That's not true. You can see plots in the 2011 and 2012 papers that give these calculations as a function of mass. It's basically impossible to make a mass-independent calculation. (Not withstanding the fact that there's no Higgsless theory to do calculations for the background).
>As an example, I'd imagine detecting the Higg's [sic] boson at 1 eV energy levels is theoretically predicted to be even less likely than billions/trillions to one odds, therefore detector noise would be a more likely explanation for such results, despite the low p-value.
You're right, the p-value is much lower. Previous experiments have long since excluded such a low Higgs mass. Also, if the Higgs mass were so low, we probably wouldn't exist.