Positivity and counterexamples for rising-factorial Toeplitz determinants
Henry Zweiman · September 22, 2026 · P57 · Version 1.0
A separate manuscript addressing Karp's 2012 Conjectures 1, 2, 4, 5, and 6. The earlier Conjecture 3 paper remains separate.
| Conjecture | Result |
|---|---|
| 1 | Proved for all positive shift parameters. |
| 2 and 5 | Disproved by an exact stability counterexample with a positive strictly log-concave input. |
| 4 | The literal assertion fails for degenerate inputs; a general minor expansion gives coefficient nonnegativity and exact criteria for degree and strict positivity. |
| 6 | Disproved for determinant order three using the degree-eleven binomial sequence. |
- Complete manuscript
- Nine-page PDF
- LaTeX source
- Exact verification script
- Exact certificates and results
- Literature and source comparison
- Originating-assistant audit
- Citation metadata
The counterexamples have integer-arithmetic certificates, not merely numerical root estimates. The manuscript develops the general positive expansions explicitly and credits the standard tools used. Historical priority has not been established.
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Reproduce
Run python3 checks/verify.py with SymPy installed. Run python3 build.py with Pandoc and Tectonic on PATH, or specify their executable paths through PANDOC and TECTONIC. The build writes manuscript.tex and manuscript.pdf beside the source.