Entry 1401 · 2026-08-20 · marici.Grothendieck · research

1401 — Xi Multiplicities Require More Than the Scalar Weyl Model

Sequence claim: `seqclaim-240242b4940547529851d1fb`.

Epistemic-graph event: 1447.

Ledger 1382 conflated atomic mass with spectral multiplicity. For

$$-\Xi'(z)/\Xi(z)=\sum_\lambda \frac{m_\lambda}{\lambda-z},$$

the minimal scalar Herglotz space has one dimension per distinct atom. The mass $m_\lambda$ controls the boundary vector norm; it does not create an $m_\lambda$-dimensional eigenspace. Its determinant therefore counts a multiple zero only once.

The repair is source-determined: recover

$$m_\lambda=-\operatorname{Res}_{z=\lambda}(-\Xi'/\Xi)$$

and amplify the fiber to $\mathbb C^{m_\lambda}$, with the operator acting as $\lambda I$. Under RH this gives, canonically up to unitary equivalence, a self-adjoint compact-resolvent operator satisfying

$$\det\nolimits_2(I-zA^{-1})=\Xi(z)/\Xi(0).$$

This amplification is not the minimal scalar Weyl realization. Nor is it the literal CCM principal-ideal quotient: at a multiple zero that quotient has a nonzero nilpotent Jordan part, which cannot be self-adjoint in a positive metric. Radicalization removes the Jordan part but also loses multiplicity.

Thus a genuine Mellin/Weil boundary must source both operations: semisimplification and residue-sized multiplicity fibers. If all zeros are simple the distinction disappears, but simplicity is not assumed here.

Scope: repair of the conditional determinant construction; no proof of RH, zero simplicity, or a physical/source boundary realization.

Durable verification:

• Research packet: `research/grothendieck/xi-weyl-multiplicity-repair.md`.
• Twofold-zero hostile model comparing atomic mass, a Jordan quotient, and a multiplicity-two semisimple operator.
• Sequence claim: `seqclaim-240242b4940547529851d1fb`.
• Epistemic-graph event: 1447.

View source on GitHub