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Strong lensing provides a powerful means of investigating the nature of darkmatter as it probes dark matter structure on sub-galactic scales. We present anextension of a forward modeling framework that uses flux ratios from quadruplyimaged quasars (quads) to measure the shape and amplitude of the halo massfunction, including line of sight (LOS) halos and main deflector subhalos. Weapply this machinery to 50 mock lenses --- roughly the number of known quads--- with warm dark matter (WDM) mass functions exhibiting free-streamingcutoffs parameterized by the half-mode mass $m_{\rm{hm}}$. Assuming cold darkmatter (CDM), we forecast bounds on $m_{\rm{hm}}$ and the corresponding thermalrelic particle masses over a range of tidal destruction severity, assuming aparticular WDM mass function and mass-concentration relation. With significanttidal destruction, at $2 \sigma$ we constrain $m_{\rm{hm}}<10^{7.9}\left(10^{8.4}\right) M_{\odot}$, or a 4.4 (3.1) keV thermal relic, with imageflux uncertainties from measurements and lens modeling of $2\%\left(6\%\right)$. With less severe tidal destruction we constrain$m_{\rm{hm}}<10^{7} \left(10^{7.4}\right) M_{\odot}$, or an 8.2 (6.2) keVthermal relic. If dark matter is warm, with $m_{\rm{hm}} = 10^{7.7} M_{\odot}$(5.1 keV), we would favor WDM with $m_{\rm{hm}} > 10^{7.7} M_{\odot}$ over CDMwith relative likelihoods of 22:1 and 8:1 with flux uncertainties of $2\%$ and$6\%$, respectively. These bounds improve over those obtained by modeling onlymain deflector subhalos because LOS objects produce additional fluxperturbations, especially for high redshift systems. These results indicatethat $\sim 50$ quads can conclusively differentiate between warm and cold darkmatter.

Probing dark matter structure down to $10^7$ solar masses: flux ratio statistics in gravitational lenses with line of sight halos