Conceptual Framework for Orbital Instability in Contact Binary Star Systems: A Rigorous Multidimensional Analysis of HR 5171 A and Implications for Stellar Mergers

Massive contact and near-contact binaries in the common-envelope (CE) phase are widely, but not universally, interpreted as short-lived way stations toward stellar coalescence, yet the field still lacks a single worked example in which Roche geometry, hydrodynamic drag, tidal instability, energy budget, and gravitational-wave (GW) consequences are treated within one transparent, reproducible, and falsifiable computational pipeline. We address this gap by developing an integrated multi-physics framework for orbital decay in such systems, unifying Roche-lobe geometry (Eggleton 1983), Bondi–Hoyle–Lyttleton (BHL) hydrodynamic drag, Darwin tidal instability, the αCE energy formalism, and quadrupolar GW emission, and applying it to the yellow hypergiant binary HR 5171 A as the primary worked example, with the red-nova progenitor V1309 Scorpii as an empirically anchored merger analog and VFTS 352 and MY Camelopardalis as extensions across the O-star mass range. The orbital-decay law is derived from energy balance applied to BHL drag, ȧ = −2a²fdṀBHLv²rel/(GMm), with every parameter explicitly justified against the literature; the full simulation, Bayesian mass inference, and Sobol' global sensitivity analysis are implemented in a single reproducible pipeline (CEDrag-1D) so that every quantitative claim in the text is generated verbatim by the accompanying code (Appendix A). For the fiducial HR 5171 A parameters (Chesneau et al. 2014), the primary is found to already overflow its Roche lobe (RA = 1315 R⊙ against RL,A ≈ 920 R⊙), corroborating the previously reported over-contact eclipsing configuration, with a semi-major axis a0 = 1688.8 R⊙ and an instantaneous orbital e-folding decay timescale τ0 ≃ 2.0 × 10⁴ yr (posterior median under primary-mass uncertainty; 95% credible interval [0.9, 5.3] × 10⁴ yr), consistent with the plunge-in timescale range 10⁴–10⁵ yr inferred independently from three-dimensional CE hydrodynamic simulations (Iaconi et al. 2019). Global Sobol' sensitivity analysis identifies the BHL drag efficiency fd and the envelope polytropic index n as the dominant, and only partially converged, sources of output variance, a limitation we quantify explicitly through multi-seed convergence diagnostics rather than reporting a single, potentially overprecise index. We further provide a dedicated scientific and technical risk assessment, a falsifiability and observational roadmap with explicit, pre-registered success/failure criteria (orbital period derivative, interferometric geometry, pre-merger photometric signature, and GW non-detection), and a critical comparison against the current CE, luminous-red-nova, and yellow-hypergiant literature, including the long-standing HR 5171 A distance controversy, which an independent 2026 Gaia cluster-membership study (Kasikov et al. 2026) now resolves in favor of the ∼3–3.5 kpc distance scale adopted here. We deliberately restrict the model's quantitative claims to the CE inspiral regime and explicitly do not extrapolate it into the terminal, GW-relevant plunge phase, for which it is not calibrated. All computations are fully reproducible from the embedded, seeded, unit-checked Python source code.