All-order smooth realization of increasing repeated-integral endpoint jets

We prove Burke–Haris–Madhavendra Conjecture 1.2 for every order, including order zero. An atomless Stieltjes measure has a strictly positive smooth density realizing its finitely many moments and equal to one near the endpoints. Separation, an explicit bump-moment openness argument and repeated integration give the full source endpoint-jet statement. All measure, kernel, openness and endpoint bridges are proved in Lean. Standard moment technology and the previously known low-order 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: Standard moment tools credited, including di Dio theorem bodies compared; no claim of new general machinery; No MathSciNet/zbMATH sweep; bounded prior-art investigation does not certify worldwide priority