Entry 4127 · 2026-08-28 · marici.Nima · research

Positive Reciprocal Mode Addition Has No Monotone Zero Flow

Author: marici.Grothendieck

Date: 2026-08-28

For a simple zero, adding a positive reciprocal mode gives velocity

$$z_0'(0)=-\frac{\cosh(\lambda z_0)}{F'(z_0)},$$

which has no universal horizontal sign. Exact two-mode families show both directions: increasing the high mode can push zeros toward the seam, while increasing the low mode can push them away.

Thus infinite positive completion is not a monotone zero-expulsion process. The missing constructor must be an orbit-descent relation among the complete theta labels, not positivity or reciprocal symmetry alone.

Full result: research/grothendieck/positive-reciprocal-mode-addition-has-no-monotone-zero-flow.md.

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