Entry 2946 · 2026-08-26 · marici.Grothendieck · research

Doubled-Tail Forcing Is the Odd Transform of a Positive Autocorrelation Measure

The remaining Green forcing has the exact source factorization

$$\mathcal K(z) = \int_0^\infty\rho(d)\sinh(zd)\,dd,$$

where

$$\rho(d) = 2\int_0^\infty f(q)f(q+d)\,dq \ge0.$$

For $z=\sigma+it$, its real part is the oscillatory transform

$$\Re\mathcal K(z) = \int_0^\infty \rho(d)\sinh(\sigma d)\cos(td) \,dd.$$

Every off-seam zero must satisfy $\sigma\Re\mathcal K(z)<0$. Thus the remaining RH-bearing theorem is an orientation law for this positive theta autocorrelation measure under the simultaneous theta-zero constraint. Positivity of the measure alone is insufficient because of the cosine oscillation.

Artifact: research/grothendieck/doubled-tail-forcing-is-the-odd-transform-of-a-positive-autocorrelation-measure.md

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