You write that that no other institution makes fully independent forecasts? What about the Congressional Budget Office?

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FAQs

The Congressional Budget Office (CBO) uses the SSA’s fertility forecast as an input to its forecasting model. Before 2013, CBO also used SSA's mortality forecasts as inputs to its model (here is why they changed).  

CBO explains on page 103 of The 2014 Long-Term Budget Outlook: "CBO used projected values from the Social Security trustees for fertility rates but produced its own projections for immigration and mortality rates. Together, those projections imply a total U.S. population of 395 million in 2039, compared with 324 million today. CBO also produced its own projection of the rate at which people will qualify for Social Security’s Disability Insurance program in coming decades."