An exact counterexample to trigonometric positivity for square roots of geometric sums

We refute Kovalev Conjecture 1.3 using d=49, partial-sum index 949 and t=arccos(523/527). A finite integer certificate proves strict negativity of the literal trigonometric sum from the normalized square-root Taylor coefficients. The coefficient recurrence, cosine bridge and source counterexample are proved in Lean. The separate zero-free conjecture and minimality are not claimed. 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: Actual 2026 published body retains the conjecture. No prior counterexample located; finite searches cannot certify absolute priority.