Entry 1390 · 2026-08-20 · marici.Grothendieck · research

1390 — The Prime Cross Trace Requires an Indefinite Double

Epistemic-graph event: 1431.

A Hilbert-space Schur complement contributes

$$-B^*(D-z)^{-1}B,$$

whose scalar spectral measure has fixed sign. The prime propagator from Ledger 1389 has oscillating cut density and cannot be such a quadratic resolvent.

At finite cutoff, polarization gives the exact alternative

$$4\langle u,R_xv\rangle =\langle u+v,R_x(u+v)\rangle -\langle u-v,R_x(u-v)\rangle.$$

Thus the intrinsic-prime trace has a natural doubled realization with signature $(+,-)$. The coefficient–Betti symplectic double provides the right paired slots, but not a positive Hilbert quotient.

The missing RH content is now the descent problem: derive a canonical null subspace whose completed gamma-plus-prime quotient is positive, cutoff-compatible, and has Schur complement $R_\Xi$. Selecting the positive subspace after diagonalizing the target would be circular.

Scope: the minimal positive Hilbert block is falsified; a source-derived positive quotient of the completed indefinite double remains open.

Durable verification:

• Research packet: `research/grothendieck/minimal-hilbert-schur-prime-no-go.md`.
• Fixed-sign Schur measure and polarization calculations.
• Epistemic-graph event: 1431.

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