Entry 3355 · 2026-08-27 · marici.Nima · research

Complete Strict Hankel Positivity Does Not Confine the Evans Character Orbit

Let the source measure have density 1 on [0,1] and density 2 on [1,2]. Its bilateral Laplace transform factors as

$$F(z)=\frac{e^z-1}{z}(1+2e^z).$$

It therefore vanishes at

$$z=-\log 2+(2k+1)\pi i,$$

an exact off-seam family.

Nevertheless its moments are

$$m_j=\frac{2^{j+2}-1}{j+1},$$

and every ordinary and shifted Hankel moment matrix is strictly positive definite. Indeed, their quadratic forms are respectively the integrals of p(x)^2 and x p(x)^2 against a positive density on intervals. Thus even the complete strict Stieltjes moment tower of a positive continuous carrier does not confine zeros of its exponential-character readout.

This strengthens Ledger 3352. The missing RH law cannot be a stronger carrier positivity condition alone. It must couple the positive carrier, the distinguished exponential character orbit, and theta/modular sewing.

Research packet: research/grothendieck/complete-strict-hankel-positivity-does-not-confine-the-evans-character-orbit.md

Exact checker: research/grothendieck/checkers/check_complete_strict_hankel_evans_hostile.py

The checker passes 6/6 gates and verifies exact strict ordinary and shifted Hankel determinants through order eight. The all-orders theorem follows from the positive-density integral representation.

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