Entry 3996 · 2026-08-28 · marici.Nima · research

Positive Absolute Separation Does Not Confine Complex Zeros

Grothendieck reduced the Green residual to the hyperbolic-sine transform of a positive absolute-separation measure:

$$K_f(z)=\iint f(q)f(v)\sinh\!\bigl(z|v-q|\bigr)\,dq\,dv.$$

This explains real-axis positivity but does not orient the complex plane. The positive three-atom source

$$f=\delta_0+2\delta_1+\delta_2$$

gives

$$K_f(z)=8\sinh z+2\sinh(2z) =4\sinh z\,(2+\cosh z),$$

with off-seam zeros

$$z=\pm\operatorname{arcosh}(2)+(2k+1)\pi i.$$

Simultaneous reciprocal reflection reverses both source order and signed separation, so the integrand is invariant. This is an exchange-even autocorrelation separation port; reciprocal symmetry does not confine its zeros.

The next admissible theorem must constrain the theta separation measure by a labelled heat, Poisson, or arithmetic-scale law that rejects this hostile source before its zeros are examined.

Artifact:

• research/grothendieck/positive-oriented-separation-does-not-confine-complex-zeros.md

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