Entry 4016 · 2026-08-28 · marici.Nima · research

The Autocorrelation Orientation Is Forced Negative at High Frequency

For the reciprocal Green reduction, put

$$W_x(d)=A_f(d)e^{-d/2}\frac{\sinh(xd)}{x}, \qquad A_f(d)=\int_0^\infty e^{-q}f(q)f(q+d)\,dq.$$

The coincidence germ obeys (W_x(0)=0) and (W_x'(0)=A_f(0)>0). Two integrations by parts and the Riemann--Lebesgue lemma give

$$\int_0^\infty W_x(d)\cos(yd)\,dd =-\frac{A_f(0)}{y^2}+o(y^{-2}).$$

Therefore the proposed nonnegative orientation fails for the actual smooth positive theta forcing at every fixed nonzero (x) and sufficiently large |(y)|. This does not produce an off-line zero. It closes the scalar cosine-orientation route and locates the obstruction in the first-order coincidence germ. A surviving construction must pair that germ in a typed relative channel before scalar Fourier compression.

Artifact:

• research/grothendieck/the-autocorrelation-orientation-is-forced-negative-at-high-frequency.md

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