Well-posedness of the TAP fixed-point characterization of α_c

Determine whether the Thouless–Anderson–Palmer (TAP) equations provide a well-posed fixed-point characterization of the binary perceptron capacity α_c; specifically, ascertain existence and uniqueness of solutions to the TAP equations that define α_c.

Background

The conjectured value α_c ≈ 0.833 is derived from a fixed-point characterization using TAP equations, originating from non-rigorous replica and cavity methods in statistical physics. While these heuristics are powerful, the mathematical status of the TAP characterization for the binary perceptron remains unsettled.

Establishing well-posedness would validate the TAP-based prediction by ensuring that the fixed point exists and is unique, thereby resolving the current ambiguity about the precise definition of α_c via TAP.