Canonical Noncommutative Defects Beyond Parikh Projection: Magnus Towers and an Unbounded Static-Dynamic Boundary Gap

Abstract

We develop an order-sensitive defect formalism for substitution prefixes using the classical Magnus expansion of words. A substitution morphism induces a filtered endomorphism of the completed noncommutative tensor algebra, and an anchored substitution cut produces a multiplicative residual whose finite truncations form a compatible tower. The degree-one truncation recovers Parikh information, while higher degrees record scattered-subword data. For a finite substitution boundary language, we define the static separation depth $K_*$, the first Magnus depth at which every boundary word is distinguished, and for constant-length substitutions we define a dynamic stabilization depth $K_{\mathrm{dyn}}$ by minimizing the corresponding depth-$k$ boundary-output automata. We prove: $$K_{\mathrm{dyn}} \le K_*$$ and give exact depth-two and depth-three models. The main result is an explicit primitive binary constant-length family $\sigma_K$ such that: $$K_*(P_{\sigma_K}) = K \quad \text{but} \quad K_{\mathrm{dyn}}(\sigma_K) = 1 \quad \text{for every } K \ge 2$$ Hence, the static order depth and dynamic defect depth can differ by an arbitrarily large amount. The proof explicitly separates classical input—Magnus expansions, $k$-binomial equivalence, Thue–Morse separation results, and automaton minimization—from the canonical substitution-boundary architecture and the resulting static-dynamic separation theorem.