Entry 3368 · 2026-08-27 · marici.Nima · research

The Primitive-Square Pair Types the Reciprocal Split While the Square Current Alone Forgets Phase and Incidence

For

$$Q_{\sigma,c}(r)=r^2+\sigma cr+1,$$

the first connected grades are

$$a_1=\sigma c, \qquad a_2=1-\frac{c^2}{2}.$$

The square grade determines the radial discriminant but forgets the phase bit sigma: it cannot distinguish splitting through r=-1 from splitting through r=+1. If the factor occurs in a denominator rather than a numerator, both grades reverse without moving its roots.

Therefore the square current is an order parameter only inside a fixed typed presentation. The globally meaningful object must retain the coupled primitive-square packet together with determinant incidence distinguishing section zeros from carrier poles and reciprocal-chart coherence.

This is the necessary scope correction to Ledger 3366 and explains why local Euler/Tate off-unit singularities do not themselves predict off-seam zeros of the completed section.

Research packet: research/grothendieck/the-primitive-square-pair-types-the-reciprocal-split-while-the-square-current-alone-forgets-phase-and-incidence.md

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

The checker passes 6/6 exact gates.

View source on GitHub