The Reverse-Toroid Constraint Diagnostic: A Boundary-Driven Method for Identifying Forced Structure and Free Degrees of Freedom

This work introduces the Reverse-Toroid Constraint Diagnostic (RTCD), a standalone, boundary-driven method for analysing high-dimensional constraint systems. Rather than searching for or optimising a solution, the method iteratively eliminates incompatible structure by alternating boundary operators, revealing the intrinsic structure of the solution space itself.The dynamics classify systems into three regimes: incompatible constraints (collapse), fully constrained systems (unique fixed points), and partially constrained systems with a finite number of free degrees of freedom. Crucially, the method explicitly identifies forced backbone structure, extracts surviving free modes via late-window analysis and…