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 singular value decomposition, and demonstrates strict contraction once these modes are quotiented out.The approach provides a constructive decomposition of constraint systems into forced truth and genuine freedom, without combinatorial search, optimisation heuristics, or domain-specific assumptions. It is applicable across satisfiability, inverse problems, identifiability analysis, physics-inspired constraint systems, and general high-dimensional diagnostics. Keywords constraint diagnosticsconstraint satisfactionbackbone detectionfree mode extractiondegeneracy analysisannealed dynamicsboundary operatorsnon-optimisation methodssolution space structureaffine solution manifoldsinverse problemsidentifiabilitygauge freedomhigh-dimensional systemsdynamical systemsmodel reductionstructural analysis