Invariant Structure in Deterministic Decoder Transformation Pipelines

This work introduces a deterministic framework for identifying, validating, and falsifying structural invariants in decoder transformation pipelines. Departing from traditional performance-centric analyses of belief propagation, the study treats decoder execution as a composition of deterministic operators and focuses on invariant properties that persist under repeated application. We formally establish several classes of invariants, including ternary closure, operator idempotence, and dark-skip cascade behavior, and validate them directly against implementation through deterministic testing. In parallel, we construct explicit counterexamples to demonstrate the failure of commonly assumed properties such as universal absorbing states and global idempotence under damping. A key contribution is the introduction of invariant-guided reasoning as a mechanism for eliminating redundant computation rather than optimizing it. By reframing decoder pipelines as dynamical systems with provable algebraic structure, this work provides a foundation for deterministic runtime reduction, modular pipeline design, and future extensions into qudit-based decoding and sonification-driven analysis. This work builds on prior results in dark-state invariant detection and forms part of an ongoing research program in invariant-driven computation and quantum-inspired information dynamics.