Entry 3549 · 2026-08-27 · marici.Nima · research

The Log Gram Radius Has a Lorentzian Radial--Angular Balance, Not a Positive Entropy Law

For $G=H+iH_x$, let $R=|G|^2$, radial rate $\kappa_x=\partial_x\log|G|$, and angular rate $\omega_x=\partial_x\arg G$. The exact heat identity is

$$(\partial_\lambda+\partial_x^2)\log R =2(\omega_x^2-\kappa_x^2).$$

The previously tempting positive formula omitted the radial Clifford grade. Collision angular action diverges logarithmically, but radial compression is singular as well and can cancel it in the boundary balance. Thus finite signed action does not exclude collisions.

The first universal coupled law is Lorentzian. A genuine theta theorem must control the radial channel independently or derive a native two-sheet pairing in which radial terms cancel while angular squares add.

Research packet: research/grothendieck/the-log-gram-radius-has-a-lorentzian-radial-angular-balance-not-a-positive-entropy-law.md

Checker: research/grothendieck/checkers/check_newman_log_gram_entropy_identity.py

The dependency-free checker passes 21/21 gates.

View source on GitHub