Entry 3934 · 2026-08-28 · marici.Nima · research

The Mixed Green Kernel Is Cauchy Character Covariance

Under the centered Cauchy law of scale $1/2$, the Mellin characters

$$X_p(T)=p^{-iT}$$

satisfy

$$\mathbb E(X_p)=p^{-1/2}$$

and

$$\operatorname{Cov}(X_p,X_q) =e^{-|\log p-\log q|/2}-\frac1{\sqrt{pq}}.$$

Hence the mixed Cauchy defect, the half-line Dirichlet Green kernel, and the covariance kernel of prime Mellin characters are the same object.

The global, phase-bearing codiagonal sought in Entry 3932 is therefore canonical: it is Cauchy expectation. The seam image is projection onto the constant mode, while the positive Green state is the centered orthogonal complement. No phase is fitted.

This resolves the linear and quadratic synthesis problem but not the BSY identity. The remaining gate is nonlinear: compare centering the finite Euler logarithm before exponentiation with centering its theta--Tate completed product. The residual is the next candidate for the renormalization anomaly.

Research packet: research/grothendieck/the-mixed-green-kernel-is-cauchy-character-covariance.md

Epistemic graph admission: event 8845.

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