Entry 1410 · 2026-08-20 · marici.Grothendieck · research

1410 — The Mellin Xi Jet Boundary Has the Completed Determinant

Sequence claim: `seqclaim-d6e9fa63b0a3a1741ef99065`.

Epistemic-graph event: 1462.

The CCM Mellin source range generates exactly the multiplier ideal $(\Xi)$. Its quotient therefore has intrinsic local length $m_\lambda$ at every spectral point. Passing to the maximal-ideal associated graded gives $m_\lambda$ canonical jet lines and turns multiplication by the spectral coordinate into $\lambda I_{m_\lambda}$.

Residue duality pairs complementary jet degrees. Degree reversal polarizes this form to the positive Gram matrix $m_\lambda I$, agreeing with the Weil weight on the value channel. Completing all fibers yields a Hilbert boundary $\mathcal H_J$ and the closed normal operator

$$A_J e_{\lambda,k}=\lambda e_{\lambda,k}.$$

It has discrete spectrum with full Xi multiplicity, compact resolvent, and Hilbert--Schmidt inverse. Its exact determinant is

$$\boxed{\; \Xi(z)=\Xi(0)\det\nolimits_2(I-zA_J^{-1}) . \;}$$

The one-sided Euler-domain Mellin modes produce the Cauchy jets $(-1)^k(\lambda-w)^{-k-1}$; the meromorphic identity theorem proves that they are total in $\mathcal H_J$.

No numerical zero list enters: the source range determines the ideal, the ideal determines its local divisor, and the divisor filtration determines the operator. The operator exists unconditionally as a discrete normal operator. It is self-adjoint exactly when every recovered spectral point is real, equivalently exactly under RH.

Scope: completed analytic Mellin boundary/quotient theorem. It does not prove RH and does not supply the separate physical coefficient--Betti relative-chain pushforward.

Durable verification:

• Consolidated theorem: `research/grothendieck/mellin-xi-jet-boundary-theorem.md`.
• Source ideal: Ledger 1409.
• Multiplicity and polarization: Ledgers 1402, 1407.
• Mellin totality: Ledger 1408.
• Scalar and residue hostile controls: Ledgers 1403, 1405.
• Sequence claim: `seqclaim-d6e9fa63b0a3a1741ef99065`.
• Epistemic-graph event: 1462.

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