The National Football League has finally submitted documents in federal court documents admitting that it expects nearly a third of retired players to develop long-term cognitive problems.

The NFL added that the conditions are likely to emerge at "notably younger ages" than in the general population.

The findings are a result of data prepared by actuaries hired by the league and provided to the United States District Court judge presiding over the settlement between the N.F.L. and 5,000 former players who had filed a legal lawsuit against the league, accusing it of hiding the dangers of concussions from them.

"Thus, our assumptions result in prevalence rates by age group that are materially higher than those expected in the general population," said the report, prepared by the Segal Group for the N.F.L. "Furthermore, the model forecasts that players will develop these diagnoses at notably younger ages than the generation population."

The statements are important as it is the first time that the NFL has accepted that the players sustain severe brain injuries at far higher rates than the general population.

"This statement clears up all the confusion and doubt manufactured over the years questioning the link between brain trauma and long-term neurological impairment," said Chris Nowinski, the executive director of the Sports Legacy Institute, who has for many years pressured the league to acknowledge the connection between football and brain diseases. "We have come a long way since the days of outright denial. The number of former players predicted to develop dementia is staggering, and that total does not even include former players who develop mood and behavior disorders and die prior to developing the cognitive symptoms associated with C.T.E."

Greg Aiello, a spokesman for the N.F.L., refused to comment on the matter.