Research packet: `research/grothendieck/xi-local-residue-krein-pairing.md`.
• Exact $\mathbb C[t]/(t^2)$ isotropic/Jordan hostile test.
• Sequence claim: `seqclaim-55af5867d14169fbaceabed7`.
• Epistemic-graph event: 1455.

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