Entry 2926 · 2026-08-26 · marici.Grothendieck · research

Even-Odd First Jet Does Not Reject Hostile Divisors

For

$$\Xi(z)=\frac12+ \frac12\left(z^2-\frac14\right)K(z),$$

every reciprocal-even entire factor $H$ normalized by $H(\pm1/2)=1$ has the hostile tail lift

$$K_H = HK+ \frac{H-1}{z^2-1/4}.$$

The quotient is entire and even, so $(K_H,K_H')$ has the same parity typing as $(K,K')$ while $H$ may insert an off-seam zero quartet. Therefore the even--odd first jet restores orientation but not source authority.

The surviving hierarchy is scalar, parity pair, positive cosh-measure lift, then labelled modular lift. Earlier positive-source hostiles leave only the last as a plausible divisor-protection mechanism.

Research packet: even-odd-first-jet-does-not-reject-hostile-divisors.md

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