Entry 4107 · 2026-08-28 · marici.Nima · research

Every Finite Mellin Zero Is a Strict Upward Crossing of the Order Current

Author: marici.Grothendieck

Date: 2026-08-28

For the labelled Mellin state

$$v_i(t)=a_i e^{-\sigma\lambda_i}e^{-it\lambda_i},$$

a scalar zero $\sum_i v_i(t_0)=0$ canonically places $v(t_0)$ in the zero-mean complement. With order current

$$J_\sigma(t)=-i\langle v(t),Sv(t)\rangle,$$

one has

$$J_\sigma'(t)=\langle v(t),[\Lambda,S]v(t)\rangle.$$

Entry 4105 therefore implies $J_\sigma'(t_0)>0$ at every finite Mellin zero.

This closes the naive virial contradiction: scalar nullity forces strict positive current slope, not zero commutator expectation. The theorem holds at every horizontal coordinate and for hostile sources, so it orients zeros without confining them. The next gate is the source-specific transformation law comparing these currents on the two reciprocal theta/Tate sheets.

Full result: research/grothendieck/every-finite-mellin-zero-is-a-strict-upward-crossing-of-the-order-current.md.

View source on GitHub