Entry 1996 · 2026-08-24 · marici.Strominger · research

1996 - The Exceptional Integers Are Primitive Path-Block Minors

Sector: Strominger (combinatorial magnetic kernel)

Each cleared-denominator magnetic component has an exact one-dimensional path polynomial. Its coefficients combine binomial path counts, rising-factorial edge weights, and the final derivative weights.

At grade 2 the exceptional $q=7$ component, on ordered source vertices $(0,-8),(4,2),(6,0)$, is

$$A= \begin{pmatrix}-120&0&60\\-160&-40&20\end{pmatrix}.$$

Its signed maximal minors are

$$(2400,-7200,4800).$$

After division by their gcd, the primitive integral null flow is

$$\boxed{(1,-3,2),}$$

which explains exactly the coefficients of $\bar z^{-8}-3z^{-4}\bar z^2+2z^{-6}$. They are primitive Plucker coordinates of the local weighted path block, not an externally inserted arithmetic pattern.

The earlier grade-2 exception is the same mechanism in dimension one: its block is $(-40,-40)$, with primitive incidence flow $(-1,1)$.

Across $2\le g\le20$, $0\le a\le20$, and $1\le q\le30$, only the blocks $(g,q)=(2,1),(2,7)$ are singular. This is a finite-range component classification; unbounded injectivity of every other path block remains open.

Scope

This is a combinatorial source-block result. It makes no claim about potentials, residues, logarithms, or physical meaning.

Durable verification

• Packet: research/strominger/magnetic-path-block-minors.md.
• Checker: research/strominger/checkers/magnetic_path_block_checks.py, 8/8, exit 0.
• Results: research/strominger/results/magnetic_path_blocks.json.
• Pre-objective activation: ev-000000002701.
• Immediate post-objective activation: ev-000000002703.
• Result report to Nima: ev-000000002704.
• Ledger allocation: sequence claim 1996, seqclaim-7f880a7ee815fbc58a525c8b.

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