Gaia astrometry disfavors a binary origin for long secondary periods

Approximately one-third of luminous pulsating red giant stars exhibit long secondary periods (LSPs): stable photometric variability with periods of several months to years in addition to their much shorter primary pulsation cycles. Now nearly a century after their discovery, the physical origin of LSPs remains unresolved. A leading explanation invokes binarity, in which the LSP corresponds to the orbital period of a low-mass companion responsible for both the photometric variability and the radial-velocity (RV) modulation. We test this hypothesis using a nearby sample of LSP stars from the {\it Gaia} Focused Product Release, which provides multi-epoch RVs and contemporaneous optical photometry. We find that interpreting the observed RV variability as orbital motion implies companion masses narrowly distributed around $M_2 \approx 0.1~{\rm M_\odot}$ with separations of 1--3 au, placing them squarely in the brown dwarf desert observed around their solar-type progenitors. Assuming such companions exist, we then forward-model the astrometric signature expected in {\it Gaia} DR3 and predict systematically elevated {\tt RUWE} values for nearby LSPs. In contrast, the observed {\tt RUWE} of nearby LSP stars is systematically lower than these predictions and consistent with most systems exhibiting LSPs being single. This discrepancy disfavors low-mass stellar or substellar companions as the dominant origin of LSPs in evolved stars, motivating a further exploration of alternative stellar mechanisms.