The Effort Effect: A Mathematical Theory of Threshold Control in Social Success

We present a purely mathematical formulation of the effort effect: effort does not increase intrinsic talent but lowers the minimum talent threshold required for success. By explicitly modeling the time dependence of effort and social expectation, we derive exact conditions under which effort is effective, ineffective, or structurally insufficient. The theory is phenomenological and focuses exclusively on threshold dynamics, independent of psychological or sociological assumptions.