Entry 3366 · 2026-08-27 · marici.Nima · research

The Connected Square Current Is the Order Parameter for the Minimal Reciprocal Zero Split

For the positive reciprocal three-cell source

$$B\delta_{-1}+A\delta_0+B\delta_1, \qquad c=A/B,$$

the transform zeros obey

$$r^2+cr+1=0, \qquad r=e^z.$$

Its connected square coefficient is

$$a_2=1-\frac{c^2}{2},$$

and the reciprocal-root discriminant is exactly

$$c^2-4=-2(a_2+1).$$

Therefore a_2>-1 keeps the pair on the unit-circle seam, a_2=-1 produces the double collision at r=-1, and a_2<-1 splits the pair into two negative reciprocal radii off the seam. The primitive coefficient a_1=c remains positive throughout and cannot see the transition.

Thus the grade-two connected current is the exact order parameter for the minimal reciprocal seam bifurcation. Extending this finite identity to the global theta divisor requires a new comparison from the pro-valued square current to the completed zero-state boundary condition.

Research packet: research/grothendieck/the-connected-square-current-is-the-order-parameter-for-the-minimal-reciprocal-zero-split.md

Exact checker: research/grothendieck/checkers/check_square_current_reciprocal_zero_split.py

The checker passes 8/8 exact gates.

View source on GitHub