Single-Site Rigidity and a Sporadic Exception for Prime-Indexed Nörlund–Stirling Quotient Polynomials

The paper continues our investigation of the arithmetic root structure of Nörlund–Stirling quotient polynomials, particularly prime-indexed pairs and simultaneous zeros at negative integer arguments. It asks whether a common zero at one specified site forces common zeros throughout the preceding negative-integer prefix. An effective large-prime rigidity theorem establishes this implication under an explicit no-cancellation condition, and an exact finite certificate verifies the condition for sites from 2 through 28. For sites from 2 through 12, the paper classifies every eligible prime and finds precisely one exceptional pair, at site 10 and prime 3617329, where isolated simultaneous vanishing does not extend to the full prefix. It also determines explicit natural densities for these common-root events using established arithmetic and Kummer-theoretic inputs. Its main novelty is the combination of an effective single-site forcing mechanism, a complete small-site classification, and a genuine sporadic counterexample, replacing an overgeneralized rigidity expectation with a sharply qualified arithmetic theorem.