A Gaussian envelope for Hermite polynomials

We prove Guo–Liu's full literal Hermite-envelope conjecture. A sine comparison gives enough strict energy loss in odd degree to attain the exact Gamma-normalized coefficient. A real-root moment inequality controls stationary peaks after the cutoff and supplies the clamped tail. Actual Rodrigues Hermite polynomials, the precise source cutoff and every assembly bridge are proved in Lean. Classical oscillator, Wallis and polynomial tools and known low/even-degree cases are credited. 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: Bounded independent prior-art screen; adjacent MDPI2026 fractional-Hermite body unread and no exhaustive subscription-index coverage; no worldwide-priority claim.