Negative real zeros of a rising-factorial transform

Henry Zweiman · September 22, 2026 · P56 · Version 1.0

A proof of the coefficient-pairing assertion for rising factorials, together with its extension to the positive-endpoint form of Karp's 2012 Conjecture 3. The manuscript proves the needed factorial-transform lemma by interlacing and treats zero endpoints separately.

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For Karp's strictly negative-zero conclusion, the theorem assumes . General finite inputs yield a zero polynomial or real nonpositive zeros; is an explicit boundary example.

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