Coordinate-Free Geometric Inversion of Stationary Multi-Harmonic Manifolds for Electron-Impact Cross-Sections in the GEC Reference Cell

Traditional compilations of electron-impact cross-sections rely on isolated crossed-beam or swarm experiments and therefore leave a transferability gap when applied to non-equilibrium, multi-frequency radio-frequency discharges. We present a dimensionless, coordinate-free alternative that treats the stationary phase-space orbit of a 13.56 MHz Gaseous Electronics Conference (GEC) reference cell as a self-consistent geometric diagnostic. After time is scaled to the RF period the discharge state becomes a closed orbit on the extended phase space M = R2 ×S1. Two intrinsic observables extracted from this abstract curve—the hydrodynamic phase lag between density and temperature responses, and the birth and growth of threshold-induced folds (or of the associated odd harmonics)—encode the elastic momentum-transfer and inelastic cross-sections, respectively. A dual-regime Design of Experiments isolates the two channels while remaining inside the validity envelope of every constitutive relation. The elastic inversion recovers the Ramsauer–Townsend minimum and the subsequent rise of σm(E) to within a few percent of the Biagi and Phelps compilations. Once earlier numerical artefacts (forced period-1 assumptions, violation of the small-signal sheath law, and omission of the explicit Heaviside sink) are removed, the same geometric method yields an inelastic cross-section σin(E) whose onsets coincide with the spectroscopic thresholds and whose high-energy magnitude tracks the established literature up to 25 eV. Because the diagnostics are properties of the embedded orbit itself, absolute laboratory calibrations never enter the final tables. The resulting pipeline converts a standard RF plasma source into a self-calibrating instrument for electron-collision spectroscopy and can be re-applied, without reformulation, to other noble gases or simple molecular mixtures. A small number of open questions remain before the framework can be regarded as fully self-consistent, principally the reconciliation of the mean-temperature Dirichlet anchor with orbit periodicity, a systematic robustness map of the elastic operating window near the Ramsauer–Townsend minimum, and explicit per-run verification of inelastic threshold crossing within the stated small-signal drive envelope.