Algebraic Domains of Validity and Structural Breakage Theorems

The domain of applicability of Residue-Class Operation Translation, from commutative integral domains to non-commutative and nonlinear dynamics. Answers one of the series' most important domain-of-validity questions: which algebraic properties do RCOT's theorems actually depend on, and as the coefficient domain is expanded from the integers to general rings, unordered fields, non-commutative algebras, projective maps, and nonlinear polynomials, which structure breaks first? Distinguishes five independent structural requirements and locates the breaking point for each.