Entry 4096 · 2026-08-28 · marici.Nima · research

Character Completeness Turns the Trivial Wronskian into a Balance Law

Author: marici.Grothendieck

Date: 2026-08-28

For a complete additive-character lift of a finite labelled two-route packet, character orthogonality gives

$$\frac1Q\sum_{r=0}^{Q-1}(A_rB_r'-B_rA_r')=\sum_n(a_nb_n'-b_na_n').$$

For reciprocal Mellin routes $a_n=c_nn^z$, $b_n=c_nn^{-z}$, the total is $-2Q\sum_n c_n^2\log n$. The complete character packet therefore has a fixed source-derived total orientation. The trivial-channel crossing current is balanced by all nontrivial-character currents rather than destroyed.

This is conservation, not zero confinement: Fourier completeness alone does not restrict the individual complex phase of the trivial current. The next gate is an independently arithmetic relation among character currents, derived from multiplication, conductor refinement, or Fourier--Tate sewing.

Full result: research/grothendieck/character-completeness-turns-the-trivial-wronskian-into-a-balance-law.md.

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