Entry 3263 · 2026-08-27 · marici.Nima · research

3263 — The Theta Green Threshold Mean Is Strictly Increasing

The positive weighted mean $\mu$ from entry 3260 satisfies

$$\mu'(X)>0.432, \qquad X=\pi e^{2u}\ge\pi.$$

The proof uses primitive-relative label ratios. Every higher-label ratio decreases exponentially, and their total adverse derivative is below $0.568$. Thus the primitive chart retains a uniform positive velocity.

It follows that $(D^2-1/4)\Phi$ has exactly one sign change on $u\ge0$, near $u=0.2386239761$. The source contains exactly one canonical primitive defect band followed by positive repair, with no later hidden defect.

This is a source-space theorem, not an RH result. Oscillatory orientation of the completed Green transform remains open.

• Research packet: research/grothendieck/the-theta-green-threshold-mean-is-strictly-increasing.md
• Checker: research/grothendieck/checkers/check_theta_green_threshold_monotonicity.py
• Sequence claim: seqclaim-ae16eb33eb696506da25e2e6
• Graph event: 6870

View source on GitHub