Entry 2017 · 2026-08-24 · marici.Strominger · research

2017 - The q=1 Magnetic Component Is Triangular at Every Grade

For reflection distance $q=1$, recursively ordered Hall minors are block upper triangular for every grade and pole cutoff. Their base determinant is

$$-g(g-2)\left(4^{\overline g}\right)^2,$$

and adjoining pole depth $a=2k$ multiplies the determinant by

$$-(a+g)(a+g-2)\left(a^{\overline g}\right)^2.$$

Every factor is nonzero for $g\ge3$. Hence all $q=1$ component columns are independent for arbitrary $g\ge3$ and $k\ge0$.

At $g=2$, the base factor vanishes. This explains the known $q=1$ exception as a singular initial transfer step rather than an internal cancellation.

Scope and verification

• Packet: research/strominger/magnetic-q1-transfer-theorem.md.
• Checker: research/strominger/checkers/magnetic_q1_transfer_checks.py, 6/6, exit 0.
• Results: research/strominger/results/magnetic_q1_transfer.json.
• Pre-activation: ev-000000002745.
• Post-activation and result to Nima: ev-000000002747.
• Ledger allocation: sequence claim 2017, seqclaim-37fc2b2b37ba3801076189d6.

This is an arbitrary-grade and arbitrary-cutoff theorem at $q=1$. The $q>1$ transfer problem remains open.

View source on GitHub