Short answers, pulled from the story.
Charles A. S. Hall introduced the energy return on investment formula in his 1984 Science paper to divide usable energy delivered from a resource by the exergy used to obtain that same resource. The equation determines if a source becomes an energy sink when the ratio drops below one or achieves net energy gain when output exceeds input.
Scientists measure this efficiency using specific units like kilowatt-hours per square meter of module area with median values around 585 kWh/m2 and minimums near 300 kWh/m2. To remain viable as a prominent fuel, a source must maintain an EROI ratio of at least three to one according to Hall's framework.
Vestas reports an EROI of 31 for its V150 model wind turbine operating under optimal conditions while oil shale typically yields only 1.4 to 1.5 due to high process heat requirements. Hydropower plants average about 110 when run for approximately 100 years and conventional oil sources vary widely depending on geologic factors ranging from 18 to 43.
Murphy and Hall proposed an extended boundary protocol in 2010 to create more realistic assessments regarding how deep supply chain probing should go. The International Energy Agency endorses different methodology that often generates more favorable values for photovoltaic solar panels while government agencies have not yet provided adequate funding for rigorous analysis by neutral observers.
Thomas Homer-Dixon argues falling EROI contributed to the collapse of the Western Roman Empire in the fifth century CE following ecological damage including deforestation and soil fertility loss beginning in the second century. Rome's population peaked under Trajan at 1.5 million but dropped to only 15,000 by 1084 as historical data shows discovery ratios declining steadily over the last century from about 1000:1 in 1919 to 5:1 in the 2010s.