Entry 1407 · 2026-08-20 · marici.Grothendieck · research

1407 — Degree Reversal Polarizes the Xi Jet Boundary

Sequence claim: `seqclaim-17a47868dc921385d10f4852`.

Epistemic-graph event: 1457.

At a real Xi zero $\lambda$ of order $m$, write

$$\Xi(z)=u_\lambda(z-\lambda)^m+\cdots .$$

On the maximal-ideal associated graded, the residue form in the canonical jet basis satisfies

$$R_\lambda(e_i,e_j) =u_\lambda^{-1}\delta_{i+j,m-1}.$$

Define degree reversal by

$$C_\lambda e_j=u_\lambda e_{m-1-j}.$$

Then the polarized form

$$h_\lambda(x,y)=mR_\lambda(x,C_\lambda y)$$

has Gram matrix $mI_m$. It is positive, retains every multiplicity direction, and restricts on the value line to $m|G(\lambda)|^2$, exactly the scalar Weil term.

Completing the associated graded of the source-generated ideal quotient $\mathcal H_{\leq1}/(\Xi)$ gives a boundary whose diagonal spectral operator is, conditional on RH, self-adjoint with compact resolvent and

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

This closes the semisimplification, multiplicity, and local-polarization gates without a numerical zero list. The remaining analytic gate is exactly RH: a nonreal Xi zero gives a nonreal eigenvalue and prevents self-adjointness in every positive metric.

Scope: conditional analytic source-ideal-saturated jet boundary; no proof of RH, no density theorem for the original compactly supported Mellin test family in all jet degrees, and no physical coefficient--Betti chain pushforward.

Durable verification:

• Research packet: `research/grothendieck/graded-residue-jet-polarization.md`.
• Exact double-zero calculation $2RC=2I_2$.
• Sequence claim: `seqclaim-17a47868dc921385d10f4852`.
• Epistemic-graph event: 1457.

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