Global Obstructions for Autonomous Regular Lagrangians in Damped Scalar Oscillators — Closed Transversals, Jacobi Multipliers, and Exactness

This technical synthesis consolidates a research sequence on global obstructions to real, autonomous, single-valued, Legendre-regular Lagrangians for scalar second-order systems. The central exposition is a concise C2 Rolle argument: an exact autonomous Euler–Lagrange scalar SODE cannot remain Legendre-regular along a closed loop that is everywhere transverse to its phase-plane vector field. The note specializes this obstruction to the linearly damped oscillator across all positive damping regimes using an explicit strict quadratic level function, and relates the result to Jacobi last multipliers, inverse integrating factors, exactness on annular domains, phase-portrait boundaries, and known local/nonstandard Lagrangian constructions. This record is a technical synthesis / sharpened formulation rather than a priority or eponym claim. Global/cohomological inverse-variational mechanisms, Jacobi-multiplier theory, inverse-integrating-factor results, and one-period exactness on annular/cylindrical domains are treated as prior art or prior-art-equivalent where applicable. Candidate Lean sources are included only as uncompiled formalization material; no proof-assistant kernel result is claimed. This technical synthesis is derived from the DL14 Foundations provenance record: **Bùi Quang Trịnh, “DL14 Foundations: Global Variational Obstructions for Damped Scalar Oscillators — Analytic Lineage, Jacobi Multipliers, Exactness, and Formalization Handoff,” Zenodo, Version 1.0.0, DOI: 10.5281/zenodo.22917936.**