Kernel-Checked Finite Zero Certification for the Riemann ξ-Function in Lean 4

We present a kernel-checked certificate-verification framework in Lean 4 for finite zero localization. Rather than attempting to evaluate transcendental special functions and numerical contour integrals inside the proof-assistant kernel, we separate numerical generation from logical verification: untrusted external oracles produce an upper bound N on the zero count alongside N independently verified distinct roots, while the Lean 4 kernel mechanically audits the formal conclusion that these constitute the complete set of zeros in the domain. The central formal abstraction is a genuinely count-only AnalyticCountCertificate (CountCert) coupled with TargetSubsetExclusion (cntTargetExcl), which mechanizes the foundational deduction: external count bound + N verified target roots ⟹ complete zero localization This deduction is mathematically independent of the underlying special function and the geometry of the target set. We instantiate this architecture on a finite certificate for the Riemann ξ-function at height T = 50: twenty disjoint critical-line root brackets are verified via the Intermediate Value Theorem, while the Backlund–Turing count bound N = 20 is supplied through an externally certified rational enclosure and verified arithmetically in Lean. All proof obligations in the formalization are kernel-checked with zero axioms and zero unproven declarations. Artifacts and Lean formalization: https://github.com/Shiviatrix/zeta-parallel-engine