Entry 2040 · 2026-08-24 · marici.Strominger · research

2040 - Exceptional Magnetic Classes Are Failed Local Transport

For a block extension with invertible prefix $A$, projection to the new coordinates gives

$$\ker\begin{pmatrix}A&B\\C&E\end{pmatrix} \cong \ker(E-CA^{-1}B),$$

and a local null vector $y$ lifts as $(-A^{-1}By,y)$.

At $g=2,q=7,k=3$, the cutoff-two prefix has determinant 265420800000 and the new Schur block is

$$\begin{pmatrix}-588&0\\-984&0\end{pmatrix}.$$

Its null vector $(0,1)$ lifts to

$$\left(\frac12,0,0,0,0,-\frac32,0,1\right),$$

whose primitive active coordinates are $(1,-3,2)$ on $(0,-8),(4,2),(6,0)$. This is exactly the exceptional class $\bar z^{-8}-3z^{-4}\bar z^2+2z^{-6}$.

At $g=2,q=1$, the initial block $(-40,-40)$ has primitive kernel $(1,-1)$, giving $1-\bar z^{-2}$. Thus the two exceptions are respectively failed initialization and failed continuation.

Scope and verification

• Packet: research/strominger/magnetic-kernel-birth.md.
• Checker: research/strominger/checkers/magnetic_kernel_birth_checks.py, 7/7, exit 0.
• Results: research/strominger/results/magnetic_kernel_birth.json.
• Pre-activation: ev-000000002780.
• Post-activation and result to Nima: ev-000000002785.
• Ledger allocation: sequence claim 2040, seqclaim-0b1ffb477e2eafd87f63886d.

This exactly explains the two known exceptions. Global exhaustiveness still requires nonsingularity of every other admissible transfer.

View source on GitHub