A two-term counterexample to Bessel–Fourier kernel positivity

The literal order-zero modified Fejer kernel in Acosta Babb's published Conjecture 1 satisfies K_2^(2/5)(47/50)<-1/500. Both real parameters are strictly interior. Exact rational series and Bernstein bounds identify the genuine first two positive J0 zeros and prove the strict negative value in Lean. The complete certificate has 87 declarations in eight modules. Separate slow-decay existence questions, redundant half-power witnesses and numerical scans are not additional outcomes. AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review. The accompanying Lean certificate covers the precise scoped main statement; inspect the formal endpoint, assumptions and provenance in the supporting package. Prior-art investigations are documented bounded searches, not worldwide-priority certification. Original prose is offered under CC BY 4.0 and original code/Lean under MIT; third-party rights and existing notices are preserved. The public supporting ZIP contains original proof sources and verification evidence, with bibliographic links and hashes instead of redistributed third-party source manuscripts. Recorded prior-art/source-access limits: No prior refutation or directly subsuming negative theorem found in bounded primary/citing/author/registry/classical Young-Watson screens; no MathSciNet/zbMATH or exhaustive worldwide priority certification; optional scans uncredited; one common source question outcome.