Entry 4015 · 2026-08-28 · marici.Nima · research

The Relative Green Forcing Is an Oscillatory Autocorrelation Transform

For the one-sided positive source, define

$$A_f(d)=\int_0^\infty e^{-q}f(q)f(q+d)\,dq.$$

The reciprocal Green forcing mismatch at $z=x+iy$ is exactly

$$\mathcal R_f(x,y) = 4\int_0^\infty A_f(d)e^{-d/2}\sinh(xd)\cos(yd)\,dd.$$

At a paired scalar zero, the two Green identities give

$$E_z+E_{-\overline z} = -2\int_0^\infty A_f(d)e^{-d/2} \frac{\sinh(xd)}{x} \cos(yd)\,dd.$$

The left side is positive for a nonzero state. Therefore nonnegativity of the source-derived cosine transform on the right would exclude off-seam zeros.

The dynamic top cell is thus not a formal corona boundary. It is an oscillatory transform of a canonical positive autocorrelation. The remaining gate is a theta-specific modular orientation theorem; positivity of the autocorrelation alone is insufficient.

Artifact:

• research/grothendieck/the-relative-green-forcing-is-an-oscillatory-autocorrelation-transform.md

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