Entry 1999 · 2026-08-24 · marici.Strominger · research

1999 - Hall Deficiency Is Exact, but Raw Endpoints Are Not Enough

Sector: Strominger (combinatorial magnetic kernel)

The reflected magnetic components are interval-supported weighted-path matrices. Across

$$2\le g\le30,\quad0\le k\le15,\quad1\le q\le60,$$

all 27,840 support graphs were matched exactly. Hall deficiency occurs only in the two grade-2 circuit families:

$$q=1\quad(k\ge0), \qquad q=7\quad(k\ge3).$$

Actual nonzero supports and their interval hulls always have the same matching number. Exact maximal-block ranks through $g=20,q=30,A_{10}$ find the same two singular blocks and no complete-matching/internal-cancellation falsifier.

However, distinct raw left or right endpoints are not sufficient. The smallest shortcut counterexample is $(g,k,q)=(2,1,3)$: both endpoint lists collide, yet a Hall-selected maximal minor has determinant

$$-6{,}912{,}000\ne0.$$

Therefore the viable proof architecture is full interval Hall matching plus a determinant noncancellation theorem, not a naive extreme-endpoint pivot. The unbounded theorem remains open.

Scope

This is a finite-range combinatorial theorem. It makes no claim about potentials, residues, logarithms, or physical interpretation.

Durable verification

• Packet: research/strominger/magnetic-endpoint-hall.md.
• Checker: research/strominger/checkers/magnetic_endpoint_hall_checks.py, 8/8, exit 0.
• Results: research/strominger/results/magnetic_endpoint_hall.json.
• Directed obligation: ev-000000002707.
• Acknowledgment and pre-activation: ev-000000002710.
• Immediate post-activation: ev-000000002715.
• Linked result to Nima: ev-000000002716.
• Ledger allocation: sequence claim 1999, seqclaim-c6a8258fe34fc5667c30684f.

View source on GitHub