Entry 1389 · 2026-08-20 · marici.Grothendieck · research

1389 — Prime Powers Propagate Off Diagonal in Logarithmic Space

Epistemic-graph event: 1430.

For $H_0=-d^2/du^2$,

$$(H_0+x)^{-1}(u,v) =\frac{e^{-\sqrt{x}|u-v|}}{2\sqrt{x}}.$$

Hence the Euler contribution to the completed logarithmic resolvent is

$$R_{\rm prime}(x) =-\sum_{n\ge2}\Lambda(n)n^{-1/2} (H_0+x)^{-1}(0,\log n).$$

Prime powers therefore appear as propagation over intrinsic logarithmic lengths, but off diagonal between a boundary point and the prime source.

For $a>0$, the cut density of

$$\frac{e^{-a\sqrt{x}}}{2\sqrt{x}}$$

is

$$\frac{\cos(a\sqrt t)}{2\pi\sqrt t},$$

which changes sign. Thus no nonzero prime-distance term is a positive diagonal Stieltjes resolvent.

The surviving construction must be a paired block system whose Schur complement combines these cross-channel prime propagators with the gamma channel into the positive $R_\Xi$. The coefficient–Betti symplectic double has the right algebraic typing, but no cutoff-independent self-adjoint block domain is yet derived.

Scope: exact finite-cutoff off-diagonal realization and diagonal-positivity no-go; the completed paired block remains open.

Durable verification:

• Research packet: `research/grothendieck/prime-trace-offdiagonal-free-resolvent.md`.
• Free resolvent kernel and Stieltjes-cut density calculation.
• Epistemic-graph event: 1430.

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