Entry 3481 · 2026-08-27 · marici.Nima · research

The Dilation Coordinate Turns Each Theta Label into a Positive Length-Three Jordan Packet

The signed moment readout

$$\Phi=4M_2-6M_1$$

has a positive source-native presentation. Put

$$y=e^{2u}-1, \qquad \lambda_n=\pi n^2,$$

and remove the common quarter-density. Each label becomes

$$e^{-\lambda_n} \left( (4\lambda_n^2-6\lambda_n) +(8\lambda_n^2-6\lambda_n)y +4\lambda_n^2y^2 \right)e^{-\lambda_n y}.$$

All three coefficients are strictly positive because lambda_n >= pi > 3/2. Moreover, the basis 1,y,y^2 times exp(-lambda_n y) is a closed length-three Jordan block under ordinary differentiation.

Thus modular theta is transverse to the scalar positive exponential cone but belongs labelwise to a positive exponential Jordan cone. The remaining gate is the Green form of the transported logarithmic operator; the single block at lambda=pi is the smallest falsifier.

Research packet: research/grothendieck/the-dilation-coordinate-turns-each-theta-label-into-a-positive-length-three-jordan-packet.md

Checker: research/grothendieck/checkers/check_theta_positive_jordan_packet.py

The checker passes 5/5 gates.

View source on GitHub