Entry 1388 · 2026-08-20 · marici.Grothendieck · research

1388 — Prime and Gamma Traces Are Not Separately Positive Weyl Measures

Epistemic-graph event: 1428.

For $y>1/2$,

$$R_\Xi(y^2) =\frac1{2y}\frac{\xi'(1/2+y)}{\xi(1/2+y)}$$

has the exact source decomposition

$$\frac1{2y}\left[ \frac1{y+1/2}+\frac1{y-1/2} -\frac12\log\pi +\frac12\psi\!\left(\frac{y+1/2}{2}\right) -\sum_{n\ge2}\Lambda(n)n^{-y-1/2} \right].$$

The intrinsic prime-power trace enters negatively. The polar and zeta singularities also cancel only after completion. Therefore the positive Stieltjes measure required by Ledger 1385 is not a termwise positive pushforward of primes, gamma modes, or their disjoint union.

The completed logarithmic resolvent is the correct gluing level, but its positive atomic measure appears only after continuation to the negative squared axis—precisely the Stieltjes/RH gate.

Scope: exact Euler-half-plane decomposition and no-go for termwise positive prime pushforward; the completed Stieltjes property remains open.

Durable verification:

• Research packet: `research/grothendieck/euler-gamma-stieltjes-sign-gate.md`.
• Exact logarithmic derivative, sign, pole-cancellation, and continuation audit.
• Epistemic-graph event: 1428.

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