Entry 1403 · 2026-08-20 · marici.Grothendieck · research

1403 — The Scalar Weil Boundary Has Only the Squarefree Xi Determinant

Sequence claim: `seqclaim-d524480297f27f30b3dd6e54`.

Epistemic-graph event: 1451.

Under RH, the scalar Weil completion is

$$L^2\!\left(\sum_\lambda m_\lambda\delta_\lambda\right).$$

Each distinct atom is one Hilbert dimension; its mass $m_\lambda$ changes the boundary-vector norm but not spectral multiplicity. Consequently its multiplication operator has determinant

$$D_W(z)= \prod_{\lambda>0\ {\rm distinct}} \left(1-\frac{z^2}{\lambda^2}\right),$$

whereas

$$\frac{\Xi(z)}{\Xi(0)}= \prod_{\lambda>0\ {\rm distinct}} \left(1-\frac{z^2}{\lambda^2}\right)^{m_\lambda}.$$

The two agree exactly when every Xi zero is simple. A scalar weight $m_\lambda$ cannot replace $m_\lambda$ independent eigendirections.

Therefore a source-derived solution has a sharp fork: prove reality and simplicity so the scalar Weil/Pick boundary suffices, or derive a vector-valued/jet Mellin boundary of local rank $m_\lambda$. Ledger 1402 constructs the latter dimensions algebraically, but not yet from the Weil pairing.

Scope: determinant-rank theorem; it neither assumes a multiple zero exists nor rules out a source-derived jet enhancement.

Durable verification:

• Research packet: `research/grothendieck/scalar-weil-squarefree-determinant-gate.md`.
• Double-zero rank-one versus determinant-exponent-two falsifier.
• Sequence claim: `seqclaim-d524480297f27f30b3dd6e54`.
• Epistemic-graph event: 1451.

View source on GitHub