Frame-Bound Mechanics / Structural Theory of Articulated Systems

This work establishes a unified mechanical framework for the analysis, interpretation, and control of articulators as frame-bound systems. The system is formalized by (C ∗ , D) ⇒ x ∗ ⇒ G(x ∗ ) = R, where admissible constraints C ∗ and dominance distribution D define the equilibrium state x ∗ , while observable motion R is its projection. The framework integrates ontology, equilibrium theory, regime theory, Atlas formalism, tool algebra, topology classification, failure theory, and operator theory. Three topological classes are resolved empirically and formally: Hanau as plateau manifold, Dentatus as ridge manifold, and Artex CN as channel manifold. The work shows that arrow is the universal default regime, straight is topology-specific and non-unique, identical traces do not imply structural equivalence, and validity exists only under full structural preservation across transformations. A dual-layer Atlas system is introduced: Atlas-A for structural encoding without load and Atlas-B for post-load validation. A formal operator algebra is then defined through regime acceptance, parameter verification, perturbation testing, carry-over, redistribution analysis, and projection-response equivalence. The resulting system replaces descriptive or parameter-based interpretation with a constrained, topology-aware, mathematically anchored methodology