The correct answers are B. discernible, E. gargantuan, and I. sporadic.

Explanation
- Blank (i): The sentence describes a "detection problem of maddening proportions." To overcome this, physicists need to generate a high concentration (flux) of neutrinos so that they can actually be detected by our instruments. B. discernible (able to be perceived or detected) fits the context perfectly. Tangible (touchable) is physically incorrect for a subatomic particle, and redundant (superfluous) makes no logical sense.
- Blank (ii): To generate such a massive flux, physicists must look to the largest, most massive, and most powerful sources in the universe, which the sentence defines as "the core-collapse of a massive star or the accretion disk of a quasar." E. gargantuan (immense or gigantic) perfectly captures the scale of these titanic cosmic events. Tempestuous (turbulent) is too focused on chaos rather than energy/scale, and superficial (shallow) is the opposite of what is needed.
- Blank (iii): The phrase "And yet" signals a contrast. Even when looking at these giant ("gargantuan") sources, the resulting signals that reach us are extremely rare and spread out over time. The clue "a detector the size of an Olympic swimming pool might register a mere handful of interactions in an entire year" describes a highly irregular, infrequent, and scattered occurence. I. sporadic (occurring at irregular intervals or only in a few places) matches this description perfectly. Infinitesimal (extremely small) describes size rather than the frequency of the events, and monumental is the opposite of what is implied by "heartbreakingly."