Uniform Burnside–Gautschi bound for interpolated gamma values

Schmidlin Conjecture 37 holds strictly for every nonnegative base point and, more generally, every nonnegative interpolation parameter. A classical Stirling remainder bound, logarithmic entropy estimate and translated positive-discriminant certificate prove the comparison. The literal gamma-function expression, source domain and all supporting inequalities are proved in Lean. Classical estimates and the previously established x>=1 range are credited; no optimal constant is 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: Wang2016 full body inaccessible; bounded independent screen found no known prior resolution, no worldwide-priority claim.